[description=state space]S$S$ [description=space of continuous functions from
$[0,\infty)$ to $S$]cf$C([0,\infty),S)$ [description=space of bounded
continuous functions from $[0,\infty)$ to $S$]cfb$C_{b}([0,\infty),S)$
[description=generator]G$G$ [description=hierarchical group of order
$N$]hiern$\Omega_{N}$ [description=hierarchical distance on
$\Omega_{N}$]dhier$d_{\Omega_{N}}$ [description=sequence of migration rates
$\underline{c}=(c_{k})_{k\in\mathbb{N}_{0}}$]mig$\underline{c}$
[description=migration rate within distance $k$]migk$c_{k}$
[description=migration kernel on
$\Omega_{N}$]migker$a^{\Omega_{N}}(\cdot,\cdot)$ [description= time-$t$
migration kernel on $\Omega_{N}$]migkert$a_{t}^{\Omega_{N}}(\cdot,\cdot)$
[description=degree of random walk]degrw$\delta$ [description=sequence of
relative seed-bank sizes
$\underline{K}=(K_{m})_{m\in\mathbb{N}_{0}}$]ssbs$\underline{K}$
[description=sequence of exchange parameters
$\underline{e}=(e_{m})_{m\in\mathbb{N}_{0}}$]sexp$\underline{e}$
[description=exchange parameter for colour $m$ ]expar$e_{m}$
[description=relative size of $m$-dormant population ]sbsm$K_{m}$
[description=total exchange rate with dormant population ]chi$\chi$
[description=relative size of dormant population ]rho$\rho$
[description=diffusion function ]g$g$ [description=class of diffusion
functions ]cldf$\mathcal{G}$ [description=standard Brownian motion at site
$\xi\in\Omega_{N}$ ]bm$(w_{\xi}(t))_{t\geq 0}$ [description=fraction of active
individuals of type $\heartsuit$ at colony $\xi$ at time $t$ ]x$x_{\xi}(t)$
[description=fraction of $m$-dormant individuals of type $\heartsuit$ at
colony $\xi$ at time $t$ ]y$y_{\xi,m}(t)$ [description=full process
$\big{(}x_{\xi}(t),(y_{\xi,m}(t))_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N},\,t\geq
0}$]fp$(X^{\Omega_{N}}(t),Y^{\Omega_{N}}(t))_{t\geq 0}$ [description=initial
density parameter active population ]thetax$\theta_{x}$ [description=initial
density parameter $m$-dormant population ]thetay$\theta_{y_{m}}$
[description=active $k$-block average at time $t$ ]xk$x_{k}^{\Omega_{N}}(t)$
[description=$m$-dormant $k$-block average at time $t$
]yk$y_{m,k}^{\Omega_{N}}(t)$ [description=domain of generator ]DF$\mathbf{F}$
[description=Fisher-Wright diffusion function ]FW$g_{\mathrm{FW}}$
[description=seed-bank coalescent ]SC$(\mathcal{C}^{(n)}(t))_{t\geqslant 0}$
[description=transition kernel corresponding to lineage in dual
]tkerd$b(\cdot,\cdot)$ [description=duality function ]H$H$ [description=block
counting process ]L$(L(t))_{t\geq 0}$ [description=full process
$(X^{\Omega_{N}}(t),Y^{\Omega_{N}}(t))_{t\geq 0}$
]fpz$(Z^{\Omega_{N}}(t))_{t\geq 0}$ [description=time lapse lineage in dual
spends in active state after becoming active for $k$-th time
]sigmak$\sigma_{k}$ [description=time lapse lineage in dual spends in active
state after becoming active for $k$-th time ]tauk$\tau_{k}$
[description=typical wake-up time of lineage in dual ]tau$\tau$
[description=function that is slowly varying at infinity ]phi$\varphi$
[description=tail exponent of typical wake-up time ]gamma$\gamma$
[description=fraction of individuals of type $\heartsuit$ in state $R_{\xi}$
in colony $\xi$ at time $t$ ]z$z_{(\xi,R_{\xi})}(t)$ [description=space of
càdlàg functions from $[0,\infty)$ to Polish space $E$]cadlag$D([0,\infty),E)$
[description=pseudopath corresponding to measurable map $v:s\to v(s)$ from
$[0,\infty)$ to $E$]pseudo$\psi_{v}$ [description=space of all
pseudopaths]pseudospace$\Psi$ [description=initial density of $\heartsuit$ in
active population and first $k+1$ dormant populations]varthetak$\vartheta_{k}$
[description=slowing-down constants]Ek$E_{k}$ [description=full process on
level $l$ with drift towards $\theta$
]zl$\Big{(}z_{l,(\theta,(y_{m,l})_{m\in\mathbb{N}_{0}})}(t)\Big{)}_{t\geq 0}$
[description=effective process on level $l$ with drift towards
$\theta$]zlf$(z^{\mathrm{eff}}_{l,\theta}(t))_{t\geq 0}$
[description=empirical average in active $l$-block at time $t$
]thetalx$\bar{\Theta}_{x}^{(l),\Omega_{N}}(t)$ [description=empirical average
in $m$-dormant $l$-block at time $t$
]thetaly$\bar{\Theta}_{y_{m}}^{(l),\Omega_{N}}(t)$ [description=empirical
average in $l$-block at time $t$ ]thetal$\bar{\Theta}^{(l),\Omega_{N}}(t)$
[description=full estimator of $l$-block at time $t$
]thetalfull$\boldsymbol{\Theta}^{(l),\Omega_{N}}(t)$ [description= effective
estimator of the $l$-block at time $t$
]thetaleff$\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(t)$ [description=
sequence with $\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$
]LN$(L(N))_{N\in\mathbb{N}}$ [description=invariant measure for full $l$-block
average process with density $\theta$ ]Gammal$\Gamma_{(\theta,y_{l})}^{(l)}$
[description=invariant measure for effective $l$-block average process with
density $\theta$ ]Gammaleff$\Gamma_{\theta}^{\mathrm{eff},(l)}$
[description=renormalisation transformation on level $l$
]Fl$\mathcal{F}^{E_{l},c_{l},K_{l},e_{l}}$ [description=$k$-th iterate of
renormalisation transformation ]Flit$\mathcal{F}^{(k)}$
[description=interaction chain ]intchain$(M^{k}_{-l})_{-l=-(k+1),-k,\ldots,0}$
[description=transition kernel of interaction chain on level $l$]Q$Q^{[l]}$
[description=effective interaction chain
]intchaineff$(M^{\mathrm{eff},k}_{-l})_{-l=-(k+1),-k,\ldots,0}$
[description=effective transition kernel of interaction chain on level
$l$]Qeff$Q^{\mathrm{eff},[l]}$ [description=first component of effective
$k$-block process]effx$\bar{x}_{k}^{\Omega_{N}}(t)$ [description=$n$-th
composition of transition kernel of interaction chain]Qn$Q^{(n)}$
[description=initial state of level-$n$ full
process]thetan$\bar{\vartheta}^{(n)}$ [description=$n$-th clustering
coefficient]An$A_{n}$ [description=$n,m$-th clustering
coefficient]Amn$A_{m}^{n}$ [description=finite geographic space
$\\{0,1,\cdots,N-1\\}$]spaceN$[N]$ [description=difference between active and
dormant estimator at time $t$]difad${\Delta}^{[N]}_{\bar{\Theta}}(t)$
[description=initial measure for infinite system obtained by periodic
continuation of finite system]mun$\mu_{N}$ [description=full infinite process
$\big{(}x_{\xi}(t),(y_{\xi,m}(t))_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N},\,t\geq
0}$ starting from initial law $\mu_{N}$
]ifp$(X^{\mu_{N}}(t),Y^{\mu_{N}}(t))_{t\geq 0}$ [description=local time at $0$
at time $t$]localtime$L_{t}^{0}$ [description=specific sequence of times
depending on $N$]barln$\bar{L}(N)$ [description=$\mathcal{D}$-semi martingale
operator]dsmop$G_{\dagger}^{[N]}$ [description=auxiliary $l$-block estimator
process at time $t$ on level
$k+1$]aux$\boldsymbol{\Theta}^{\mathrm{aux},(l),\Omega_{N}^{k+1}}(t)$
[description=hierarchical group truncated at level
$k+1$]omegatrunc$\Omega_{N}^{k+1}$ [description=equilibrium measure of
auxiliary process with density $\theta$ at level
$1$]gammaaux$\Gamma_{\theta}^{\mathrm{aux},(1)}$

# Spatial populations with seed-bank:
renormalisation on the hierarchical group

Andreas Greven1, Frank den Hollander2, Margriet Oomen3

(6 October 2021)

###### Abstract

We consider a system of interacting diffusions labeled by a geographic space
that is given by the hierarchical group $\Omega_{N}$ of order
$N\in\mathbb{N}$. Individuals live in colonies and are subject to resampling
and migration as long as they are active. Each colony has a seed-bank into
which individuals can retreat to become dormant, suspending their resampling
and migration until they become active again. The migration kernel has a
hierarchical structure: individuals hop between colonies at a rate that
depends on the hierarchical distance between the colonies. The seed-bank has a
layered structure: when individuals become dormant they acquire a colour that
determines the rate at which they become active again. The latter allows us to
model seed-banks whose wake-up times have a fat tail. We analyse a system of
coupled stochastic differential equations that describes the population in the
large-colony-size limit.

For fixed $N\in\mathbb{N}$, the system exhibits a dichotomy between
_coexistence_ (= locally multi-type equilibria) and _clustering_ (= locally
mono-type equilibria). To identify the _clustering regime_ , i.e., the range
of parameters controlling the migration and the seed-bank for which clustering
prevails, we apply a necessary and sufficient criterion derived in [GHO20]
that is valid for any geographic space given by a countable Abelian group
endowed with the discrete topology. The diffusion function controlling the
resampling does not play a role in this criterion.

We carry out a _multi-scale renormalisation analysis_ in the _hierarchical
mean-field limit_ $N\to\infty$. In particular, we show that block averages on
hierarchical space-time scale $k\in\mathbb{N}$ perform a diffusion with a
renormalised diffusion function that depends on $k$. In the clustering regime,
after an appropriate scaling with $k$, this diffusion function converges to
the Fisher-Wright diffusion function as $k\to\infty$, irrespective of the
diffusion function controlling the resampling. Thus, the system exhibits _full
universality_ on large space-time scales in terms of the scaling limit. For
several subclasses of parameters we identify the speed at which the scaled
renormalised diffusion function converges to the Fisher-Wright diffusion as
$k\to\infty$. This speed in turn determines the speed at which _mono-type
clusters grow in space and time_. We show that the seed-bank reduces the speed
compared to the model without seed-bank. This reduction is the result of a
delicate interplay between migration and seed-bank, which in the limit as
$N\to\infty$ can be worked out explicitly.

_Keywords:_ Fisher-Wright diffusion, resampling, migration, seed-bank, multi-
scale renormalisation, universality.

_MSC 2010:_ Primary 60J70, 60K35; Secondary 92D25.

_Acknowledgements:_ AG was supported by the Deutsche Forschungsgemeinschaft
(grant DFG-GR 876/6-1,2). FdH and MO were supported by the Netherlands
Organisation for Scientific Research (NWO Gravitation Grant
NETWORKS-024.002.003). FdH visited Bonn and Erlangen in the Fall of 2019 and
2020, supported by a Research Award of the Alexander von Humboldt Foundation.
AG visited Leiden in February 2020, as Kloosterman Chair.

1) Department Mathematik, Universität Erlangen-Nürnberg, Cauerstr. 11, D-91058
Erlangen, Germany
<EMAIL_ADDRESS>
2) Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA
Leiden, NL
<EMAIL_ADDRESS>
3) Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA
Leiden, NL
<EMAIL_ADDRESS>
###### Contents

  1. 1 Background, goals and outline
    1. 1.1 Background
    2. 1.2 Goals
    3. 1.3 Outline
  2. I MODEL AND MAIN RESULTS
    1. 2 Introduction of model and basic properties
      1. 2.1 Model: geographic space $\Omega_{N}$, hierachical group of order $N$
        1. 2.1.1 Hierarchical migration
        2. 2.1.2 Layered seed-bank
        3. 2.1.3 Resampling rate
      2. 2.2 Evolution equations
        1. 2.2.1 Evolution of single colonies
        2. 2.2.2 Evolution of block averages
      3. 2.3 Well-posedness
      4. 2.4 Duality
      5. 2.5 Clustering criterion
        1. 2.5.1 Clustering criterion for Fisher-Wright diffusion function
        2. 2.5.2 Clustering criterion for general diffusion function
    2. 3 Main results: $N<\infty$, identification of clustering regime
      1. 3.1 Finite mean wake-up time
      2. 3.2 Infinite mean wake-up time
      3. 3.3 Clustering regime
    3. 4 Main results: $N\to\infty$, renormalisation and multi-scale limit
      1. 4.1 Intermezzo: Meyer-Zheng topology
      2. 4.2 Main ingredients for the hierarchical multi-scale limit
      3. 4.3 Hierarchical multi-scale limit theorems
        1. 4.3.1 Effective process
        2. 4.3.2 Full process
      4. 4.4 Heuristics behind the multi-scale limit
        1. 4.4.1 Evolution of the block-averages
        2. 4.4.2 Limiting evolution of the block-averages
        3. 4.4.3 Formation of the interaction chain
    4. 5 Main results $N\to\infty$: Orbit and cluster formation
      1. 5.1 Orbit of renormalisation transformations
      2. 5.2 Growth of mono-type clusters
      3. 5.3 Rates of scaling for renormalised diffusion function
  3. II PREPARATIONS AND PROOFS
    1. 6 Proofs: $N<\infty$, identification of clustering regime
      1. 6.1 Explanation of clustering criterion for infinite seed-bank
      2. 6.2 Scaling of wake-up time and migration kernel for infinite seed-bank
      3. 6.3 Hierarchical clustering
    2. 7 Preparation: $N\to\infty$, McKean-Vlasov process and mean-field system
      1. 7.1 McKean-Vlasov process
      2. 7.2 Mean-field system and McKean-Vlasov limit
      3. 7.3 Proof of equilibrium and ergodicity
      4. 7.4 Proof of McKean-Vlasov limit
    3. 8 Proofs: $N\to\infty$, mean-field finite-systems scheme
      1. 8.1 Mean-field finite-systems scheme
      2. 8.2 Abstract scheme behind finite-systems scheme
    4. 9 Proofs: $N\to\infty$, mean-field, proof of abstract scheme
      1. 9.1 Proof of step 1. Equilibrium of the single components
      2. 9.2 Proof of step 2. Convergence of the estimator
      3. 9.3 Proof of step 3. Convergence of the $1$-blocks in the Meyer-Zheng topology
      4. 9.4 Proof of step 4. Mean-field finite-systems scheme
    5. 10 Preparation: $N\to\infty$, two-colour mean-field
      1. 10.1 Two-colour mean-field finite-systems scheme
      2. 10.2 Proof of the two-colour mean-field finite-systems scheme
    6. 11 Preparation: $N\to\infty$, two-level three-colour
      1. 11.1 Two-level three-colour mean-field finite-systems scheme
      2. 11.2 Scheme for the two-level three-colour mean-field analysis.
      3. 11.3 Proof of two-level three-colour mean-field finite-systems scheme
        1. 11.3.1 Tightness of the $2$-block estimators
        2. 11.3.2 Stability of the $2$-block estimators
        3. 11.3.3 Tightness of the 1-block estimators
        4. 11.3.4 Stability of the $1$-block estimators
        5. 11.3.5 Limiting evolution for the single components
        6. 11.3.6 Limiting evolution of the $1$-block estimator process
        7. 11.3.7 Convergence of 2-block process
        8. 11.3.8 State of the slow seed-banks
        9. 11.3.9 Limiting evolution of the estimator processes
        10. 11.3.10 Convergence in the Meyer-Zheng topology
        11. 11.3.11 Proof of the two-level three-colour mean-field finite-systems scheme
    7. 12 Proofs: $N\to\infty$, hierarchical multi-scale limit theorems
      1. 12.1 Finite-level mean-field finite-systems scheme and interaction chain
      2. 12.2 Proof of the mean-field finite-systems scheme: finite-level
      3. 12.3 Proof: of the hierarchical multi-scale limit theorems.
    8. 13 Proofs: $N\to\infty$, orbit of renormalisation transformation
      1. 13.1 Moment relations
      2. 13.2 Iterate moment relations
      3. 13.3 Clustering
      4. 13.4 Dichotomy finite versus infinite seed-bank
    9. A Computation of scaling coefficients
      1. A.1 Regularly varying coefficients
      2. A.2 Pure exponential coefficients
      3. A.3 Slowly varying functions
    10. B Meyer-Zheng topology
      1. B.1 Basic facts about the Meyer-Zheng topology
      2. B.2 Pseudopaths of stochastic processes on a general metric separable space
      3. B.3 Proof of key lemmas

###

###  1 Background, goals and outline

####  1.1 Background

###### Single colony with seed-bank.

In populations with a _seed-bank_ , individuals can temporarily become dormant
and refrain from reproduction, until they can become active again. In
[BGCEK15] and [BGCKWB16] the evolution of a population evolving according to
the two-type Fisher-Wright model with seed-bank was studied. Individuals move
in and out of the seed-bank at prescribed rates. Outside the seed-bank
individuals are subject to _resampling_ , while inside the seed-bank their
resampling is _suspended_. Both the long-time behaviour and the genealogy of
the population were analysed in detail.

Seed-banks are observed in many taxa, including plants, bacteria and other
micro-organisms. Typically, they arise as a response to unfavourable
environmental conditions. The dormant state of an individual is characterised
by low metabolic activity and interruption of phenotypic development (see e.g.
[LJ11]). After a varying and possibly large number of generations, a dormant
individual can be resuscitated under more favourable conditions and reprise
reproduction after having become active again. This strategy is known to have
important implications for population persistence, maintenance of genetic
variability and stability of ecosystems. It acts as a _buffer_ against
evolutionary forces such as genetic drift, selection and environmental
variability.

###### Multiple colonies with seed-bank.

In [GHO20] we considered a _spatial_ version of the two-type Fisher-Wright
model with seed-bank in which individuals can _migrate_ between colonies,
organised into a _geographic space_ , each having a seed-bank consisting of
_multiple layers_ , each with their own rate of moving in (becoming dormant)
and moving out (waking up). We found that the presence of the seed-bank
_enhances genetic diversity_ compared to the spatial model without seed-bank.
Interestingly, we found that the seed-bank can affect the longtime behaviour
of the system both qualitatively and quantitatively.

In [GHO20] we settled existence and uniqueness of the spatial model when the
geographic space is $\mathbb{Z}^{d}$, $d\in\mathbb{N}$. We proved convergence
to equilibrium, showed that there is a dichotomy between _coexistence_ (=
locally multi-type equilibria) and _clustering_ (= locally mono-type
equilibria), and identified the parameter regime for both. We found a change
of the dichotomy due to the presence of the seed-bank. Without seed-bank, for
migration in the domain of attraction of Brownian motion, clustering occurs in
$d=1,2$ and coexistence in $d\geq 3$, i.e., the _critical dimension_ for the
dichotomy is $d=2$. With seed-bank, however, clustering becomes more difficult
and occurs in $d=2$ only when the wake-up time of a typical individual in the
seed-bank has finite mean, and in $d=1$ only when the wake-up time has a
sufficiently thin tail. In other words, the seed-bank has a tendency to
_lower_ the critical dimension.

In fact, in [GHO20] we found that our technique of proof works for geographic
spaces that are _arbitrary_ countable Abelian groups endowed with the discrete
topology. The reason is that the dichotomy can be formulated in terms of how
the _degree_ of the random walk that underlies the migration balances with the
exponent of the _tail_ of the typical wake-up time. This raises the question
how we can better understand the behaviour of spatial models with seed-bank
close to criticality.

In [GHO21] we established the so-called _finite-systems scheme_ , i.e., we
identified how a finite truncation of the system (both in the geographic space
and in the seed-bank) behaves as both the time and the truncation level tend
to infinity, properly tuned together. We found that if the wake-up time has
finite mean, then the scaling time is proportional to the volume of the system
and there is a _single universality class_ for the scaling limit, namely, the
system moves through a succession of equilibria of the infinite system with a
density that evolves according to a Fisher-Wright diffusion. On the other
hand, we found that if the wake-up time has infinite mean, then the scaling
time grows faster than the volume of the system, and there are _two
universality classes_ depending on how fast the truncation level of the seed-
bank grows compared to the truncation level of the geographic space.

####  1.2 Goals

In the present paper we take as geographic space the _hierarchical group
$\Omega_{N}$_ of order $N$. The reason for this choice is that $\Omega_{N}$
allows for more detailed computations. At the same time, migration on
$\Omega_{N}$ can be used to _approximate_ migration on $\mathbb{Z}^{d}$ in the
_hierarchical mean-field limit_ $N\to\infty$. In particular, by playing with
the migration kernel we can approximate two-dimensional migration in the sense
of potential theory. We consider migration kernels that in the limit as
$N\to\infty$ are _critically recurrent_ , i.e., the degree of the class of
hierarchical migrations that we consider in the present paper converges to
$0$, either from above or from below.

The present paper has three goals:

  * (1)

We apply the results obtained in [GHO20] to $\Omega_{N}$ with $N<\infty$
fixed. We again find that part of the coexistence regime without seed-bank
shifts into the clustering regime with seed-bank when the average wake-up time
of a typical individual is infinite.

  * (2)

We analyse a _space-time renormalised_ system in the limit as $N\to\infty$.
Namely, we show that the block averages on successive space-time scales each
perform a diffusion with a _renormalised diffusion function_. In other words,
we establish a _multi-scale version of the finite-systems scheme_. Also, we
compare the behaviour of the space-time renormalised system with seed-bank to
the one analysed in [DG93b] and [DG93a] without seed-bank.

  * (3)

We exhibit _universal behaviour_ in the clustering regime close to
criticality. To do so, we analyse the attracting orbits of the renormalisation
transformation, acting on the space of diffusion functions, that connects
successive hierarchical levels. We show that, in the _clustering regime_ and
after appropriate scaling, the renormalised diffusion function converges to
the Fisher-Wright diffusion function as we move up in the hierarchy,
irrespective of the diffusion function controlling the resampling. This
convergence shows that the hierarchical system exhibits _universality_ on
large space-time scales in terms of the scaling limit. For several subclasses
of parameters we identify the scaling of the renormalised diffusion function,
which reveals a delicate interplay between the parameters controlling the
migration and the seed-bank. This rate in turn determines the speed at which
_mono-type clusters grow in space and time_.

In the _coexistence regime_ , universality does _not_ hold and the equilibrium
depends on the diffusion function. Since the seed-bank enhances genetic
diversity, it may be expected that equilibrium correlations between far away
colonies decay faster with seed-bank than without seed-bank, an issue that
will not be addressed.

######  Remark 1.1.

[More general types] Throughout the paper we consider the _two-type_ Fisher-
Wright model with seed-bank, in the _continuum limit_ where the number of
individuals per colony tends to infinity. The extension to a general type
space, called the Fleming-Viot model (see [DGV95]), requires only standard
adaptations and will not be considered here. In what follows, we _only_ work
with continuum models. However, we motivate these models by viewing them as
the _large-colony-size limit_ of individual-based models. For earlier work on
hierarchically interacting Fisher-Wright diffusions without seed-bank we refer
the reader to [DG93a, DG93b, DGV95, DG96] and [BCGdH95, BCGdH97, dH01].
$\blacksquare$

####  1.3 Outline

The present paper consist of two parts:

  * •

Part I: Model and main results. Sections 2–5 collect the main propositions and
theorems. In Section 2 we define the hierarchical model and state some basic
properties: the _well-posedness_ of the associated martingale problem
(Proposition 2.6), the _duality relation_ (Proposition 2.7), and the
_clustering criterion_ via duality (Proposition 2.12). These properties were
all derived in [GHO20]. In Section 3 we state our main results for $N<\infty$.
In particular, we compute the scaling of the wake-up time and the migration
kernel (Theorem 3.2) and identify the _clustering regime_ in terms of the
coefficients controlling the migration and the seed-bank under the assumption
that these are asymptotically polynomial or pure exponential (Theorem 3.3)).
In Section 4 we state our main results for $N\to\infty$, the hierarchical
mean-field limit. In particular, we introduce _block averages on successive
hierarchical space-time scales_ , analyse their limiting dynamics (Theorems
4.2 and 4.4), offer a heuristic explanation how this limiting dynamics arises,
introduce a path topology called the Meyer-Zheng topology that is needed for a
proper formulation, and introduce an object called the interaction chain,
which describes how the diffent hierarchical levels interact with each other.
In Section 5 we identify the _orbit of the renormalisation transformation_ in
the clustering regime (Theorem 5.1), identify the rate of scaling for the
renormalised diffusion function (Theorem 5.3), and link this scaling to the
rate of growth of mono-type clusters.

  * •

Part II: Preparations and proofs. Sections 6–13 provide the proofs of the
theorems stated in Part I. These proofs consist of a long series of
propositions and lemmas needed to build up the argument. In Section 6 we prove
our main results for $N<\infty$. In Sections 7–9 we focus on the _mean-field
model_ (consisting of a single hierarchy) and, respectively, state and prove a
number of results that serve as preparation. In Sections 10–12 we consider
extensions of the mean-field model (consisting of finitely hierarchies), which
serve as further preparation. In Section 12 we use the results in Sections
7–12 to deal with the full hierarchical model (consisting of infinitely many
hierarchies), and prove our main results for $N\to\infty$. In Section 13 we
analyse the orbit of the renormalisation transformation controlling the multi-
scaling. Appendix A contains a technical computation needed for the
identification of the clustering regime. Appendix B contains a basic
introduction to convergence of paths in the Meyer-Zheng topology, which is
needed for the main theorems.

Part I contains all the main results and their interpretations, and can be
read without reference to Part II.

##  Part I MODEL AND MAIN RESULTS

###  2 Introduction of model and basic properties

Section 2.1 introduces the model ingredients, Section 2.2 gives the evolution
equations, Section 2.3 states the well-posedness, Section 2.4 introduces the
dual and states the duality relation, while Section 2.5 formulates the
dichotomy between clustering versus coexistence in terms of the dual.

####  2.1 Model: geographic space $\Omega_{N}$, hierachical group of order $N$

###### Single colony.

Our building block is the single-colony Fisher-Wright model with seed-bank
defined in [BGCKWB16]. In that model, each individual in the population
carries one of two types, $\heartsuit$ or $\diamondsuit$, and each individual
can be either active or dormant. Active individuals _resample_ until they
become dormant. Dormant individuals suspend resampling until they become
actieve again. The repository for the dormant individuals is called the _seed-
bank_. When an active individual resamples, it randomly chooses another active
individual and _adopts its type_. When an active individual becomes dormant,
it randomly chooses a dormant individual and forces it to becomes active,
i.e., the active and the dormant population _exchange_ individuals (see Fig.
1). This exchange guarantees that the sizes of the active and the dormant
population stay fixed over time. During the swap both the active and the
dormant individual _retain their type_.

The types of the active population evolve through resampling and through
exchange with the dormant population. The types of the the dormant population
evolve only through exchange with the active population. It was shown in
[BGCKWB16] that in the large-colony-size limit, i.e., as the number of
individuals per colony tends to infinity and time is speeded up by the size of
the colony, the two quantities

  * •

$x(t)$ = the fraction of active individuals of type $\heartsuit$ at time $t$,

  * •

$y(t)$ = the fraction of dormant individuals of type $\heartsuit$ at time $t$,

satisfy the following system of coupled SDEs:

(2.1) $\displaystyle{\rm d}x(t)$ $\displaystyle=Ke\,[y(t)-x(t)]\,{\rm
d}t+\sqrt{x(t)(1-x(t))}\,{\rm d}w(t),$ $\displaystyle{\rm d}y(t)$
$\displaystyle=e\,[x(t)-y(t)]\,{\rm d}t.$

Here, $e$ denotes the rate at which an active individual _exchanges_ with a
dormant individual from the seed-bank, $K$ denotes the relative size of the
dormant population with respect to the active population, and $(w(t))_{t\geq
0}$ is a Brownian motion. The first term in the first equation describes the
flow from the dormant population to the active population, the term in the
second equation describes the flow from the active population to the dormant
population, while the second term in the first equation describes the effect
of resampling on the active population (see Fig. 1). Active individuals
resample at rate $1$. Since dormant individuals do not resample, we do not see
such a term in the second equation. The formal derivation of the continuum
equations can be found in [BGCKWB16] and in [GHO20, Appendix A].

$A$$D$exchangeresampling$Ke$$e$1 Figure 1: Active individuals resample at rate
$1$. Active and dormant individuals exchange at rate $e$. The extra factor $K$
arises from the fact that the dormant population is $K$ times as large as the
active population. Dormant individuals suspend resampling.

###### Multi-colony.

The present paper focuses on a _multi-colony_ setting of the model described
above, where the underlying geographic space is the hierarchical lattice of
order $N$, given by ($\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$)

(2.2)
$=\left\\{\xi=(\xi_{k})_{k\in\mathbb{N}_{0}}\colon\,\xi_{k}\in\\{0,1,\ldots,N-1\\},\
\sum_{k\in\mathbb{N}_{0}}\xi_{k}<\infty\right\\},$

which with addition modulo $N$ becomes the hierarchical group of order $N$
(see Fig. 2). The _hierarchical distance_ on $\Omega_{N}$ is defined by

(2.3)
$(\xi,\eta)=d_{\Omega_{N}}(0,\xi-\eta)=\min\left\\{k\in\mathbb{N}_{0}\colon\,\xi_{l}=\eta_{l}\
\forall\,l\geq k\right\\},\qquad\xi,\eta\in\Omega_{N},$

and is an ultra-metric, i.e.,

(2.4)
$d_{\Omega_{N}}(\xi,\eta)\leq\max\big{\\{}d_{\Omega_{N}}(\xi,\zeta),d_{\Omega_{N}}(\eta,\zeta)\big{\\}}\qquad\forall\,\xi,\eta,\zeta\in\Omega_{N}.$

The choice of $\Omega_{N}$ as geographic space plays an important role for
population models, and was first exploited in [SF83] in an attempt to
formalise ideas coming from ecology. One interpretation is that the sequence
$(\xi_{k})_{k\in\mathbb{N}_{0}}$ encodes the ‘address’ of colony $\xi$:
$\xi_{0}$ is the ‘house’, $\xi_{1}$ is the ‘street’, $\xi_{2}$ is the
‘village’, $\xi_{3}$ is the ‘province’, $\xi_{4}$ is the ‘country’, and so on.
To describe the system on the hierarchical group we need three ingredients:

  1. 1.

Hierarchical migration (Section 2.1.1).

  2. 2.

Layered seed-bank (Section 2.1.2).

  3. 3.

Resampling rate (Section 2.1.3).

Figure 2: Close-ups of a 1-block, a 2-block and a 3-block in the hierarchical
group of order $N=3$. The elements of the group are the leaves of the tree
(indicated by $\Box$’s). The hierarchical distance between two elements in the
group is the graph distance to the most recent common ancestor in the tree:
$d_{\Omega_{3}}(\eta,\zeta)=2$ for $\eta$ and $\zeta$ in the picture.

#####  2.1.1 Hierarchical migration

We construct a migration kernel $a^{\Omega_{N}}(\cdot,\cdot)$ on the
hierarchical group $\Omega_{N}$ built from a sequence of migration rates

(2.5) $=()_{k\in\mathbb{N}_{0}}\in(0,\infty)^{\mathbb{N}_{0}}$

that do not depend on $N$. Individuals migrate as follows:

  * •

For all $k\in\mathbb{N}$, each individual chooses at rate $c_{k-1}/N^{k-1}$
the block of radius $k$ around its present location and selects a colony
uniformly at random from that block. Subsequently it selects an individual in
this colony uniformly at random and adopts its type.

Note that the block of radius $k$ contains $N^{k}$ colonies, and that the
migration kernel is therefore given by

(2.6) $a^{\Omega_{N}}(\eta,\xi)=\sum\limits_{k\geq
d_{\Omega_{N}}(\eta,\xi)}\frac{c_{k-1}}{N^{k-1}}\frac{1}{N^{k}},\quad\eta,\xi\in\Omega_{N},\eta\neq\xi,\qquad
a^{\Omega_{N}}(\eta,\eta)=0,\quad\eta\in\Omega_{N}.$

Throughout the paper, we assume that

(2.7) $\limsup_{k\to\infty}\frac{1}{k}\,\log c_{k}<\log N.$

This guarantees that the total migration rate per individual is finite.
Indeed, note that for every $\eta\in\Omega_{N}$,

(2.8)
${}\sum_{\xi\in\Omega_{N}}a^{\Omega_{N}}(\eta,\xi)=\sum_{\xi\in\Omega_{N}}\sum_{k\geq
d_{\Omega_{N}}(\eta,\xi)}\frac{c_{k-1}}{N^{2k-1}}=\sum_{k\in\mathbb{N}}\left[\sum_{\xi\in\Omega_{N}}\textbf{1}_{\\{d_{\Omega_{N}}(\eta,\xi)\leq
k\\}}\right]\frac{c_{k-1}}{N^{2k-1}}=\sum_{k\in\mathbb{N}}\frac{c_{k-1}}{N^{k-1}},$

which is finite because of (2.7).

######  Remark 2.1.

[Degree of random walk] For a random walk on an Abelian group with time-$t$
transition kernel , the _degree_ is defined as (see [DGW05])

(2.9) $=\sup\left\\{\zeta\in(-1,\infty)\colon\,\int^{\infty}_{0}{\rm
d}t\,t^{\zeta}a^{\Omega_{N}}_{t}(0,0)<\infty\right\\}.$

The degree is said to be $\delta^{+}$, respectively, $\delta^{-}$ when the
integral is finite, respectively, infinite _at_ the degree. If $\delta>0$,
then $\delta$ is called the _degree of transience_. If $\delta\in(-1,0)$, then
$-\delta$ is called the _degree of recurrence_. If the degree is $0^{-}$, then
the random walk is called _critically recurrent_. (It would be interesting to
have a version of (2.9) that includes a slowly varying function in front of
the power $t^{\zeta}$. However, such an extension appears not to have been
explored in the literature.) $\blacksquare$

By playing with $\underline{c}$ and letting $N\to\infty$, we can approximate
migration for which the corresponding random walk is critically recurrent,
i.e., $\delta^{-}=0$. In that case both the potential theory and the Green
function for the hierarchical random walk have the same asymptotics as the
potential theory and the Green function for a critically recurrent random walk
on $\mathbb{Z}^{2}$ in the domain of attraction of Brownian motion. Therefore,
by tuning $\underline{c}$ properly, we can mimic migration on the geographic
space $\mathbb{Z}^{2}$ (for which $\delta^{-}=0$), an idea that was exploited
in [DGV95], [DG96], [DG14], [GdHKK14], [GdHK18].

#####  2.1.2 Layered seed-bank

To create a layered seed-bank, dormant individuals are labeled with a colour
$m\in\mathbb{N}_{0}$. An active individual that becomes dormant is assigned a
colour $m\in\mathbb{N}_{0}$. When an active individual becomes dormant with
colour $m$, it exchanges with a dormant individual of colour $m$. This dormant
individual becomes active, _loses_ its colour, but _retains_ its type. To
describe the layered seed-bank we need two sequences

(2.10)
$\displaystyle=(K_{m})_{m\in\mathbb{N}_{0}}\in(0,\infty)^{\mathbb{N}_{0}},$
$\displaystyle=()_{m\in\mathbb{N}_{0}}\in(0,\infty)^{\mathbb{N}_{0}},$

both not depending on $N$, which we interpret as follows:

  * •

is the relative size of the dormant population of colour $m$ with respect to
the active population, i.e.,

(2.11) $K_{m}=\frac{\text{size $m$-dormant population}}{\text{size active
population}}.$

  * •

At rate $K_{m}\frac{e_{m}}{N^{m}}$ an active individual becomes dormant, is
assigned colour $m$, and retains its type. At the same time a dormant
individual with colour $m$ becomes active, loses its colour, and retains its
type. By defining the rates in this way, the layered structure of the seed-
bank is tuned to the hierarchical structure of the geographic space.

By giving the seed-bank a layered structure, we are able to tune the
distribution of the wake-up time, i.e., the time an individual spends in the
seed-bank before waking up. In particular, we will see that a layered seed-
bank enables us to model wake up times with a _fat tail_ , while at the same
time preserving the Markov property of the evolution.

Since active and dormant individuals exchange, $K_{m}$ remains constant over
time for all $m\in\mathbb{N}_{0}$. Throughout the paper we assume that

(2.12) $\limsup_{m\to\infty}\frac{1}{m}\,\log(K_{m}e_{m})<\log N.$

This guarantees that the total rate of exchange per individual, given by

(2.13) $=\sum_{m\in\mathbb{N}_{0}}K_{m}\frac{e_{m}}{N^{m}},$

is finite. On the other hand, the relative size of the dormant population with
respect to the active population

(2.14) $=\sum_{m\in\mathbb{N}_{0}}K_{m}$

can be either finite or infinite. We will see that $\rho<\infty$ and
$\rho=\infty$ represent two _different regimes_.

#####  2.1.3 Resampling rate

To describe the resampling we use a diffusion function that is taken from the
set

(2.15)
$=\Big{\\{}g(x)\colon\,[0,1]\to[0,\infty)\colon\,g(0)=g(1)=0,\,g(x)>0\,\,\forall\,x\in(0,1),\,g\mbox{
Lipschitz}\Big{\\}},$

and think of $h(x)=g(x)/x(1-x)$ as the rate of resampling at type frequency
$x$. The choice $g=dg_{\mathrm{FW}}$, $d\in(0,\infty)$, with
$g_{\mathrm{FW}}(x)=x(1-x)$, $x\in[0,1]$, corresponds to Fisher-Wright
resampling at rate $d$. We use a collection of independent Brownian motions

(2.16) $W=\big{(}\big{)}_{\xi\in\Omega_{N}}$

to describe the fluctuations of the type frequencies caused by the resampling
in each colony.

####  2.2 Evolution equations

#####  2.2.1 Evolution of single colonies

With the above three ingredients, we can now describe the evolution of the
system. For $\xi\in\Omega_{N}$, define

(2.17) $\displaystyle=\text{ the fraction of active individuals of type
$\heartsuit$ at colony $\xi$ at time $t$},$ $\displaystyle(t)$
$\displaystyle=\text{ the fraction of $m$-dormant individuals of type
$\heartsuit$ at colony $\xi$ and time $t$}.$

Note that $x_{\xi}(t)\in[0,1]$ and $y_{\xi,m}(t)\in[0,1]$ for all
$\xi\in\Omega_{N}$, $m\in\mathbb{N}_{0}$, $t\geq 0$. Therefore the state space
of a single colony is $\mathfrak{s}=[0,1]\times[0,1]^{\mathbb{N}_{0}}$, and
the state space of the system is

(2.18) $=\mathfrak{s}^{\Omega_{N}}.$

Our object of interest is the random process taking values in $S$, written

(2.19)
$=,\qquad(X^{\Omega_{N}}(t),Y^{\Omega_{N}}(t))=\big{(}x_{\xi}(t),(y_{\xi,m}(t))_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}},$

whose components evolve according to the following SSDE (= system of
stochastic differential equations):

(2.20) $\displaystyle{\rm d}x_{\xi}(t)$
$\displaystyle=\sum_{\eta\in\Omega_{N}}a^{\Omega_{N}}(\xi,\eta)[x_{\eta}(t)-x_{\xi}(t)]\,{\rm
d}t+\sqrt{g(x_{\xi}(t))}\,{\rm d}w_{\xi}(t)$
$\displaystyle\qquad+\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}e_{m}}{N^{m}}\,[y_{\xi,m}(t)-x_{\xi}(t)]\,{\rm
d}t,$ $\displaystyle{\rm d}y_{\xi,m}(t)$
$\displaystyle=\frac{e_{m}}{N^{m}}\,[x_{\xi}(t)-y_{\xi,m}(t)]\,{\rm d}t,\quad
m\in\mathbb{N}_{0},\qquad\xi\in\Omega_{N}.$

The first term in the first equation describes the evolution of the active
population at colony $\xi$ due to migration, the second term due to the
resampling. The third term in the first equation and the term in the second
equation describe the exchange between the active and the dormant population
at colony $\xi$ (see Fig. 3). Since dormant individuals are not subject to
resampling or migration, the dynamics of the dormant population is completely
determined by the exchange with the active population. For the _initial state_
we assume that

(2.21)
$\displaystyle\mathcal{L}(X^{\Omega_{N}}(0),Y^{\Omega_{N}}(0))=\mu^{\otimes\Omega_{N}}$
$\displaystyle\text{with
}\mathbb{E}^{\mu}[x_{\xi}(0)]=,\,\mathbb{E}^{\mu}[y_{\xi,m}(0)]=\text{ with
}\lim_{m\to\infty}\theta_{y_{m}}=\theta\text{ for some }\theta\in[0,1].$

The last assumption in (2.21), which in [GHO20] was referred to as $\mu$ being
_colour regular_ , guarantees that for finite $N$ the system in (2.20)
converges to an ergodic equilibrium.

######  Remark 2.2.

[Notation] Throughout the sequel we use lower case letters for _single
components_ and upper case letters for _systems of single components_. We
exhibit the geographic space for the system, but suppress it from the
components. $\blacksquare$

A$D_{0}$$D_{1}$$D_{m}$exchangeresamplingmigration$K_{0}e_{0}$$e_{0}$$K_{1}e_{1}/N$$e_{1}/N$$K_{m}e_{m}/N^{m}$$e_{m}/N^{m}$$h(x)$
Figure 3: Active individuals ($A$) are subject to migration, resampling and
exchange with dormant individuals ($D$). When active individuals become
dormant they are assigned a colour ($D_{m}$, $m\in\mathbb{N}_{0}$), which they
lose when they become active again. The resampling rate in the active state at
type-$\heartsuit$ frequency $x$ equals $h(x)=g(x)/x(1-x)$ with
$g\in\mathcal{G}$ (e.g. for the standard Fisher-Wright diffusion the
resampling rate is $1$).

#####  2.2.2 Evolution of block averages

The choice of the migration kernel in (2.6) implies that, for every
$k\in\mathbb{N}$, at rate $\asymp\frac{1}{N^{k}}$ individuals choose a space
horizon of distance $k+1$ and subsequently choose a random colony from that
space horizon. Therefore, in order to see interactions over a distance $k+1$,
we need to speed up time by a factor $N^{k}$. A similar observation applies to
the interaction with the seed-bank. Dormant individuals with colour $k$ become
active at rate $\asymp\frac{1}{N^{k}}$. Therefore, in order to see
interactions with the $k$-dormant population, we need to speed up time by a
factor $N^{k}$. To analyse the effective interaction on time scale $N^{k}$, we
introduce _successive block averages_ labelled by $k\in\mathbb{N}_{0}$.

######  Definition 2.3.

[Block averages] For $k\in\mathbb{N}_{0}$, let
$B_{k}(0)=\\{\eta\in\Omega_{N}\colon\,d_{\Omega_{N}}(0,\eta\leq k\\}$ denote
the $k$-block around $0$. Define the $k$-block average around $0$ at time
$N^{k}t$ by

(2.22) $\displaystyle=\frac{1}{N^{k}}\sum_{\eta\in B_{k}(0)}x_{\eta}(N^{k}t),$
$\displaystyle=\frac{1}{N^{k}}\sum_{\eta\in B_{k}(0)}y_{\eta,m}(N^{k}t),\qquad
m\in\mathbb{N}_{0}.$

The $k$-block average represents the dynamics of the system on space-time
scale $k$. $\blacksquare$

By translation invariance of the SSDE in (2.20), each $\xi\in\Omega_{N}$ can
serve as the origin. In the remainder of the paper we consider without loss of
generality the $k$-block average around $\xi=0$, and suppress the center $0$
from the notation.

######  Remark 2.4.

[Notation] We use lower case letters for the block averages because they live
in the space of components $\mathfrak{s}=[0,1]\times[0,1]^{\mathbb{N}_{0}}$.
At the same time we exhibit the geographic space $\Omega_{N}$ for the block
averages because they are functionals of the system of components (recall
Remark 2.2). $\blacksquare$

Using Definition 2.3 and inserting the specific choice of the migration kernel
defined in (2.6), we can rewrite (2.20) for $\xi=0$ as follows ($0$-blocks are
single components):

(2.23) $\displaystyle{\rm d}x_{0}^{\Omega_{N}}(t)$
$\displaystyle=\sum_{l\in\mathbb{N}}\frac{c_{l-1}}{N^{l-1}}\big{[}x_{l}^{\Omega_{N}}(N^{-l}t)-x_{0}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t+\sqrt{g\big{(}x_{0}^{\Omega_{N}}(t)\big{)}}\,{\rm d}w(t)$
$\displaystyle\qquad+\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}e_{m}}{N^{m}}\big{[}y_{m,0}^{\Omega_{N}}(t)-x_{0}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{\rm d}y_{m,0}^{\Omega_{N}}(t)$
$\displaystyle=\frac{e_{m}}{N^{m}}\big{[}x_{0}^{\Omega_{N}}(t)-y_{m,0}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,\qquad m\in\mathbb{N}_{0}.$

From (2.23) we see that migration between colonies can be expressed as a drift
towards block averages at a higher hierarchical level.

The SSDE for the $k$-block average on time scale $N^{k}$ reads as follows
(recall (2.22)):

(2.24) $\displaystyle{\rm d}{x}_{k}^{\Omega_{N}}(t)$
$\displaystyle=\sum_{l\in\mathbb{N}}\frac{c_{k+l-1}}{N^{l-1}}\big{[}{x}_{k+l}^{\Omega_{N}}(N^{-l}t)-{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t+\sqrt{\frac{1}{N^{k}}\sum_{i\in
B_{k}(0)}g\big{(}x_{i}(N^{k}t)\big{)}}\,\,{\rm d}w_{k}(t)$
$\displaystyle\qquad+\sum_{m\in\mathbb{N}_{0}}N^{k}\frac{K_{m}e_{m}}{N^{m}}\big{[}{y}_{m,k}^{\Omega_{N}}(t)-{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{\rm d}{y}_{m,k}^{\Omega_{N}}(t)$
$\displaystyle=N^{k}\frac{e_{m}}{N^{m}}\big{[}{x}_{k}^{\Omega_{N}}(t)-{y}_{m,k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,\qquad m\in\mathbb{N}_{0}.$

To deduce these equations from (2.20), we sum over $\xi\in B_{k}$, speed up
time by a factor $N^{k}$, insert the specific choice of the migration kernel
in (2.6), and use the standard scaling properties of Brownian motion:
$w(ct)=^{d}\sqrt{c}\,w(t)$ and
$\sqrt{a}w(t)+\sqrt{b}w^{\prime}(t)=^{d}\sqrt{a+b}\,w^{\prime\prime}(t)$, with
$w(t)$ and $w^{\prime}(t)$ independent Brownian motions, and with $=^{d}$
denoting equality in distribution. This computation is spelled out in Section
12.

######  Remark 2.5.

The _block averages_ and their evolution equations in (2.24) will be key
objects in the analysis of the hierarchical mean-field limit $N\to\infty$. We
will see that the limit $N\to\infty$ brings about considerable
simplifications. In Section 4 we discuss these simplifications in detail. In
particular, a _complete separation of space-time scales_ takes places, in
which each block average lives on its own time scale, effectively interacts
with only one seed-bank, and effectively feels a drift towards the block
average one hierarchical level up. $\blacksquare$

####  2.3 Well-posedness

Let denote the space of continuous function form $[0,\infty)$ to the state
space $S$ and let be the space of bounded continuous functions form
$[0,\infty)$ to $S$. The generator of the system in (2.20) is given by

$\displaystyle=$
$\displaystyle\sum_{\xi\in\Omega_{N}}\Bigg{(}\sum_{\eta\in\Omega_{N}}a^{\Omega_{N}}(\xi,\eta)[x_{\eta}(t)-x_{\xi}(t)]\frac{\partial}{\partial
x_{\xi}}+\frac{1}{2}g(x_{\xi}(t))\frac{\partial^{2}}{\partial x_{\xi}^{2}}$
$\displaystyle\quad+\sum_{m\in\mathbb{N}_{0}}\left[\frac{K_{m}e_{m}}{N^{m}}[y_{\xi,m}(t)-x_{\xi}(t)]\frac{\partial}{\partial
x_{\xi}}+\frac{e_{m}}{N^{m}}[x_{\xi}(t)-y_{\xi,m}(t)]\frac{\partial}{\partial
y_{\xi,m}}\right]\Bigg{)}.$

Let

(2.26) $\displaystyle=\Big{\\{}$ $\displaystyle f\in
C_{b}([0,\infty),S)\colon\,f\mbox{ depends on finitely many components}$
$\displaystyle\mbox{and is twice continuously differentiable in each
component}\Big{\\}}.$

######  Proposition 2.6.

[Well-posedness]

  * (a)

The SSDE in (2.20) has a unique strong solution in , whose law is the unique
solution of the $(G,\mathbf{F},\delta_{u})$-martingale problem for all $u\in
S$.

  * (b)

The process starting from $u\in S$ is Feller and strong Markov. Consequently,
the SSDE in (2.20) defines a unique Borel Markov process starting from any
initial law on $S$.

###### Proof.

Comparing with what is called model 2 in [GHO20], we see that the Abelian
group is chosen as in (2.2), the transition kernel is chosen as in (2.6), and
the rates in and out of the seed-bank are $\frac{e_{m}}{N^{m}}$ and
$\frac{K_{m}e_{m}}{N^{m}}$ for colour $m$. Hence the claim follows from
[SS80], in the same way as shown in the proof of [GHO20, Theorem 2.1]. ∎

Henceforth we write $\mathbb{P}$ and $\mathbb{E}$ to denote probability and
expectation with respect to the random process in (2.19).

####  2.4 Duality

If $g=d$, then our model has a tractable dual, which turns out to play a
crucial role in the analysis of the long-time behaviour. In this section we
introduce the dual process following the same line of argument as in [GHO20,
Section 2.4]. There it was shown that the spatial Fisher-Wright diffusion with
seed-bank is dual to a so-called _block-counting process_ of a seed-bank
coalescent. The latter describes the ancestral lines of $n\in\mathbb{N}$
individuals sampled from the current population backwards in time in terms of
partition elements. At time zero the ancestral line of each individual is
represented by a partition element. Traveling backwards in time, two partition
elements merge as soon as their ancestral lines coalesce, i.e., two
individuals have the same ancestor from that time onwards. Hence the seed-bank
coalescent divides the ancestral lines of the $n\in\mathbb{N}$ individuals
into subgroups of individuals with the same ancestor (i.e., individuals that
are identical by descent). Therefore the seed-bank coalescent generates the
ancestral lineages of the individuals evolving according to a Fisher-Wright
diffusion with seed-bank, i.e., generates their full genealogy. The
corresponding block-counting process counts the number of partition elements
that are left when we travel backwards in time.

coalescencemigration$e_{m}K_{m}/N^{m}$$e_{m}/N^{m}$$m$-dormantactive Figure 4:
Transition scheme for an ancestral lineage in the dual, which moves according
to the transition kernel $b(\cdot,\cdot)$ defined in (2.30). Two active
ancestral lineages that are at the same colony coalesce at rate $d$.

Formally, the spatial seed-bank coalescent is described as follows. Let
$\mathbb{S}=\Omega_{N}\times\\{A,(D_{m})_{m\in\mathbb{N}_{0}}\\}$ be the
_effective geographic space_. For $n\in\mathbb{N}$ the state space of the _$n$
-spatial seed-bank coalescent_ is the set of partitions of $\\{1,\ldots,n\\}$,
where the partition elements are marked with a position vector giving their
locations. A state is written as $\pi$, where

(2.27)
$\displaystyle\pi=((\pi_{1},\eta_{1}),\ldots,(\pi_{\bar{n}},\eta_{\bar{n}})),\quad\bar{n}=|\pi|,$
$\displaystyle\pi_{\ell}\subset\\{1,\ldots,n\\},\quad\\{\pi_{1},\cdots\pi_{\bar{n}}\\}\
\mbox{ is a partition of }\\{1,\ldots,n\\},$
$\displaystyle\eta_{\ell}\in\mathbb{S},\quad\ell\in\\{1,\ldots,\bar{n}\\},\quad
1\leq\bar{n}\leq n.$

A marked partition element $(\pi_{\ell},\eta_{\ell})$ is called active if
$\eta_{\ell}=(\xi,A)$ and $m$-dormant if $\eta_{\ell}=(\xi,D_{m})$ for some
$\xi\in\Omega_{N}$. The $n$-spatial seed-bank coalescent is denoted by

(2.28) ,

and starts from

(2.29)
$\mathcal{C}^{(n)}(0)=\pi(0),\qquad\pi(0)=\\{(\\{1\\},\eta_{\ell_{1}}),\ldots,(\\{n\\},\eta_{\ell_{n}})\\},\qquad\eta_{\ell_{1}},\ldots,\eta_{\ell_{n}}\in\mathbb{S}.$

$t$ Figure 5: Picture of the evolution of lineages in the spatial coalescent.
The purple blocks depict the colonies, the black lines the active lineages,
and the coloured lines the dormant lineages. Blue lineages can migrate. Two
black lineages can coalesce when they are at the same colony. Red dormant
lineages first have to become black and active before they can migrate or
coalesce with other black and active lineages. Note that the dual runs
_backwards in time_.

The $n$-spatial seed-bank coalescent is the Markov process that evolves
according to the following two rules (see Figs. 4–5).

  1. 1.

Each partition element moves independently of all other partition elements
according to the transition kernel

(2.30)
$b((\xi,R_{\xi}),(\eta,R_{\eta}))=\left\\{\begin{array}[]{ll}a^{\Omega_{N}}(\xi,\eta),&\text{
if }R_{\xi}=R_{\eta}=A,\\\ K_{m}\frac{e_{m}}{N^{m}},&\text{ if }\xi=\eta,\
R_{\xi}=A,\ R_{\eta}=D_{m},\text{ for some }m\in\mathbb{N}_{0},\\\
\frac{e_{m}}{N^{m}},&\text{ if }\xi=\eta,\ R_{\xi}=D_{m},\ R_{\eta}=A,\text{
for some }m\in\mathbb{N}_{0},\\\ 0,&\mbox{ otherwise},\end{array}\right.$

where $a^{\Omega_{N}}(\cdot,\cdot)$ is the migration kernel defined in (2.6),
$K_{m}$, $m\in\mathbb{N}_{0}$ are the relative sizes of the $m$-dormant
population and the active population defined in (2.11), and $e_{m}$,
$m\in\mathbb{N}_{0}$ are the coefficients controlling the exchange between the
active and the dormant population defined in (LABEL:defKem). Thus, an active
partition element migrates according to the transition kernel
$a^{\Omega_{N}}(\cdot,\cdot)$ and becomes $m$-dormant at rate
$K_{m}\frac{e_{m}}{N^{m}}$, while an $m$-dormant partition element can only
become active and does so at rate $\frac{e_{m}}{N^{m}}$.

  2. 2.

Independently of all other partition elements, two partition elements that are
at the same colony and are both active coalesce with rate $d$, i.e., the two
partition elements merge into one partition element.

Fig. 4 gives a schematic overview of the possible transitions of a single
lineage, while Fig. 5 gives an example of the evolution in the dual. The
_spatial seed-bank coalescent_ $(\mathcal{C}(t))_{t\geq 0}$ is defined as the
projective limit of the $n$-spatial seed-bank coalescents
$(\mathcal{C}^{(n)}(t))_{t\geq 0}$ as $n\to\infty$. This object is well-
defined by Kolmogorov’s extension theorem (see [BGCKWB16, Section 3]).

For $n\in\mathbb{N}$ we define the block-counting process corresponding to the
$n$-spatial seed-bank coalescent as the process that counts at each site
$(\xi,R_{\xi})\in\Omega_{N}\times\\{A,(D_{m})_{m\in\mathbb{N}_{0}}\\}$ the
number of partition elements of $\mathcal{C}^{(n)}(t)$, i.e.,

(2.31) $\displaystyle
L(t)=\big{(}L_{(\xi,A)}(t),\left(L_{(\xi,D_{m})}(t)\right)_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}},$
$\displaystyle
L_{(\xi,A)}(t)=L_{(\xi,A)}(\mathcal{C}^{(n)}(t))=\sum_{\ell=1}^{\bar{n}}1_{\\{\eta_{\ell}(t)=(\xi,A)\\}},$
$\displaystyle
L_{(\xi,D_{m})}(t)=L_{(\xi,D_{m})}(\mathcal{C}^{(n)}(t))=\sum_{\ell=1}^{\bar{n}}1_{\\{\eta_{\ell}(t)=(\xi,D_{m})\\}},\qquad
m\in\mathbb{N}_{0}.$

The state space of $(L(t))_{t\geq 0}$ is
$S^{\prime}=(\mathbb{N}_{0}\times\mathbb{N}_{0}^{\mathbb{N}_{0}})^{\Omega_{N}}$.
We denote the elements of $S^{\prime}$ by sequences
$(m_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}$, and
define $\delta_{(\eta,R_{\eta})}\in S^{\prime}$ to be the element of
$S^{\prime}$ that is $0$ at all sites
$(\xi,R_{\xi})\in\Omega_{N}\times\\{A,(D_{m})_{m\in\mathbb{N}_{0}}\\}\backslash(\eta,R_{\eta})$,
and $1$ at the site $(\eta,R_{\eta})$. From the evolution of
$\mathcal{C}^{(n)}(t)$ described below (2.28) we see that the block-counting
process has the following transition kernel:

(2.32)
$\displaystyle(m_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}\rightarrow$
$\displaystyle\qquad\begin{cases}({m}_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}-\delta_{(\eta,A)}+\delta_{(\zeta,A)},\
&\text{at rate }m_{\eta}a(\eta,\zeta)\text{ for }\eta,\zeta\in\Omega_{N},\\\
({m}_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}-\delta_{(\eta,A)},\
&\text{at rate }d{m_{\eta}\choose 2}\text{ for }\eta\in\Omega_{N},\\\
({m}_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}-\delta_{(\eta,A)}+\delta_{(\eta,D_{m})},\
&\text{at rate }m_{\eta}K_{m}\frac{e_{m}}{N^{m}}\text{ for
}\eta\in\Omega_{N},\\\
({m}_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}+\delta_{(\eta,A)}-\delta_{(\eta,D_{m})},\
&\text{at rate }n_{\eta,m}\frac{e_{m}}{N^{m}}\text{ for }\eta\in\Omega_{N}.\\\
\end{cases}$

The process $(Z(t))_{t\geq 0}$ defined in (2.20) is dual to the block-counting
process $(L(t))_{t\geq 0}$ with _duality function_ $\colon\,S\times
S^{\prime}\to\mathbb{R}$ defined by

(2.33)
$H\Big{(}\big{(}x_{\xi},(y_{\xi,m})_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}},\big{(}m_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}}\Big{)}=\prod_{\xi\in\Omega_{N}}x_{\xi}^{m_{\xi}}\prod_{m\in\mathbb{N}_{0}}y_{\xi,m}^{n_{\xi,D_{m}}}.$

######  Proposition 2.7.

[Duality relation]. Let $H$ be as in (2.33). Then, for all
$(x_{\xi},(y_{\xi,m})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}$ $\in S$ and
$(m_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}})_{\xi\in\Omega_{N}}\in
S^{\prime}$,

(2.34)
$\displaystyle\mathbb{E}_{\big{(}x_{\xi},(y_{\xi,m})_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}}}\Big{[}H\Big{(}\big{(}x_{\xi}(t),(y_{\xi,m}(t))_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}},(m_{\xi},n_{\xi})_{\xi\in\Omega_{N}}\Big{)}\Big{]}$
$\displaystyle=\mathbb{E}_{\big{(}m_{\xi},(n_{\xi,D_{m}})_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}}}\Big{[}H\Big{(}\big{(}x_{\xi},(y_{\xi,m})_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}},\big{(}L_{(\xi,A)}(t),(L_{(\xi,D_{m})}(t))_{m\in\mathbb{N}_{0}}\big{)}_{\xi\in\Omega_{N}}\Big{)}\Big{]}$

with $\mathbb{E}$ the generic symbol for expectation (on the left over the
original process, on the right over the dual process).

Proposition 2.7 was proved in [GHO20, Section 2.4]. Since the duality function
$H$ captures all the mixed moments of $(Z(t))_{t\geq 0}$, the duality relation
is that of a _moment dual_.

######  Remark 2.8.

[Duality relation in terms of the effective geographic space] Interpreting
$(Z(t))_{t\geq 0}$ as a process on the effective geographic space
$\mathbb{S}=\Omega_{N}\times\\{A,(D_{m})_{m\in\mathbb{N}_{0}}\\}$, we can
rewrite (2.20) as

(2.35) $\displaystyle{\rm d}$
$\displaystyle=\sum_{(\xi,R_{\xi})\in\mathbb{S}}b((\xi,R_{\xi}),(\eta,R_{\eta}))[z_{(\eta,R_{\eta})}(t)-z_{(\xi,R_{\xi})}(t)]\,{\rm
d}t$
$\displaystyle\qquad+1_{\\{R_{\xi}=A\\}}\sqrt{g(z_{(\xi,R_{\xi})}(t))}\,{\rm
d}w_{\xi}(t),\qquad(\xi,R_{\xi})\in\mathbb{S},$

where $b(\cdot,\cdot)$ is the transition kernel defined in (2.30). If
$g=dg_{\mathrm{FW}}$, then we can write its dual process as follows. Let
$(L(t))_{t\geq 0}=(L(\mathcal{C}(t))_{t\geq 0}$ be the block-counting process
that at each site $(\xi,\mathbb{R}_{\xi})\in\mathbb{S}$ counts the number of
partition elements of $\mathcal{C}(t)$, i.e.,

(2.36) $\displaystyle L(t)$
$\displaystyle=(L_{(\xi,R_{\xi})}(t))_{{(\xi,R_{\xi})}\in\mathbb{S}},$
$\displaystyle L_{(\xi,R_{\xi})}(t)$
$\displaystyle=L_{(\xi,R_{\xi})}(\mathcal{C}(t))=\sum_{\ell=1}^{\bar{n}}1_{\\{\eta_{\ell}(t)={(\xi,R_{\xi})}\\}}.$

Rewrite the duality function $H$ in (2.33) as

(2.37)
$H\Big{(}(z_{(\xi,R_{\xi})},l_{(\xi,R_{\xi})})_{{(\xi,R_{\xi})}\in\mathbb{S}}\Big{)}=\prod_{{(\xi,R_{\xi})}\in\mathbb{S}}z_{(\xi,R_{\xi})}^{l_{(\xi,R_{\xi})}}.$

Then, for $z\in S$ and $l\in S^{\prime}$, the duality relation in (2.34) reads

(2.38)
$\mathbb{E}_{z(\xi,R_{\xi})}\big{[}H(z_{(\xi,R_{\xi})}(t),l_{(\xi,R_{\xi})})\big{]}=\mathbb{E}_{l(\xi,R_{\xi})}\big{[}H(z_{(\xi,R_{\xi})},L_{(\xi,R_{\xi})}(t))\big{]}.$

Interpreting the duality relation in terms of the effective geographic space
$\mathbb{S}$, we see that each ancestral lineage in the dual is a Markov
process moving according to the transition kernel $b(\cdot,\cdot)$ defined in
(2.30). Interpreting the duality relation in terms of the geographic space
$\Omega_{N}$, we see that an ancestral lineage is a random walk on
$\Omega_{N}$, with internal states $A$ and $(D_{m})_{m\in\mathbb{N}_{0}}$.
Both interpretations turn out to be useful when we analyse the long-time
behaviour of the system. $\blacksquare$

$\sigma_{1}$$\tau_{1}$$\sigma_{2}$$\tau_{2}$$\sigma_{3}$$\tau_{3}$ Figure 6:
Renewal process induced by a lineage in the dual moving according to the
transition kernel $b(\cdot,\cdot)$. For $k\in\mathbb{N}$, $\sigma_{k}$ denotes
the $k$th active period and $\tau_{k}$ the $k$th dormant period.

######  Remark 2.9.

[The renewal process induced by the dual process] The partition elements
describing the dual process give rise to a renewal process on the active state
$A$ and the dormant state $D=\bigcup_{m\in\mathbb{N}_{0}}D_{m}$. Since the
only transition a dormant lineage can make is to become active, irrespectively
of its colour, each dual lineage induces a sequence of active and dormant time
lapses. Let $(\sigma_{k})_{k\in\mathbb{N}}$ denote the successive active time
periods and $(\tau_{k})_{k\in\mathbb{N}}$ the successive dormant time periods
(see Fig. 6). Then $()_{k\in\mathbb{N}}$ and $()_{k\in\mathbb{N}}$ are
sequences of i.i.d. random variables with marginal laws (recall (2.13))

(2.39) $\mathbb{P}(\sigma_{1}>t)={\rm e}^{-\chi
t},\quad\mathbb{P}(\tau_{1}>t)=\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}\frac{e_{m}}{N^{m}}}{\chi}\,{\rm
e}^{-\frac{e_{m}}{N^{m}}t},\qquad t\geq 0.$

######  Remark 2.10.

[Wake up times] The renewal process in Fig. 6 is key to understanding the
long-time behaviour of the model (as we will see in Section 3). Note that

(2.40) $=\tau_{1}$

represents _the typical wake-up time of a lineage in the dual_. By choosing
specific sequences $(K_{m})_{m\in\mathbb{N}_{0}}$ and
$(e_{m})_{m\in\mathbb{N}_{0}}$ we can mimic different wake-up time
distributions. In particular, if we allow
$\rho=\sum_{m\in\mathbb{N}_{0}}K_{m}=\infty$ (recall (2.14)), then $\tau$ may
have a fat-tail (examples are given in Section 3). In other words, _the
internal structure of the seed-bank allows us to model fat-tailed wake-up
times without loosing the Markov property of the evolution._ $\Box$

Note that even when there is no dual, i.e., $g\in\mathcal{G}$ with $g\neq
dg_{\mathrm{FW}}$, we can still define $\tau$ by (2.40), since $\tau_{1}$ in
(2.39) is a random variable that depends only on the sequences
$(K_{m})_{m\in\mathbb{N}_{0}}$ and $(e_{m})_{m\in\mathbb{N}_{0}}$, and we can
still interpret $\tau$ as the typical wake-up time of an individual in the
population. $\blacksquare$

####  2.5 Clustering criterion

#####  2.5.1 Clustering criterion for Fisher-Wright diffusion function

In [GHO20] we showed that the system exhibits a dichotomy between
_coexistence_ (= locally multi-type equilibria) and _clustering_ (= locally
mono-type equilibria). The clustering criterion is based on the dual and
requires the notion of colour regularity. We call a law translation invariant
when it is _invariant under the group action_.

######  Definition 2.11.

[Colour regular initial measures] We say that a translation invariant initial
measure $\mu(0)$ is _colour regular_ when

(2.41) $\lim_{m\to\infty}\mathbb{E}_{\mu(0)}[y_{0,m}]\quad\text{ exists}.$

This condition is needed because, as time progresses, lineages starting from
slower and slower seed-banks become active and bring new types into the active
population. Without control on the initial states of the slow seed-banks,
there may be no convergence to equilibrium. $\blacksquare$

The key clustering criterion is the following.

######  Proposition 2.12.

[Clustering criterion] Suppose that $\mu(0)$ is translation invariant. If
$\rho=\infty$ (recall (2.14)), then additionally suppose that $\mu(0)$ is
colour regular. Let $d\in(0,\infty)$. Then the system with
$g=dg_{\mathrm{FW}}$ clusters if and only if in the dual two partition
elements coalesce with probability $1$.

The idea behind Theorem 2.12 is as follows. If in the dual two partition
elements coalesce with probability 1, then a random sample of $n$ individuals
drawn from the current population has a common ancestor some finite time
backwards in time. Since individuals inherit their type from their parent
individuals, this means that all $n$ individuals have the same type. A formal
proof was given in [GHO20, Section 4.3]. The proof is valid for any geographic
space given by a countable Abelian group endowed with the discrete topology,
of which $\Omega_{N}$ is an example.

#####  2.5.2 Clustering criterion for general diffusion function

For $g\in\mathcal{G}$ with $g\neq dg_{\mathrm{FW}}$ no dual is available and
hence we cannot use the clustering criterion in Proposition 2.12. However, as
shown in [GHO20], we can argue by duality comparison arguments (see [GHO20,
Lemma 5.5 and Lemma 6.3]) that the system evolving according to (2.20) with
$g\in\mathcal{G}$ clusters if and only if the system with $g=dg_{\mathrm{FW}}$
for some $d\in(0,\infty)$ clusters. In particular, for $g=dg_{\mathrm{FW}}$,
$d\in(0,\infty)$, whether or not the system clusters does not depend on the
resampling rate $d$.

###  3 Main results: $N<\infty$, identification of clustering regime

In this section we identify the _clustering regime_ , i.e., the range of
_parameters_ for which the clustering criterion in Proposition 2.12 is met. In
[GHO20, Section 3.2, Theorem 3.3] we derived a necessary and sufficient
condition for when clustering prevails, for any geometric space given by a
countable Abelian group endowed with the discrete topology. Recall $\chi$ in
(2.13), $\rho$ in (2.14) and $\tau$ in (2.40),. From (2.39) it follows that

(3.1)
$\mathbb{E}[\tau]=\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}}{\chi}=\frac{\rho}{\chi},$

and hence the mean wake-up time is finite if $\rho<\infty$ and infinite if
$\rho=\infty$. In Section 3.1 we look at $\rho<\infty$ and in Section 3.2 at
$\rho=\infty$. In Section 3.3 we summarise our findings and identify the
clustering regime.

####  3.1 Finite mean wake-up time

Suppose that the system evolving according to (2.20) has a translation
invariant initial measure $\mu(0)$ with density $\theta\in(0,1)$. Then [GHO20,
Theorem 3.3] says that for $\rho<\infty$ clustering occurs if and only if

(3.2) $\int_{1}^{\infty}{\rm d}t\,a^{\Omega_{N}}_{t}(0,0)=\infty.$

It is known that (3.2) holds for the hierarchical migration defined in (2.6)
if and only if [DGW05, Section 3]

(3.3) $\sum_{k\in\mathbb{N}_{0}}\frac{1}{c_{k}}=\infty.$

Hence, for $\rho<\infty$, the clustering criterion depends on the migration
kernel _only_ and the seed-bank has no effect.

$\sigma_{1}$$\tau_{1}$$\sigma_{2}$$\tau_{2}$$\sigma_{3}$$\tau_{3}$$\sigma^{\prime}_{1}$$\tau^{\prime}_{1}$$\sigma^{\prime}_{2}$$\tau^{\prime}_{2}$$\sigma^{\prime}_{3}$$\tau^{\prime}_{3}$$\ldots$
Figure 7: Successive periods during which the two random walks are active and
dormant (recall (2.39) and Fig. 6). The time lapses between successive pairs
of dotted lines represent periods of _joint activity_.

In view of Proposition 2.12, if $g=dg_{\mathrm{FW}}$, then clustering prevails
if and only if two lineages in the dual coalesce with probability 1. Recall
that two lineages in the dual can only coalesce when they are at the same site
and are both active. Since the rate of coalescence is $d\in(0,\infty)$, each
time this happens the two lineages have a positive probability to coalesce
before moving or becoming dormant. Therefore, clustering prevails if and only
if two lineages meet infinitely often while being active. This happens exactly
when (3.2) holds. The fact that the seed-bank plays no role can be seen from
the dual. Each lineage in the dual moves according to the transition kernel
$b(\cdot,\cdot)$ (recall (2.30)). Looking at the renewal process induced by
the dual process (recall Remark 2.9 and Fig. 7), we see that for $\rho<\infty$
the probability that a lineage in the dual is active at time $t$ is
approximately $\frac{1}{1+\rho}$ for large $t$. The total activity time of a
lineage up to time $t$ is therefore approximately $\frac{1}{1+\rho}t$ for
large $t$. Hence the total time the two lineages in the dual are at the same
site and are both active is approximately

(3.4) $\int_{1}^{\infty}{\rm
d}t\left(\frac{1}{1+\rho}\right)^{2}a_{2\frac{1}{1+\rho}t}(0,0),$

By Polya’s argument, if the integral in (3.4) is infinite, then two lineages
in the dual meet infinitely often while being active. After a variable
transformation, (3.4) becomes the integral in (3.2) up to a constant. (For a
formal proof of the criterion in (3.2), we refer to [GHO20].) By the above
argument, we can think of the integral in (3.2) as the _total hazard of
coalescence_ of two dual lineages. To get the result for general
$g\in\mathcal{G}$ we must invoke the duality comparison arguments mentioned in
Section 2.5.2.

In terms of the degree of the random walk (recall Remark 2.1), (3.2)
corresponds to hierarchical migration with degree $0^{-}$. The same criterion
as in (3.2) was found in [FG96] for interacting Fisher-Wright diffusions on
the hierarchical lattice without seed-bank ($\rho=0$). Hence we conclude that
for $\rho<\infty$ the seed-bank does not affect the dichotomy.

####  3.2 Infinite mean wake-up time

If $\rho=\infty$, then the seed-bank does affect the dichotomy. To apply the
criterion in [GHO20, Theorem 3.3], we assume that the system evolving
according to (2.20) has a translation-invariant initial measure $\mu(0)$ with
density $\theta\in(0,1)$ that is colour regular.

The criterion for clustering that was derived in [GHO20] for $\rho=\infty$
applies to wake-up times $\tau$ (recall (2.40)) of the form

(3.5) $\frac{\mathbb{P}(\tau\in{\rm d}t)}{{\rm
d}t}\sim\varphi(t)\,t^{-(1+\gamma)},\quad t\to\infty,\quad\gamma\in(0,1],$

with _slowly varying at infinity_. Define

(3.6)
$\hat{\varphi}(t)=\left\\{\begin{array}[]{ll}\varphi(t),&\gamma\in(0,1),\\\\[5.69046pt]
\mathbb{E}\left[\tau\wedge t\right]&\gamma=1.\end{array}\right.$

As shown in [BGT87, Section 1.3], every slowly varying function $\varphi$ may
be assumed to be infinitely differentiable and to be represented by the
integral

(3.7) $\varphi(t)=\exp\left[\int_{(\cdot)}^{t}\frac{{\rm
d}u}{u}\,\psi(u)\right]$

for some $\psi\colon[0,\infty)\to\mathbb{R}$ such that
$\lim_{u\to\infty}|\psi(u)|=0$. From (3.5) we see that $\hat{\varphi}(t)$ is
also slowly varying. If we assume that $|\psi(u)|\leq C/\log u$ for some
$C<\infty$, then the system clusters if and only if (see [GHO20, Section 3.2])

(3.8) $\int_{(\cdot)}^{\infty}{\rm
d}t\,\hat{\varphi}(t)^{-1/\gamma}\,t^{-(1-\gamma)/\gamma}\,\hat{a}_{t}(0,0)=\infty.$

Note that in (3.5) is the _tail exponent_ of the typical wake-up time $\tau$
(recall Remark 2.9) and depends on the sequences $\underline{e},\underline{K}$
in (LABEL:defKem) governing the exchange with the seed-bank. If
$g=dg_{\mathrm{FW}}$, then in view of Theorem 2.12 the criterion in (3.8)
determines whether two lineages in the dual coalesce with probability 1.

In Section 6 we will use the renewal process induced by the dual (recall
Remark 2.9) to show that (3.8) indeed gives the _total hazard of coalescence_
of two dual lineages. Therefore the integral in (3.8) is the counterpart of
(3.2). The rate of coalescence again does not affect the dichotomy: in [GHO20]
a duality comparison argument was used to show that (3.8) gives the clustering
criterion also for $g\in\mathcal{G}$ with $g\neq dg_{\mathrm{FW}}$. The effect
of the seed-bank on the dichotomy is embodied by the term
$\hat{\varphi}(t)^{-1/\gamma}\,t^{-(1-\gamma)/\gamma}$ in (3.8). The criterion
in (3.8) shows that there is a competition between migration and exchange with
the seed-bank.

For the special case where $\hat{\varphi}(t)\asymp 1$, the criterion in (3.8)
says that (recall Remark 2.1)

(3.9) $\text{clustering}\quad\Longleftrightarrow\quad\text{ either
}\delta^{-}\leq-\frac{1-\gamma}{\gamma}\text{ or
}\delta^{+}<-\frac{1-\gamma}{\gamma}.$

Condition (3.8) implies that for $\gamma\in(0,\tfrac{1}{2})$ no clustering is
possible: the typical wake-up time has such a heavy tail that with a positive
probability two dual lineages do not meet, irrespective of the migration.

######  Definition 3.1.

In what follows we will focus on the following two specific parameter regimes:

  1. $\bullet$

_Asymptotically polynomial_ , i.e.,

(3.10) $K_{k}\sim Ak^{-\alpha},\quad e_{k}\sim Bk^{-\beta},\quad c_{k}\sim
Fk^{-\phi},\quad
k\to\infty,\,A,B,F\in(0,\infty),\,\alpha,\beta,\phi\in\mathbb{R}.$

  2. $\bullet$

_Pure exponential_ , i.e.,

(3.11) $K_{k}=K^{k},\quad e_{k}=e^{k},\quad c_{k}=c^{k},\quad
k\in\mathbb{N}_{0},\,K,e,c\in(0,\infty).$

Note that both (2.7) and (2.12) are satisfied for $N\to\infty$. Also note that
an infinite seed-bank corresponds to $\alpha\in(-\infty,1]$, respectively,
$K\in[1,\infty)$. $\Box$

The scaling of the wake-up time and the migration kernel in these parameter
regimes are as follows.

######  Theorem 3.2.

[Infinite seed-bank: Scaling of wake-up time and migration kernel] Suppose
that $\rho=\infty$. Then

  * (a)

Subject to (3.10),

(3.12) $\displaystyle\gamma=1,\quad\varphi(t)\asymp(\log
t)^{-\alpha},\quad\hat{\varphi}(t)\asymp\left\\{\begin{array}[]{ll}(\log
t)^{1-\alpha},&\alpha\in(-\infty,1),\\\\[5.69046pt] \log\log
t,&\alpha=1,\end{array}\right.$ $\displaystyle a_{t}^{\Omega_{N}}(0,0)\asymp
t^{-1}\log^{\phi}t.$

  * (b)

Subject to (3.11),

(3.13)
$\displaystyle\gamma=\gamma_{N,K,e}=\frac{\log(N/Ke)}{\log(N/e)},\quad\varphi(t)\asymp
1,\quad\hat{\varphi}(t)\asymp\left\\{\begin{array}[]{ll}1,&K\in(1,\infty),\\\\[5.69046pt]
\log t,&K=1,\end{array}\right.$ $\displaystyle a_{t}^{\Omega_{N}}(0,0)\asymp
t^{-1-\delta_{N,c}},$

where

(3.14) $\delta_{N,c}=\frac{\log c}{\log(N/c)}.$

Theorem 3.2 will be proved in Section 6.

Note that, by (3.13), $\gamma_{N,K,e}=1$ for all $N$ when $K=1$, while
$\gamma_{N,K,e}<1$ for all $N$ when $K>1$, but with $\gamma_{N,K,e}\uparrow 1$
as $N\to\infty$. Also note that, subject to (3.10), (3.12) says that the
degree of the random walk is $0^{-}$ for $\phi\geq-1$ and $0^{+}$ for
$\phi<-1$, while subject to (3.11), by (3.13), the degree of the random walk
is $\delta_{N,c}^{-}$, which is $0$ for all $N$ when $c=1$, and tends to $0$
as $N\to\infty$ from above when $c>1$ and from below as $c<1$. Thus, both
(3.10) and (3.11) with $N\to\infty$ correspond to a _critically recurrent
migration_ and a _critically infinite seed-bank_.

####  3.3 Clustering regime

Summarising the above discussion, we can now identify the clustering regime
for both finite and infinite seed-banks.

######  Theorem 3.3.

[Clustering regime]

  1. (1)

If $\rho<\infty$, then clustering prevails if and only if

(3.15) $\sum_{k\in\mathbb{N}_{0}}\frac{1}{c_{k}}=\infty.$

  2. (2)

If $\rho=\infty$, then clustering prevails for $N$ large enough

     * (a)

Subject to (3.10) if and only if

(3.16) $-\phi\leq\alpha\leq 1.$

     * (b)

Subject to (3.11) if and only if

(3.17) $Kc\leq 1\leq K.$

Also Theorem 3.3 will be proved in Section 6.

Note that for $\rho<\infty$ the clustering regime follows by combining (3.2)
and (3.3), while for $\rho=\infty$ the clustering regime follows by
substituting into (3.8) the scaling of the wake-up times and the migration
kernel stated in Theorem 3.2.

######  Remark 3.4.

Note that subject to (3.10), respectively, (3.11), $\rho<\infty$ implies that
$\alpha>1$, respectively, $K<1$, and so the clustering regime is $-\phi\leq
1$, respectively, $c\leq 1$ (recall (3.15)), which are less stringent than
(3.16), respectively, (3.17). $\blacksquare$

###  4 Main results: $N\to\infty$, renormalisation and multi-scale limit

This section contains our multi-scale hierarchical limit theorems. The multi-
scale hierarchical limit theorems analyse the evolution of the block averages
defined in Definition 2.3. In Section 4.1 we recall a path topology referred
to as the _Meyer-Zheng topology_ , which we will need in part of our multi-
scale hierarchical limit theorems. In Section 4.2 we present the conceptual
ingredients needed for our theorems. In Section 4.3 we state two versions of
the hierarchical multi-scale limit (Theorems 4.2 and 4.4), and comment on how
they are related to each other. In Section 4.4 we explain how they arise from
a heuristic analysis of the SSDE in (2.20).

####  4.1 Intermezzo: Meyer-Zheng topology

Recall the block averages defined in (2.3) and their evolution equations in
(2.24). In the limit as $N\to\infty$, some of the pre-factors in (2.24)
diverge as a result of the speeding up of time. This makes the processes
increasingly more volatile: paths becomes rougher and rougher during rarer and
rarer times. Therefore we cannot work with weak convergence on path space
$C([0,\infty),E)$ w.r.t. the topology generated by the _sup-norm_ on compacts,
or on path space w.r.t. the Skorohod metric on compacts. Rather we must follow
the methodology used in [DG14, pp. 792–794] and employ the so-called _Meyer-
Zheng topology_ on _pseudopaths_ , (see [MZ84]), which is based on the
following idea. Consider functions $f\colon\,[0,\infty)\to E$, with $(E,d)$ a
Polish space, and sequences of functions $(f_{n})_{n\in\mathbb{N}}$ that are
càdlàg paths, i.e., functions in the Skorohod space $D([0,\infty),E)$. Then
$(f_{n})_{n\in\mathbb{N}}$ converges to $f$ in the Meyer-Zheng topology if and
only if

(4.1) $\lim_{n\to\infty}\int_{a}^{b}{\rm d}t\,\big{[}1\wedge
d(f(t),f_{n}(t))\big{]}=0\qquad\forall\,\,0\leq a<b<\infty.$

However, the topology induced by the metric in (4.1) does not turn
$D([0,\infty),E)$ into a _closed_ space (while in order to apply the classical
theory of weak convergence of probability laws on path space we need the path
space to be Polish).

To turn the idea from (4.1) into a manageable topology, we proceed by defining
a _space of pseudopaths_ equipped with the _Meyer-Zheng topology_. If $(E,d)$
is a Polish space and $s\mapsto v(s)$ is a measurable map from $[0,\infty)$ to
$E$, then the pseudopath is the _probability measure_ $\rho$ on
$[0,\infty)\times E$, defined by

(4.2) $\rho((a,b)\times B)=\int_{a}^{b}{\rm d}s\,{\rm
e}^{-s}\,1_{B}(v(s)),\qquad B\in\mathcal{B}(E),$

where $\mathcal{B}(E)$ is the Borel $\sigma$-algebra over $E$. Hence
$\psi_{v}$ is the image measure of ${\rm e}^{-t}{\rm d}t$ under the mapping
$t\to(t,v(t))$. In other words, we consider the _weighted occupation measure_
of the path in $E$ in order to describe paths that are _regular
representatives_ in the space of functions once we take into account (4.1).
Note that a piece-wise constant càdlàg path is uniquely determined by its
occupation measure. So is a continuous path with continuous local times. The
space of all pseudopaths is denoted by .

Since pseudopaths are measures on $[0,\infty]\times E$, convergence of
pseudopaths is defined as _weak convergence_ of _probability measures on
$[0,\infty]\times E$_. A sequence $(v_{n})_{n\in\mathbb{N}}$ of measurable
maps from $[0,\infty)\times E$ is said to converge in the Meyer-Zheng topology
to a measurable map $v$ if $\lim_{n\to\infty}\psi_{v_{n}}=\psi_{v}$, i.e.,
$\lim_{n\to\infty}\psi_{v_{n}}f=\psi_{v}f$ for all continuous bounded
functions $f$ on $[0,\infty]\times E$.

######  Remark 4.1.

[Pseudopaths] The space $\Psi$ of pseudopaths endowed with the Meyer-Zheng
topology is Polish, but the space $D([0,\infty),E)$ endowed with the Meyer-
Zheng topology is not Polish (see [MZ84, p. 372]). $\blacksquare$

In what follows, each time convergence holds in the Meyer-Zheng topology we
will say so explicitly. If no topology is mentioned, then we mean convergence
in $\mathcal{C}_{b}([0,\infty),[0,1])$. In Appendix B we collect some basic
facts about the Meyer-Zheng topology taken from [MZ84] and [Kur91].

####  4.2 Main ingredients for the hierarchical multi-scale limit

Recall the definition of $\theta_{x}$ and $\theta_{y_{m}}$ in (2.21). Define

(4.3)
$=\frac{\theta_{x}+\sum_{m=0}^{k}K_{m}\theta_{y_{m}}}{1+\sum_{m=0}^{k}K_{m}},\qquad
k\in\mathbb{N}_{0}.$

For $\rho<\infty$, and for $\rho=\infty$ under the additional assumption of
colour regularity (recall Proposition 2.12), we have

(4.4) $\lim_{k\to\infty}\vartheta_{k}=\theta\quad\text{ for some
}\theta\in[0,1].$

Define the _slowing-down constants_ ($E_{0}=1$)

(4.5) $=\frac{1}{1+\sum_{m=0}^{k-1}K_{m}},\qquad k\in\mathbb{N}_{0}.$

For $l\in\mathbb{N}_{0}$, let

(4.6) $\big{(}\theta,(y_{m,l})_{m\in\mathbb{N}_{0}}\big{)}$

be a sequence of random variables taking values in $[0,1]$, and let

(4.7) $=\Big{(}x_{l}(t),(y_{m,l}(t))_{m\in\mathbb{N}_{0}}\Big{)}_{t\geq 0}$

be the _full process_ on level $l$ evolving according to

(4.8) $\displaystyle{\rm d}x_{l}(t)=E_{l}\Bigg{[}c_{l}[\theta-x_{l}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(l)}g)(x_{l}(t))}\,{\rm
d}w(t)+K_{l}e_{l}\,[y_{l,l}(t)-x_{l}(t)]\,{\rm d}t\Bigg{]},$
$\displaystyle\begin{array}[]{lll}&y_{m,l}(t)=x_{l}(t),&0\leq m<l,\\\ &{\rm
d}y_{l,l}(t)=e_{l}\,[x_{l}(t)-y_{l,l}(t)]\,{\rm d}t,&m=l,\\\
&y_{m,l}(t)=y_{m,l},&m>l.\end{array}$

where $\mathcal{F}^{(l)}g$ is an element of $\mathcal{G}$, (recall (2.15)),
that will be defined in (4.17) below. By [YW71] the above SSDE has a unique
solution for every initial measure. For $l\in\mathbb{N}_{0}$, let

(4.9) $=\left(x^{\mathrm{eff}}_{l}(t),y^{\mathrm{eff}}_{l,l}(t)\right)_{t\geq
0}$

be the _effective process_ evolving according to

(4.10) $\displaystyle{\rm
d}x^{\mathrm{eff}}_{l}(t)=E_{l}\left[c_{l}\,[\theta-x^{\mathrm{eff}}_{l}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(l)}g)(x^{\mathrm{eff}}_{l}(t))}\,{\rm
d}w(t)+K_{l}e_{l}\,[y^{\mathrm{eff}}_{l,l}(t)-x^{\mathrm{eff}}_{l}(t)]\,{\rm
d}t\right],$ $\displaystyle{\rm
d}y^{\mathrm{eff}}_{l,l}(t)=e_{l}\,[x^{\mathrm{eff}}_{l}(t)-y^{\mathrm{eff}}_{l,l}(t)]\,{\rm
d}t.$

Comparing (4.8) with (4.10), we see that the effective process looks at the
non-trivial components of the full process.

Apart from (4.7) and (4.9), we need the following list of _ingredients_ to
formally state the multi-scale limit:

  1. 1.

For $l\in\mathbb{N}_{0}$ and $t>0$, define the _estimators_ for the finite
system by

(4.11) $\displaystyle=\frac{1}{N^{l}}\sum_{\xi\in
B_{l}}\frac{x^{\Omega_{N}}_{\xi}(t)+\sum_{m=0}^{l-1}K_{m}y^{\Omega_{N}}_{\xi,m}(t)}{1+\sum_{m=0}^{l-1}K_{m}},$
$\displaystyle=\frac{1}{N^{l}}\sum_{\xi\in B_{l}}x^{\Omega_{N}}_{\xi}(t),$
$\displaystyle=\frac{1}{N^{l}}\sum_{\xi\in
B_{l}}y^{\Omega_{N}}_{\xi,m}(t),\qquad m\in\mathbb{N}_{0},$

and put

(4.12)
$\displaystyle=\big{(}\Theta^{(l),\Omega_{N}}_{x}(t),\big{(}\Theta^{(l),\Omega_{N}}_{y_{m}}(t)\big{)}_{m\in\mathbb{N}_{0}}\big{)},$
$\displaystyle=\big{(}\bar{\Theta}^{(l),\Omega_{N}}(t),\Theta^{(l),\Omega_{N}}_{y_{l}}(t)\big{)}.$

We call $(\boldsymbol{\Theta}^{(l),\Omega_{N}}(t))_{t\geq 0}$ the _full
estimator process_ and
$(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(t))_{t\geq 0}$ the
_effective process_. Note that $\Theta_{x}^{(l),\Omega_{N}}(t)$ is the
empirical average of the active components in the $l$-block, while
$\Theta_{y_{m}}^{(l),\Omega_{N}}(t)$, is the empirical average of the
$m$-dormant components in the $l$-block, both without scaling of time. Note
that $\Theta^{(l),\Omega_{N}}_{x}(N^{l}t)=x^{\Omega_{N}}_{l}(t)$. The _level-
$l$ estimator_ $\bar{\Theta}^{(l),\Omega_{N}}(t)$ will play an important role
in our analysis. Using (2.24), we can derive the evolution equations of
$\bar{\Theta}^{(l),\Omega_{N}}(N^{l}t)$. We see that in the evolution of
$\bar{\Theta}^{(l),\Omega_{N}}(N^{l}t)$ no rates appear that tend to infinity
as $N\to\infty$. However, in the evolution of
$\Theta^{(l),\Omega_{N}}_{x}(N^{l}t)$ and
$\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{l}t)$ for $m<l$ the rates describing the
interaction between the active and the dormant population tend to infinity as
$N\to\infty$.

  2. 2.

For $l\in\mathbb{N}_{0}$, consider _time scales_ $N^{l}t_{l}$ such that

(4.13)
$\mathcal{L}\big{[}\bar{\Theta}^{(l),\Omega_{N}}(N^{l}t_{l}-L(N)N^{l-1})-\bar{\Theta}^{(l),\Omega_{N}}(N^{l}t_{l})\big{]}=\delta_{0}$

for all sequences satisfying $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$, but not for $L(N)=N$. In words, $N^{l}t_{l}$ is
the time scale on which
$(\bar{\Theta}^{(l),\Omega_{N}}(N^{l}t_{l}))_{t_{l}>0}$ is no longer a fixed
process.

  3. 3.

For $l\in\mathbb{N}_{0}$ the _invariant measure_ for the limiting evolution of
the full process on level $l$ in (4.8) is denoted by

(4.14) $,\qquad y_{l}=(y_{m,l})_{m\in\mathbb{N}_{0}}.$

(The existence of and convergence to this equilibrium will be proved in
Section 12.2.) Note that $\Gamma_{(\theta,y_{l})}^{(l)}$ depends on choice of
the rates $E_{l},c_{l},K_{l},e_{l}$ in (4.8). The _invariant measure_ of the
limiting evolution for the effective $l$-block process in (4.10) is denoted by

(4.15) .

Also $\Gamma_{\theta}^{\mathrm{eff},(l)}$ depends on the choice of the rates
$E_{l},c_{l},K_{l},e_{l}$.

  4. 4.

For $l\in\mathbb{N}_{0}$, let denote the _renormalisation transformation_
acting on $\mathcal{G}$ defined by

(4.16)
$(\mathcal{F}^{E_{l},c_{l},K_{l},e_{l}}g)(\theta)=\int_{[0,1]^{2}}g(x)\,\Gamma_{\theta}^{\mathrm{eff},(l)}({\rm
d}x),\qquad\theta\in[0,1].$

(In Section 9 we show that $\mathcal{F}g\in\mathcal{G}$.) For
$k\in\mathbb{N}_{0}$, define the _iterate_ of the renormalisation
transformation as the composition

(4.17)
$=\mathcal{F}^{E_{k-1},c_{k-1},K_{k-1},e_{k-1}}\circ\cdots\circ\mathcal{F}^{E_{0},c_{0},K_{0},e_{0}}.$

  5. 5.

For $k\in\mathbb{N}_{0}$, define the _interaction chain_ [DGV95]

(4.18)

as the _time-inhomogeneous_ Markov chain on
$[0,1]\times[0,1]^{\mathbb{N}_{0}}$ with initial state

(4.19)
$M^{k}_{-(k+1)}=(\vartheta_{k},\overbrace{\vartheta_{k},\cdots,\vartheta_{k}}^{k+1\text{
times }},\theta_{y_{k+1}},\theta_{y_{k+2}},\cdots)$

that evolves from time $-(l+1)$ to time $-l$ according to the transition
kernel on $[0,1]\times[0,1]^{\mathbb{N}_{0}}$ given by

(4.20) $Q^{[l]}(u,{\rm d}v)=\Gamma_{u}^{(l)}({\rm d}v).$

(See Fig. 9.) For $k\in\mathbb{N}_{0}$, define the _effective interaction
chain_

(4.21)

as the _time-inhomogeneous_ Markov chain on $[0,1]\times[0,1]$ with initial
state

(4.22) $M^{\mathrm{eff},k}_{-(k+1)}=(\vartheta_{k},\theta_{y_{k+1}})$

that evolves from time $-(l+1)$ to time $-l$ according to the transition
kernel on $[0,1]\times[0,1]$ given by

(4.23) $Q^{\mathrm{eff},[l]}(u,{\rm
d}v)=\Gamma^{\mathrm{eff},(l)}_{u_{x}}({\rm d}v),$

where $u_{x}$ denotes the first component of $u=(u_{x},u_{y})$.(See Fig. 8.)
We denote the components of $\left(M^{\mathrm{eff},k}_{-l}\right)$ by

(4.24)
$M^{\mathrm{eff},k}_{-l}=\big{(}M^{\mathrm{eff},k}_{-l,x},M^{\mathrm{eff},k}_{-l,y}\big{)}.$

####  4.3 Hierarchical multi-scale limit theorems

First we present and discuss the scaling of the effective process. Afterwards
we do the same for the full process.

[width=.75]oefenplaatje

Figure 8: Effective interaction chain.

#####  4.3.1 Effective process

We present one of our main theorems, the hierarchical mean-field limit for the
effective process. We will use the process and notation introduced in Section
4.2.

######  Theorem 4.2.

[Hierarchical mean-field: the effective process] Suppose that the initial
state of the hierarchical system is given by (2.21). Let $L(N)$ be such that
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$. For
$k\in\mathbb{N}$ and $t_{k},\ldots,t_{0}\in(0,\infty)$, set
$\bar{t}=N^{k}L(N)+\sum_{n=0}^{k}N^{n}t_{n}$.

  * (a)

For $k\in\mathbb{N}$,

(4.25)
$\lim_{N\to\infty}\mathcal{L}\left[\big{(}\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(\bar{t}\,)\big{)}_{l=k+1,k,\ldots,0}\right]=\mathcal{L}\left[(M^{\mathrm{eff},k}_{-l})_{-l=-(k+1),-k,\ldots,0}\right].$

  * (b)

For $k\in\mathbb{N}$,

(4.26) $\displaystyle
l>k\colon\lim\limits_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(\bar{t}+N^{k}t)\right)_{t>0}\right]=\delta_{(\vartheta_{l},\theta_{y_{l}})},$
$\displaystyle
l=k\colon\lim\limits_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(\bar{t}+N^{l}t)\right)_{t>0}\right]=\mathcal{L}\left[\left(z^{\mathrm{eff}}_{k,M^{\mathrm{eff},k}_{-(k+1),x}}(t)\right)_{t>0}\right],$
$\displaystyle
l<k\colon\lim\limits_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(\bar{t}+N^{l}t)\right)_{t>0}\right]=\mathcal{L}\left[\left(z^{\mathrm{eff}}_{l,M^{\mathrm{eff},k}_{-(l+1),x}}(t)\right)_{t>0}\right],$

where the initial laws of the limiting processes are given by (see Fig. 8)

(4.27)
$\displaystyle\mathcal{L}\left[z^{\mathrm{eff}}_{k,M^{k}_{-(k+1),x}}(0)\right]=\Gamma^{\mathrm{eff},(k)}_{M^{k}_{-(k+1),x}},$
$\displaystyle\mathcal{L}\left[z^{\mathrm{eff}}_{k,M^{k}_{-(k+1),x}}(0)\right]=\Gamma^{\mathrm{eff},(k)}_{M^{k}_{-(l+1),x}},$
$\displaystyle\Gamma^{\mathrm{eff},(l)}_{M^{k}_{-(l+1)}}=\int_{[0,1]^{2}}\cdots\int_{[0,1]^{2}}\Gamma^{\mathrm{eff},(k)}_{M^{k}_{-(k+1)}}({\rm
d}u_{k})\cdots\Gamma^{\mathrm{eff},(l+1)}_{u_{l+2}}({\rm
d}u_{l+1})\Gamma^{\mathrm{eff},(l)}_{u_{l+1}}$

Theorem 4.2 can be interpreted as follows. The statement in (a) shows that if
we look at the effective process on multiple space-time scales simultaneously,
then the joint distribution of the different block averages is the law of the
two-dimensional interaction chain defined in (4.21) and depicted in Fig. 8.
Note that the process
$(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(\bar{t}+N^{l}t))_{t>0}$
has at each level a different colour seed-bank average as second component,
which is called _the effective seed-bank_. The statement in (b) describes the
law of the path on different time scales.

  * •

On time scale $\bar{t}+N^{k}t$ the $l$-block averages with $l>k$ are not
moving, i.e.,
$(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(\bar{t}+N^{k}t))_{t>0}$
converges to the constant process taking the value
$(\vartheta_{l},\theta_{y_{l}})=\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(0)$.

  * •

On time scale $\bar{t}+N^{k}t$ the $k$-block averages have reached
equilibrium. The full $k$-block average feels a drift towards the full
$(k+1)$-block average, which is still in its initial state $\vartheta_{k}$.
Therefore migration between the $k$-blocks in the hierarchical mean-field
limit is replaced by a drift towards $\vartheta_{k}$, and the $k$-blocks
become independent. This phenomenon is called _decoupling_ (or ‘propagation of
chaos’). The resampling function for the full estimator converges to
$\mathcal{F}^{(k)}g$ (see (4.16)), the average diffusion function of the
$k$-blocks. Finally, the full $k$-block exchanges individuals with the
$k$-dormant population. Hence the $k$-dormant population is the effective
seed-bank on space-timescale $k$ Both the migration and the renormalisation
are qualitatively similar to that of the hierarchical system without seed-bank
[DG93b]. However, the seed-bank still quantitatively affects the migration and
the resampling through the slowing-down factor $E_{k}$. (In Section 4.4 we
will see how the latter arises.)

  * •

On time scale $\bar{t}+N^{l}t$ the $l$-block averages with $l<k$ are in a
_quasi-equilibrium_. The full $l$-block averages feel a drift towards the
instantaneous value of the $(l+1)$-block average at time $\bar{t}$. Therefore
also the $l$-block averages decouple. The $(l+1)$-block average is not moving
on time scale $\bar{t}+N^{l}t$, and so for $t=L(N)$ we see that the $l$-block
averages equilibrate faster than the $(l+1)$-block averages moves. The
resampling function is given by $\mathcal{F}^{(l)}g$, which is to be
interpreted as the average diffusion function of the $l$-blocks. The full
average interacts with the $l$-blocks of the $l$-dormant population, which is
the effective seed-bank on level $l$. Again the full $l$-block average feels a
slowing-down factor $E_{l}$.

Note that Theorem 4.2 only describes the limiting process of the combined
block average $\bar{\Theta}^{(l),\Omega_{N}}$ and the effective seed-bank
$\Theta_{y_{l}}^{(l),\Omega_{N}}$. It does not provide a full description of
the system, which is in Theorem 4.4 below. We will see later that Theorem 4.2
does describe all the _non-trivial_ components of the system.

######  Remark 4.3.

[Quasi equilibria] Note that Theorem 4.2 does not depend on the choice of
$t_{k},\ldots,t_{0}\in(0,\infty)$. Since at each level $0\leq l\leq k$ we
start from time $\bar{t}$, the $l$-block averages have already reached a
quasi-equilibrium.  $\blacksquare$

[width=1]interactionchain

Figure 9: Full interaction chain.

#####  4.3.2 Full process

To state our second main theorem, we will again use the process and the
notation as defined in Section 4.2.

######  Theorem 4.4.

[Hierarchical mean-field: full process] Suppose that the initial state is
given by (2.21). Let $L(N)$ be such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$. For $k\in\mathbb{N}$ and
$t_{k},\ldots,t_{0}\in(0,\infty)$, set
$\bar{t}=N^{k}L(N)+\sum_{n=0}^{k}N^{n}t_{n}$.

  * (a)

For $k\in\mathbb{N}$,

(4.28)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{(l),\Omega_{N}}\left(\bar{t}\,\right)\right)_{l=k+1,k,\ldots,0}\right]=\mathcal{L}\left[(M^{k}_{-l})_{-l=-(k+1),-k,\ldots,0}\right].$

  * (b)

For $k\in\mathbb{N}$,

(4.29) $\displaystyle
l>k\colon\lim\limits_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{k}t)\right)_{t>0}\right]=\delta_{(M^{k}_{-(k+1)})},$
$\displaystyle
l=k\colon\lim\limits_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{l}t)\right)_{t>0}\right]=\mathcal{L}\left[\left(z_{k,M^{k}_{-(k+1)}}(t)\right)_{t>0}\right],$
$\displaystyle
l<k\colon\lim\limits_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{l}t)\right)_{t>0}\right]=\mathcal{L}\left[\left(z_{l,M^{k}_{-(l+1)}}(t)\right)_{t>0}\right],$
$\displaystyle\text{in the Meyer-Zheng topology},$

where the initial laws of the limiting processes are given by (see Fig. 9)

(4.30)
$\displaystyle\mathcal{L}\left[z_{k,M^{k}_{-(k+1),x}}(0)\right]=\Gamma^{(k)}_{M^{k}_{-(k+1)}},$
$\displaystyle\mathcal{L}\left[z_{k,M^{k}_{-(k+1),x}}(0)\right]=\Gamma^{(l)}_{M^{k}_{-(l+1)}},$
$\displaystyle\Gamma^{(l)}_{M^{k}_{-(l+1)}}=\int_{\mathfrak{s}}\cdots\int_{\mathfrak{s}}\int_{\mathfrak{s}}\Gamma^{(k)}_{M^{k}_{-(k+1)}}({\rm
d}u_{k})\Gamma^{(k-1)}_{u_{k}}({\rm d}u_{k-1})\cdots\Gamma^{(l+1)}_{l+2}({\rm
d}u_{1})\Gamma^{(l)}_{u_{l+1}}$

######  Remark 4.5.

[convergence in the Meyer-Zheng topology] Note that Theorem 4.4(b) is stated
in the Meyer-Zheng topology. This topology is needed because at time-scales
$N^{l}t$ rates occur in(2.24) that tend to infinity as $N\to\infty$. In
Section 4.4 we define the Meyer-Zheng topology and explain why it is neeeded.
$\blacksquare$

The statement in (a) shows that if we look at multiple space-time scales
simultaneously, then the joint distribution of the different block averages
behaves like the infinite-dimensional interaction chain defined in (4.18). The
statement in (b) describes the law of the path on different times scales.

  * •

On time scale $N^{k}t$, the $l$-block averages with $l>k$ are not moving,
i.e., $(\boldsymbol{\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{k}t))_{t>0}$ is a
constant process. However, there is a difference between seed-banks with
colour $m>k$ and seed-banks with colour $0\leq m\leq k$ in the way they
interact with the active population. For $m>k$, even the $m$-dormant single
colonies have not yet moved at time $\bar{t}+N^{k}t$, and hence are still in
their initial states, with expectations $(\theta_{y_{m}})_{m=l+1}^{\infty}$.
Therefore, also the $l$-block averages of $m$-dormant populations are still in
their initial states, with expectations $(\theta_{y_{m}})_{m=l+1}^{\infty}$.
For $l\leq k$ the $m$-dormant single colonies with $m\leq k$ at time
$\bar{t}+N^{k}t$ have already interacted with the active population. Due to
this interaction, for $l>k$ the $l$-block averages of $m$-dormant populations
with $m\leq k$ are in state $\vartheta_{k}$ instead of their initial state
$\theta_{y_{m}}$. However, on time scale $\bar{t}+N^{k}t$ $l$-block averages
of $m$-dormant populations are not moving. (In Section 4.4 we explain how the
shift from $\theta_{y_{m}}$ to $\vartheta_{k}$ occurs.) Also the $l$-block
averages of the active population are in state $\vartheta_{k}$.

  * •

On time scale $\bar{t}+N^{k}t$, the $k$-block averages have reached
equilibrium. We see that the active $k$-block average and the $k$-dormant
$k$-block average evolve together like the effective $k$-block process in
Theorem 4.2. Therefore the evolution of the active $k$-block average is slowed
down by a factor $E_{k}$, the active $k$-block feels a drift towards
$\vartheta_{k}$ (the value of the active $(k+1)$-block average at time
$\bar{t}$), resamples with diffusion function $\mathcal{F}^{(k)}g$, and
exchanges individuals with the $k$-dormant $k$-block. The $k$-dormant
$k$-block average evolves only via interaction with the active $k$-block. The
$m$-dormant $k$-block averages with colour $0\leq m<k$ equal the active
$k$-block average and hence follow their evolution. The $m$-dormant $k$-blocks
with colour $m>k$ are still in their initial states, since on time scale
$\bar{t}+N^{k}t$ even single colony seed-banks with colour $m>k$ have not yet
started to interact with the active population.

  * •

On time scale $\bar{t}+N^{l}t$, for $0\leq l<k$, the $l$-block averages are in
a quasi-equilibrium. Again, the active $l$-block and the $l$-dormant $l$-block
average, which is the effective seed-bank, behave as the effective process in
4.2. Hence, the active $l$-block average feels a drift towards the
instantaneous value of the active $(l+1)$-block average, which is given by the
first component of the interaction chain $M^{k}_{-(l+1)}$, resamples according
to the renormalised diffusion function $\mathcal{F}^{(l)}g$, and exchanges
with the $l$-block of the $l$-dormant population. The evolution of the active
$l$-block average is slowed down by a factor $E_{l}$. The $l$-block of the
$l$-dormant population exchanges individuals with the active population. The
$l$-blocks of the $m$-dormant population with colours $0\leq m<l$ follow the
active population. The states of the $m$-dormant population with colour $m>l$
are given by the corresponding components in the interaction chain
$M^{k}_{-(l+1)}$. Hence the $l$-block averages with colours $m>k$ are still in
their initial states $\theta_{y_{m}}$, because on time scale $\bar{t}+N^{l}t$
even the single dormant colonies with colour $m>k$ have not yet interacted
with the active population. However, something interesting is happening with
the colours $l<m\leq k$: they are in a state obtained on the time scale in
which they where effective, i.e., for $l<m\leq k$ the $m$-dormant $l$-block
average is in state $y_{m,m}(\bar{t}\,)$. This happens because at time
$\bar{t}$ the single colonies have already interacted with the active
population, but on time scale $N^{l}t$ they do not interact anymore with the
active population. (Also this effect will be further explained in Section
4.4.)

######  Remark 4.6.

[Comparison to the hierachical multi-scale limit without seed-bank] Comparing
Theorem 4.4 with the multi-scale limit theorems derived for the hierarchical
system without seed-bank [DG93b], [DG93a], [DGV95], we see that the seed-bank
affects the system both quantitatively and qualitatively. First, the active
population is _slowed down_ by the total size of the seed-banks it has
interacted with, represented by the slowing-down factors
$(E_{l})_{l\in\mathbb{N}_{0}}$. Second, the interaction with the _effective
seed-bank_ on each space-time scale is special to the system with seed-bank.
Still, the _decoupling_ of the active component and the _renormalisation
transformation_ for the diffusion function are similar as in the system
without seed-bank.

######  Remark 4.7.

[$k\to\infty$ limit of the interaction chain] The result in (4.26) raises the
question how the hierarchical multi-scale limit behaves for large $k$. We find
the following dichotomy:

(4.31)
$\lim_{k\to\infty}\mathcal{L}\left[(M^{k}_{-(k+1),-k,\cdots,0})\right]=\mathcal{L}\left[(M^{\infty}_{k})_{k\in\mathbb{Z}^{-}}\right],$

where in the _clustering regime_

(4.32)
$\mathcal{L}\left[(M^{\infty}_{k})_{k\in\mathbb{Z}^{-}}\right]=\theta\delta_{(1,1^{\mathbb{N}_{0}})^{\mathbb{Z}^{-}}}+(1-\theta)\delta_{(0,0^{\mathbb{N}_{0}})^{\mathbb{Z}^{-}}}$

and in the _coexistence regime_
$M^{\infty}=(M_{k}^{\infty})_{k\in\mathbb{Z}^{-}}$ is a realisation of the
unique entrance law of the interaction chain at time $-\infty$ with

(4.33) $\lim_{l\to\infty}M_{-l}^{\infty}=(\theta,\theta).$

In the latter case, $M^{\infty}$ corresponds to the equilibrium vector of
block averages around site $0$, whose law agrees with that of the equilibrium
block averages for the mean-field model after we take the limit $N\to\infty$
(see [DGV95, Proposition 6.2 and 6.3]). $\blacksquare$

######  Remark 4.8.

[Interaction field] Theorem 4.4 looks at the tower of block averages over a
fixed site, namely, $0$. In order to study the cluster formation in the
clustering regime or the equilibria in the coexistence regime, we must analyse
the dependence structure between the towers of block averages over _different_
sites. We can show that, in the limit as $N\to\infty$, an interacting random
field emerges, indexed by a tree with countably many edges coming out of every
site at every level. This random field has the property that the averages over
any two points $\eta,\eta^{\prime}$ with $d(\eta,\eta^{\prime})=l$, follow a
single interaction chain in equilibrium from $k+1$ until $l$ (or from
$-\infty$ until $l$ in the entrance law) and, conditional on the state in $l$,
evolve independently as the interaction chain beyond $l$. This corresponds to
what is called _propagation of chaos_ of the $(l-1)$-block averages given the
$l$-block average. For the model without seed-bank such results are described
in [DGV95, Section 0(e)]. Using our results for the model with seed-bank
above, we can in principle follow an analogous line of argument. We refrain
from spelling out the details. $\blacksquare$

####  4.4 Heuristics behind the multi-scale limit

The proofs of Theorems 4.2 and 4.4 written out in Sections 7–12, are long and
technical. In order to help the reader appreciate these proofs, we provide the
heuristics in Sections 4.4.1–4.4.3.

#####  4.4.1 Evolution of the block-averages

Recall the block averages introduced in Definition 2.3 and their evolution
defined in (2.24). In the limit as $N\to\infty$, we heuristically obtain from
the SSDE in (2.24) the following results for the $k$-block process

(4.34)
$({x}_{k}^{\Omega_{N}}(t),({y}_{m,k}^{\Omega_{N}}(t))_{m\in\mathbb{N}_{0}})_{t\geq
0}.$

  * •

Migration. Recall that the migration is captured by the first term of the
first equation in (2.24), i.e., the first term of the evolution of the active
part of the population. Letting $N\to\infty$, we see that in the sum over $l$
only the term $l=1$ contributes. Hence we expect that the effective migration
felt by the active $k$-block average is towards ${x}_{k+1}^{\Omega_{N}}(0)$,
the initial state of the active $(k+1)$-block average. Note that the migration
term in (2.24) can be written as

(4.35)
$\sum_{l\in\mathbb{N}}\frac{c_{k+l-1}}{N^{l-1}}\big{[}{x}_{k+l}^{\Omega_{N}}(N^{-l}t)-{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t=\sum_{l\in\mathbb{N}}\frac{c_{k+l-1}}{N^{l-1}}\left[\frac{1}{N^{l}}\sum_{k=0}^{N^{l}-1}{x}_{k}^{\Omega_{N}}(t)-{x}_{k}^{\Omega_{N}}(t)\right]\,{\rm
d}t.$

The drift towards the $(k+1)$-block average is therefore also a drift towards
the current average of the $k$-blocks in the $(k+1)$-block. In the limit as
$N\to\infty$, the latter can be approximated by $\mathbb{E}[x_{k}(t)]$.
Effectively, as $N\to\infty$, the $k$-blocks become independent given the the
value of ${x}_{k+1}^{\Omega_{N}}(0)$, i.e., there is _decoupling_.

  * •

Resampling. Recall that the diffusion term in the evolution equation of the
active population represents the resampling. Therefore we see that the active
$k$-block resamples at a rate that is the average resampling rate over the
$k$-block. For $k=1$, the resampling rate of the $1$-block is the average of
the resampling ratse of the single colonies. Therefore, in the limit
$N\to\infty$, due to the decoupling described above, we expect that the
resampling rate for the $1$ block is given by $\mathbb{E}[g]$, where the
expectation is w.r.t. the quasi-equilibrium of the single colonies. This
expectation is exaclty the renormalised diffusion function $\mathcal{F}g$ (see
(4.16)). For the $k$-block, we may interpret the diffusion function to be the
average of the diffusion function for the $(k-1)$-blocks. By “induction” we
assume that the $(k-1)$ blocks resample at rate $\mathcal{F}^{(k-1)}g$. Hence,
due to the decoupling of the $(k-1)$-blocks as $N\to\infty$, we expect the
resampling rate for the $k$ blocks to equal
$\mathbb{E}[\mathcal{F}^{(k-1)}g]$, where the expectation is w.r.t. the quasi-
equlibrium of the $(k-1)$-blocks. This yields another iteration of the
renormalisation transformation (see (4.17)). Hence, we expect the diffusion
function for the $k$-blocks to converge to $\mathcal{F}^{(k)}g$.

  * •

Exchange with the seed-bank. Recall that the last term of the first equation
in (2.24) and the second equation in (2.24) together describe the exchange of
the active $k$-block with the $m$-dormant $k$-block. To describe the limiting
behaviour as $N\to\infty$, we distinguish three cases: $0\leq m<k$, $m=k$,
$m>k$.

    * –

If $0\leq m<k$, then we see that the rate of exchange between the active
$k$-block and the $m$-dormant $k$-block tends to infinity as $N\to\infty$. We
therefore expect them to equalise, i.e.,

(4.36)
$\lim_{N\to\infty}\mathcal{L}\left[\left({x}_{k}^{\Omega_{N}}(t)-{y}_{m,k}^{\Omega_{N}}(t)\right)_{t>0}\right]=\delta_{0},$

where $0$ should be interpreted as the process equal to $0$, $(0)_{t>0}$.
Hence we see that $m$-dormant $k$-block follows the active $k$-block
immediately. (To formalise this fact, we need the _Meyer-Zheng topology_
[MZ84].)

    * –

If $m=k$, then there is a non-trivial exchange between the active $k$-block
and the $k$-dormant $k$-block.

    * –

If $m>k$, then the exchange rate between the active $k$-block and the
$m$-dormant $k$-block tends to zero as $N\to\infty$.

Thus, only the $k$-dormant $k$-block has a non-trivial interaction with the
active $k$-block. We express this by saying that on space-time scale $k$ the
$k$-dormant population plays the role of the _effective seed-bank_.

#####  4.4.2 Limiting evolution of the block-averages

To determine the limiting evolution of the full block-averages process, we
first have a look at the limiting evolution of the effective process.

###### The effective process.

To determine the limit as $N\to\infty$ of (2.24), we need to get rid of the
diverging rates. Instead of only looking at the $k$-block process
$({x}_{k}^{\Omega_{N}}(t),({y}_{m,k}^{\Omega_{N}}(t))_{m\in\mathbb{N}_{0}})_{t\geq
0}$, which evolves according to (2.24), we look at the _effective $k$-block
process_ defined as

(4.37)
$\left(\bar{x}_{k}^{\Omega_{N}}(t),y_{k,k}^{\Omega_{N}}(t)\right)_{t\geq 0},$

where we abbreviate

(4.38)
$=\frac{x_{k}^{\Omega_{N}}(t)+\sum_{m=0}^{k-1}K_{m}y_{m,k}^{\Omega_{N}}(t)}{1+\sum_{m=0}^{k-1}K_{m}}.$

By (4.36) and the heuristic discussion given above, the process in (4.37)
equals $(x_{k}^{\Omega_{N}}(t),y_{k,k}^{\Omega_{N}}(t))_{t\geq 0}$ in the
limit as $N\to\infty$, i.e., it describes the joint distribution of the active
$k$-block and the effective dormant $k$-block, which is the $k$-dormant
$k$-block. Using (2.24), we see that the process in (4.37) evolves according
to the SSDE

(4.39) $\displaystyle{\rm d}\bar{x}_{k}^{\Omega_{N}}(t)$
$\displaystyle=E_{k}\sum_{l\in\mathbb{N}}\frac{c_{k+l-1}}{N^{l-1}}\big{[}{x}_{k+l}^{\Omega_{N}}(N^{-l}t)-{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t$ $\displaystyle\qquad+E_{k}\sqrt{\frac{1}{N^{k}}\sum_{\xi\in
B_{k}(0)}g(x_{\xi}(N^{k}t))}\,\,{\rm d}w_{k}(t)$
$\displaystyle\qquad+E_{k}\sum_{m=k}^{\infty}N^{k}\frac{K_{m}e_{m}}{N^{m}}\big{[}{y}_{m,k}^{\Omega_{N}}(t)-{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{\rm d}{y}_{k,k}^{\Omega_{N}}(t)$
$\displaystyle=e_{k}\big{[}{x}_{k}^{\Omega_{N}}(t)-{y}_{k,k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t.$

In (4.39) no infinite rates appear anymore. In the limit as $N\to\infty$, by
(4.36) we can approximate

(4.40) $x_{k}^{\Omega_{N}}(t)\approx y_{m,k}^{\Omega_{N}}(t),\qquad 0\leq
m<k,$

such that

(4.41) $x_{k}^{\Omega_{N}}(t)\approx\bar{x}_{k}^{\Omega_{N}}(t).$

We can therefore approximate (4.39) by

(4.42) $\displaystyle{\rm d}\bar{x}_{k}^{\Omega_{N}}(t)$
$\displaystyle=E_{k}\sum_{l\in\mathbb{N}}\frac{c_{k+l-1}}{N^{l-1}}\big{[}\bar{x}_{k+l}^{\Omega_{N}}(N^{-l}t)-\bar{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t$ $\displaystyle\qquad+E_{k}\sqrt{\frac{1}{N^{k}}\sum_{\xi\in
B_{k}(0)}g\big{(}\bar{x}_{\xi}(N^{k}t)\big{)}}\,\,{\rm d}w_{k}(t)$
$\displaystyle\qquad+E_{k}\sum_{m=k}^{\infty}N^{k}\frac{K_{m}e_{m}}{N^{m}}\big{[}{y}_{m,k}^{\Omega_{N}}(t)-\bar{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{\rm d}{y}_{k,k}^{\Omega_{N}}(t)$
$\displaystyle=e_{k}\big{[}\bar{x}_{k}^{\Omega_{N}}(t)-{y}_{k,k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t.$

Hence, in the limit as $N\to\infty$, the process in (4.37) becomes autonomous.
Moreover, assuming that
$\lim_{N\to\infty}\bar{x}_{l+1}^{\Omega_{N}}(0)=\vartheta_{l}$, we see that
(4.42) approaches the effective process defined in (4.10), with

(4.43) $\theta=\vartheta_{l},\quad E=E_{k},\quad c=c_{k},\quad e=e_{k},\quad
K=K_{k},\quad g=\mathcal{F}^{(k)}g.$

In particular, we see that the _slowing-down constant_ $E_{k}$ arises because
the active population is the only part of the first component of (4.37). Note
that therefore only a part, the active part, from the first component
migrates, resamples and exchanges with the seed-bank. Due to the infinite
rates, the active population “drags along all the fast seed-banks with total
size $\sum_{m=0}^{k-1}K_{m}$”. This causes the slowing down factors $E_{k}$.

Since there are no infinite rates in the evolution of the effective process,
we can use the classical path space topology. This allows us in the proof in
Sections 7–12 to build on techniques developed for the hierarchical mean-field
model without seed-bank in [DG93a], [DGV95]. It turns out that the _effective
process_ is very useful in our analysis.

###### From the effective process to the full process.

For large $N$, by (4.40) and (4.41), the evolution of our original process
$({x}_{k}^{\Omega_{N}}(t),({y}_{m,k}^{\Omega_{N}}(t))_{m\in\mathbb{N}_{0}})_{t\geq
0}$ can be approximated by

(4.44) $\displaystyle{\rm d}\bar{x}_{k}^{\Omega_{N}}(t)$ $\displaystyle\approx
E_{k}\sum_{l\in\mathbb{N}}\frac{c_{k+l-1}}{N^{l-1}}\big{[}\bar{x}_{k+l}^{\Omega_{N}}(N^{-l}t)-\bar{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t$ $\displaystyle\qquad+E_{k}\sqrt{\frac{1}{N^{k}}\sum_{\xi\in
B_{k}(0)}g(\bar{x}_{\xi}(N^{k}t))}\,\,{\rm d}w_{k}(t)$
$\displaystyle\qquad+E_{k}\sum_{m=k}^{\infty}N^{k}\frac{K_{m}e_{m}}{N^{m}}\big{[}{y}_{m,k}^{\Omega_{N}}(t)-\bar{x}_{k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{y}_{k,k}^{\Omega_{N}}(t)$
$\displaystyle=\bar{x}_{k}^{\Omega_{N}}(t),$ $\displaystyle{\rm
d}{y}_{k,k}^{\Omega_{N}}(t)$
$\displaystyle=e_{k}\big{[}\bar{x}_{k}^{\Omega_{N}}(t)-{y}_{k,k}^{\Omega_{N}}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{y}_{k,k}^{\Omega_{N}}(t)$
$\displaystyle={y}_{m,k}^{\Omega_{N}}(0).$

By the ergodic theorem for exchangeable measures, we can assume that
$\lim_{N\to\infty}\bar{x}_{k+1}^{\Omega_{N}}(0)=\vartheta_{k}\,a.s.$. We
expect that (4.44) approaches (4.8) with

(4.45) $\theta=\vartheta_{k},\quad E=E_{k},\quad c=c_{k},\quad e=e_{k},\quad
K=K_{k},\quad g=\mathcal{F}^{(k)}g.$

To prove that

(4.46)
$\lim_{N\to\infty}\mathcal{L}[{y}_{m,k}^{\Omega_{N}}(t)]=\lim_{N\to\infty}\mathcal{L}[\bar{x}_{k}^{\Omega_{N}}(t)],\qquad
0\leq m<k-1,$

we need _the Meyer-Zheng topology_ explained in Section 4.1. In the proof in
Sections 8–12 we show how the above approximations can be made rigorous.

###### Conserved quantities.

Note that, by (2.24) and (4.35), for each $k\in\mathbb{N}_{0}$

(4.47)
$\mathbb{E}\left[\frac{x^{\Omega_{N}}_{k}(t)+\sum_{m\in\mathbb{N}_{0}}K_{m}y^{\Omega_{N}}_{k}(t)}{1+\sum_{m\in\mathbb{N}_{0}}K_{m}}\right]=\theta,\qquad
t\geq 0,$

is a conserved quantity. For each $k\in\mathbb{N}$ we obtain that for $l\geq
k$

(4.48)
$\lim_{N\to\infty}\mathbb{E}\left[\frac{x^{\Omega_{N}}_{k}(t)+\sum_{m=0}^{l}K_{m}y^{\Omega_{N}}_{k}(t)}{1+\sum_{m=0}^{l}K_{m}}\right]=\vartheta_{l},$

is a conserved quantity. For the effective process, (4.42) implies that

(4.49) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]$
$\displaystyle=E_{k}K_{k}e_{k}\,\Big{(}\mathbb{E}[y_{k,k}^{\Omega_{N}}(t)]-\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]\Big{)},$
$\displaystyle\frac{{\rm d}}{{\rm d}t}\mathbb{E}[y_{k,k}^{\Omega_{N}}(t)]$
$\displaystyle=e_{k}\,\Big{(}\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]-\mathbb{E}[y_{k,k}^{\Omega_{N}}(t)]\Big{)}.$

Recall that

(4.50)
$\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(0)]=\mathbb{E}\left[\frac{x_{k}^{\Omega_{N}}(0)+\sum_{m=0}^{k-1}K_{m}y_{m,k}^{\Omega_{N}}(0)}{1+\sum_{m=0}^{k-1}K_{m}}\right]=\vartheta_{k-1},\qquad\mathbb{E}[y_{k,k}^{\Omega_{N}}(0)]=\theta_{y_{k}}.$

Therefore we can solve (4.49) explicitly as

(4.51) $\displaystyle\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]$
$\displaystyle=\vartheta_{k}+\frac{E_{k}K_{k}}{1+E_{k}K_{k}}(\vartheta_{k-1}-\theta_{y_{k}})\,{\rm
e}^{-(E_{k}K_{k}+1)e_{k}t},$
$\displaystyle\mathbb{E}[y_{k,k}^{\Omega_{N}}(t)]$
$\displaystyle=\vartheta_{k}-\frac{1}{1+E_{k}K_{k}}(\vartheta_{k-1}-\theta_{y_{k}})\,{\rm
e}^{-(E_{k}K_{k}+1)e_{k}t}.$

The above computation shows what happens to
$\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]$ if we move one space-time scale up
in the hierarchy, namely, a new seed-bank starts interacting with the active
population. This causes that $\vartheta_{k-1}$ is pulled a bit towards
$\theta_{y_{k}}$, so that also $\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]$
changes a bit. Each new seed-bank that opens up changes the expectation of the
active population, which results in the sequence
$(\theta_{x},\vartheta_{0},\vartheta_{1},\vartheta_{2},\ldots)$ for the
expectation of the active population on space-time scales
$\\{0,1,2,3,\cdots\\}$. From (4.35) we see that the drift of
$\bar{x}_{k}^{\Omega_{N}}(t)$ is towards

(4.52)
$\bar{x}_{k+1}^{\Omega_{N}}(N^{-1}t)=\frac{1}{N^{k}}\sum_{k=0}^{N^{k}-1}{x}_{k}^{\Omega_{N}}(t)\approx\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)],$

where the last approximation can be made because the $k$-blocks decouple.
Hence, in the limit as $N\to\infty$, once the $k$-blocks are in a quasi-
equilibrium we can replace the drift towards
$\bar{x}_{k+1}^{\Omega_{N}}(N^{-1}t)$ by a drift towards
$\mathbb{E}[\bar{x}_{k}^{\Omega_{N}}(t)]=\vartheta_{k}$.

###### Shifting averages.

Recall the full estimator process
$(\mathbf{\Theta}^{(l),\Omega_{N}}(t))_{t>0}$ defined in (4.12). Equation 2.23
implies that the evolution of the estimator process is given by

(4.53) $\displaystyle{\rm d}\Theta^{(l),\Omega_{N}}_{x}(t)$
$\displaystyle=\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1}}[\Theta^{(n),\Omega_{N}}_{x}(t)-\Theta^{(l)}_{x}(t)]\,{\rm
d}t$ $\displaystyle\qquad+\sqrt{\frac{1}{N^{2l}}\sum_{\xi\in
B_{l}}g\big{(}x_{\xi}(t)\big{)}}\,{\rm d}w(t)$
$\displaystyle\qquad+\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}e_{m}}{N^{m}}[\Theta^{(l),\Omega_{N}}_{y_{m}}(t)-\Theta^{(1),\Omega_{N}}_{x}(t)]\,{\rm
d}t,$ $\displaystyle{\rm d}\Theta^{(l),\Omega_{N}}_{y_{m}}(t)$
$\displaystyle=\frac{e_{m}}{N^{m}}[\Theta^{(l),\Omega_{N}}_{x}(t)-\Theta^{(l),\Omega_{N}}_{y_{m}}(t)]\,{\rm
d}t,\,m\in\mathbb{N}_{0}.$

Looking at the estimator process $(\mathbf{\Theta}^{(l),\Omega_{N}}(t))_{t>0}$
on time scale $N^{k}t$, we see that

(4.54) $\displaystyle{\rm d}\Theta_{x}^{(l),\Omega_{N}}(N^{k}t)$
$\displaystyle=\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1-k}}\left[\Theta_{x}^{(n),\Omega_{N}}(N^{k}t)-\Theta_{x}^{(l),\Omega_{N}}(N^{k}t)\right]\,{\rm
d}t$ $\displaystyle\qquad+\sqrt{\frac{N^{k}}{N^{2l}}\sum_{\xi\in
B_{l}}g\big{(}x_{\xi}(N^{k}t)\big{)}}\,{\rm d}w(t)$
$\displaystyle\qquad+\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}e_{m}}{N^{m-k}}\left[\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)-\Theta^{(l),\Omega_{N}}_{x}(N^{k}t)\right]\,{\rm
d}t,$ $\displaystyle{\rm d}\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)$
$\displaystyle=\frac{e_{m}}{N^{m-k}}\left[\Theta^{(l),\Omega_{N}}_{x}(N^{k}t)-\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)\right]\,{\rm
d}t,\qquad m\in\mathbb{N}_{0}.$

From (4.54) we get that, on time scale $N^{k}t$, for all $l\geq k+1$,

(4.55)
$\Theta^{(l),\Omega_{N}}(N^{k}t)=\frac{\Theta^{(l),\Omega_{N}}_{x}(N^{k}t)+\sum_{m=0}^{l-1}K_{m}\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)}{1+\sum_{m=0}^{l-1}K_{m}},\qquad
t\geq 0,$

is a conserved quantity in the limit as $N\to\infty$, and for $t\geq 0$,

(4.56)
$\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}(N^{k}t)=\frac{\theta_{x}+\sum_{m=0}^{l-1}K_{m}\theta_{y_{m}}}{1+\sum_{m=0}^{l-1}K_{m}}=\vartheta_{l},\text{
in probability}.$

For $m>k$, also $\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)$ is a conserved
quantity, and

(4.57)
$\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)=\theta_{y_{m}},\qquad
t\geq 0.$

However, for $l\geq k+1$, $\Theta^{(l),\Omega_{N}}_{x}(N^{k}t)$ and
$(\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t))_{m=0}^{k}$ are not conserved
quantities in the limit as $N\to\infty$. Note that from 2.24 we heuristically
see that the full $l$-block estimator process
$({\bf\Theta}^{(l),\Omega_{N}}(N^{k}t))_{t\geq 0}$ with $l>k$ converges to the
process

(4.58)
$\left(\Theta^{(l)}_{x}(t),\left(\Theta^{(l)}_{y_{m}}(t)\right)_{m\in\mathbb{N}_{0}}\right)_{t>0},$

which evolves according to

(4.59) $\displaystyle{\rm
d}\Theta^{(l)}_{x}(t)=E_{k-1}K_{k}e_{k}[\Theta^{(l)}_{y_{k}}(t)-\Theta^{(l),\Omega_{N}}_{x}(t)]\,{\rm
d}t,$
$\displaystyle\begin{array}[]{llll}\Theta^{(l)}_{y_{m}}(t)&=&\Theta^{(l)}_{x}(t),&m<k,\\\
{\rm
d}\Theta^{(l)}_{y_{k}}(t)&=&e_{k}[\Theta^{(l)}_{x}(t)-\Theta^{(l)}_{y_{k}}(t)]\,{\rm
d}t,&m=k,\\\ \Theta^{(l)}_{y_{m}}(t)&=&\theta_{y_{m}},&m>k.\end{array}$

This system can be explicitly solved as

(4.60)
$\displaystyle\Theta^{(l)}_{x}(t)=\frac{\Theta^{(l)}_{x}(0)+E_{k-1}K_{k}\Theta^{(l)}_{y_{k}}(0)}{1+E_{k-1}K_{k}}$
$\displaystyle\qquad\qquad\qquad+\frac{E_{k-1}K_{k}}{1+E_{k-1}K_{k}}[\Theta^{(l),\Omega_{N}}_{x}(0)-\Theta^{(l)}_{y_{k}}(0)]\,{\rm
e}^{-(E_{k-1}K_{k}e_{k}+e_{k})t},$
$\displaystyle\begin{array}[]{llll}\Theta^{(l)}_{y_{m}}(t)&=&\Theta^{(l)}_{x}(t),&m<k,\\\\[5.69046pt]
\Theta^{(l)}_{y_{k}}(t)&=&\frac{\Theta^{(l)}_{x}(0)+E_{k-1}K_{k}\Theta^{(l)}_{y_{k}}(0)}{1+E_{k-1}K_{k}}\\\\[5.69046pt]
&&\qquad-\frac{1}{1+E_{k-1}K_{k}}[\Theta^{(l),\Omega_{N}}_{x}(0)-\Theta^{(l)}_{y_{k}}(0)]\,{\rm
e}^{-(E_{k-1}K_{k}e_{k}+e_{k})t},&m=k,\\\
\Theta^{(l)}_{y_{m}}(t)&=&\theta_{y_{m}},&m>k.\end{array}$

The latter shows that, each time we enter a new space-time scale, all the
large active blocks interact with the large effective dormant blocks until
they equalise. Thus, on each space-time scale, all the active $l$-blocks and
the dormant $l$-blocks of colour $m\leq l$ move for a short period of time. As
a consequence, the value of $\Theta_{x}^{(l)}(0)$ depends on the scaling we
choose. To illustrate this, we note that

(4.61) $\displaystyle\lim_{N\to\infty}\Theta_{x}^{(l),\Omega_{N}}(0)$
$\displaystyle=\theta_{x},$
$\displaystyle\lim_{N\to\infty}\Theta_{x}^{(l),\Omega_{N}}(L(N)+t)$
$\displaystyle=\vartheta_{0},$
$\displaystyle\lim_{N\to\infty}\Theta_{x}^{(l),\Omega_{N}}((L(N)N^{n}+N^{n}t))$
$\displaystyle=\vartheta_{n},\quad 0\leq n\leq k.$

However, the scaling $N^{k}t$ would imply that

(4.62) $\displaystyle\lim_{N\to\infty}\Theta_{x}^{(l),\Omega_{N}}(0)$
$\displaystyle=\theta_{x},$
$\displaystyle\lim_{N\to\infty}\Theta_{x}^{(l),\Omega_{N}}\left(N^{k}\frac{L(N)+t}{N^{k}}\right)$
$\displaystyle=\vartheta_{0},$
$\displaystyle\lim_{N\to\infty}\Theta_{x}^{(l),\Omega_{N}}\left(N^{k}\frac{L(N)N^{n}+N^{n}t}{N^{k}}\right)$
$\displaystyle=\vartheta_{n},\quad 0\leq n\leq k.$

Hence, if $(t(N))_{N\in\mathbb{N}}$ is a sequence such that
$\lim_{N\to\infty}N^{k}t(N)=0$, then the value of the limit

(4.63) $\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}_{x}(N^{k}t(N))$

depends on $(t(N))_{N\in\mathbb{N}}$. Moreover, to obtain a limiting process
for $({\bf\Theta}^{(l),\Omega_{N}}(N^{k}t))_{t\geq 0}$ we need convergence
also at time $0$, while it is not clear what $N^{k}t\downarrow 0$ means. To
circumvent these subtleties, we look at the process at times $t>0$ and use as
starting time $\bar{t}$ defined in Theorem 4.2.

From (4.60) it follows that, for $l\geq k$,

(4.64) $\displaystyle\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}_{x}(\bar{t}\,)$
$\displaystyle=\vartheta_{k},\text{ in probability},$
$\displaystyle\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}\,)$
$\displaystyle=\vartheta_{k},\quad m\leq k,\text{ in probability},$
$\displaystyle\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}\,)$
$\displaystyle=\theta_{y_{m}},\quad m>k\text{ in probability}.$

Note that the shifting of averages mentioned earlier is closely related to the
conserved quantities discussed in Section 4.4.2 because, for large $N$,

(4.65)
$\lim_{N\to\infty}\Theta^{(l),\Omega_{N}}_{x}(\bar{t}\,)\approx\mathbb{E}[x_{k}^{\Omega_{N}}],$

where the expectation is taken in the quasi-equilibrium the $k$-blocks have
attained after scaling with time $\bar{t}$.

#####  4.4.3 Formation of the interaction chain

In Section 4.4.1 we saw how subsequent space-time scales are connected via the
migration term. In this section we show how the interaction chain arises from
this connection. We first show how the effective interaction chain arises for
the effective process. Then we show how the full interaction chain is formed,
by studying the slow seed-banks.

###### Connections between different space-time scales

Let $\bar{t}$ be as in Theorem 4.2. From (4.53) it follows that the process
$({\bf\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{l}t))_{t>0}$ evolves according to

(4.66) $\displaystyle{\rm d}\Theta^{(l),\Omega_{N}}_{x}(\bar{t}+N^{l}t)$
$\displaystyle=\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1-l}}[\Theta^{(n),\Omega_{N}}_{x}(\bar{t}+N^{l}t)-\Theta^{(l)}_{x}(\bar{t}+N^{l}t)]\,{\rm
d}t$ $\displaystyle\qquad+\sqrt{\frac{1}{N^{l}}\sum_{\xi\in
B_{l}}g\big{(}x_{\xi}(\bar{t}+N^{l}t)\big{)}}\,{\rm d}w(t)$
$\displaystyle\qquad+\sum_{m\in\mathbb{N}_{0}}\frac{K_{m}e_{m}}{N^{m-l}}[\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}+N^{l}t)-\Theta^{(1),\Omega_{N}}_{x}(\bar{t}+N^{l}t)]\,{\rm
d}t,$ $\displaystyle{\rm d}\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}+N^{l}t)$
$\displaystyle=\frac{e_{m}}{N^{m-l}}[\Theta^{(l),\Omega_{N}}_{x}(\bar{t}+N^{l}t)-\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}+N^{l}t)]\,{\rm
d}t,\qquad m\in\mathbb{N}_{0}.$

Therefore, in the limit as $N\to\infty$, the active population
$\Theta^{(l),\Omega_{N}}_{x}(\bar{t}+N^{l}t)$ feels a drift towards the
$(l+1)$-block average $\Theta^{(l+1),\Omega_{N}}_{x}(\bar{t}+N^{l+1}t)$. If
$l=k$, then

(4.67)
$\lim_{N\to\infty}\mathcal{L}[{\bf\Theta}^{(k+1),\Omega_{N}}(\bar{t}+N^{k}t)]=\lim_{N\to\infty}\mathcal{L}[{\bf\Theta}^{(k+1),\Omega_{N}}(\bar{t})],$

since the $(k+1)$-block has not yet started to move at time $\bar{t}+N^{k}t$.
From (4.64) it follows that

(4.68)
$\lim_{N\to\infty}\Theta^{(k+1),\Omega_{N}}_{x}(\bar{t}+N^{k}t)=\vartheta_{k}\,in\,probability.$

Therefore the drift of the active population
$\Theta^{(l),\Omega_{N}}_{x}(\bar{t}+N^{l}t)$ is towards $\vartheta_{k}$.
Since $\bar{t}>L(N)N^{k}$, the process
${\bf\Theta}^{(k),\Omega_{N}}(\bar{t}+N^{k}t)$ has, in the limit $N\to\infty$,
already reached its equilibrium, which is denoted by
$\Gamma_{\bar{\vartheta_{k}}}$, where

(4.69)
$\bar{\vartheta_{k}}=(\vartheta_{k},\overbrace{\vartheta_{k},\cdots,\vartheta_{k}}^{k+1\text{
times }},\theta_{y_{k+1}},\theta_{y_{k+2}},\cdots),$

so that we recognise $(\bar{\vartheta}_{k})=M^{k}_{-(k+1)}$. From (4.64) with
$l=k+1$ we see that $(\bar{\vartheta}_{k})=M^{k}_{-(k+1)}$ represents the
state of ${\bf\Theta}^{(k+1),\Omega_{N}}(\bar{t})$.

If we look on time scale $\bar{t}+N^{k-1}t$, then we see that the active
$(k-1)$-block averages feels a drift towards the active $k$-block average. The
active $k$-block does not move on time scale $N^{k-1}$, but it has already
moved at time $\bar{t}$. At time $\bar{t}$ the active $k$-block has even
reached its quasi-equilibrium, given by $\Gamma^{(k)}_{\vartheta_{k}}$. Thus,
the drift of the active $(k-1)$-block average is towards the instantaneous
state of the active $k$-block average, which has distribution
$\Gamma^{(k)}_{\bar{\vartheta_{k}}}$. This explains the first step in the
interaction chain.

For $0\leq l<k$, the active $l$-block average feels a drift towards the
$(l+1)$-block average. The latter does not evolve on time scale $N^{l}t$, but
it has already moved at time $\bar{t}$. Therefore it is no longer in its
initial state, but in a quasi-equilibrium $\Gamma^{(l+2)}_{u}$, where $u$ is
the value of the active $(l+1)$-block averages determined via the interaction
chain, recall Figure 9. This explains how the different space-time scales are
connected via the active block averages. The states of the different seed-bank
averages is a little bit more complicated. Below we give a very short
heuristic explanation of the different seed-banks in the interaction chain.

For the effective process
$(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}}(t))_{t>0}$ instead of the
full process $(\boldsymbol{\Theta}^{(l),\Omega_{N}}(t))_{t>0}$, we can
consider in (4.53) only the active block average and the effective seed-bank
average with $m=l$ (recall that the full block average equals the active block
average). According to the above explanation, we have to replace
$\Gamma^{(k)}_{\bar{\vartheta}_{k}}$ by
$\Gamma^{\mathrm{eff},(k)}_{\vartheta_{k}}$ and $\Gamma^{(l)}_{u}$ by
$\Gamma^{\mathrm{eff},(l)}_{u}$. Hence we find the effective interaction chain
defined in (4.21) and depicted in Figure 8.

###### Slow seed-banks.

From (4.64) we see that if $l\geq k$ and we use the scaling $\bar{t}$, then
all $l$-blocks of seed-banks with colour $0\leq m\leq k$ equal
$\vartheta_{k}$, and all $l$-blocks of seed-banks with colour $m>k$ equal
their initial values $\theta_{y_{m}}$. Something interesting happens when we
choose $0\leq l<k$ and use the scaling $\bar{t}+N^{l}t$. The single colonies
of seed-banks with colour $0\leq m<l$ on time scale $\bar{t}+N^{l}t$ follow
the active population, and hence their $l$-block averages equal the $l$-block
average of the active population. The $l$-block average of the seed-bank with
colour $l$ has a non-trivial interaction with the active $l$-block. The
$l$-blocks of seed-banks with colour $m>k$ have not yet moved and hence are
still in their initial states $(\theta_{y_{m}})_{m=k+1}^{\infty}$. However,
(4.66) implies that the single colonies of the seed-banks with colour $k\geq
m>l$ are not moving on time scale $\bar{t}+N^{l}t$, even though they had
already moved at time $\bar{t}$. Therefore the $l$-blocks averages of seed-
banks with colour $l<m\leq k$ are no longer in their initial state at time
$\bar{t}$. Note that they are also not in the state $\vartheta_{k}$, since
this is the state of their $k$-block averages and not of their $l$-block
averages. The single colony seed-banks with colour $l<m\leq k$ are in the
state given by

(4.70) $y_{m,0}(\bar{t}\,)=\int_{0}^{\bar{t}}{\rm
d}s\,\frac{e_{m}}{N^{m}}[x_{0}(s)-y_{m,0}(s)].$

Hence, for large $N$,

(4.71)
$y_{m,0}(\bar{t})\approx\int_{0}^{L(N)N^{k}+N^{k}t_{k}+\cdots+N^{m}t_{m}}{\rm
d}s\,\frac{e_{m}}{N^{m}}[x_{0}(s)-y_{m,0}(s)].$

Similarly, for the $l$-block average with colour $m$ we have

(4.72) $\displaystyle\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}\,)$
$\displaystyle=\int_{0}^{\bar{t}}{\rm
d}s\,\frac{e_{m}}{N^{m}}[\Theta^{(l),\Omega_{N}}_{x}(s)-\Theta^{(l),\Omega_{N}}_{y_{m}}(s)]$
$\displaystyle\approx\int_{0}^{L(N)N^{k}+N^{k}t_{k}+\cdots+N^{m}t_{m}}{\rm
d}s\,\frac{e_{m}}{N^{m}}[\Theta^{(l),\Omega_{N}}_{x}(s)-\Theta^{(l),\Omega_{N}}_{y_{m}}(s)].$

Thus, we see that the state of $\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t})$ is
completely determined at time $L(N)N^{k}+N^{k}t_{k}+\cdots+N^{m}t_{m}$, i.e.,
the last time before $\bar{t}+N^{l}t$ that the single colony seed-banks of
colour $m$ had an opportunity to move. Up to time
$L(N)N^{k}+N^{k}t_{k}+\cdots+N^{m}t_{m}$, the single colony seed-banks with
colour $m$ interact at a very slow rate with the active single colonies, and
similarly for the $l$-blocks. Therefore effectively the colour-$m$ seed-bank
interacts with a “time-average on scale $N^{m}t_{m}$” of the active
population. On time scale $N^{m}t_{m}$, a single active colony migrates very
fast in its $(m-1)$-block. As a consequence at time $\bar{t}+N^{m}t_{m}$
individuals that start from a particular colony, e.g. site $0$, are spread
uniformly over the $m$-block containing this site. Hence the interaction of a
single $m$-dormant colony with the active population can be intuitively
interpreted as an interaction with the active $m$-block, and similarly for an
$m$-dormant $l$-block. Once we move to lower time scales, the $m$-dormant
single colonies do not interact with the active colony anymore. In the
detailed proofs we show that one consequence of this is that, for $l<m$,

(4.73) $\displaystyle\Theta^{(l),\Omega_{N}}_{y_{m}}(\bar{t}+N^{l}t)$
$\displaystyle\approx\Theta^{(l),\Omega_{N}}_{y_{m}}(L(N)N^{k}+N^{k}t_{k}+\cdots+N^{m}t_{m})$
$\displaystyle\approx\Theta^{(m),\Omega_{N}}_{y_{m}}(L(N)N^{k}+N^{k}t_{k}+\cdots+N^{m}t_{m}).$

Thus, the $l$-block averages of colours $l\leq m\leq k$ equal the state of the
corresponding $m$-block. This is the $(m+2)$-th component of the interaction
chain at level $l$.

###### Conclusion.

Combining the intuitive descriptions in Sections 4.4.1-4.4.3, we see how
Theorems 4.2 and Theorems 4.4 come about. Their proofs will rely on coupling