# Durrett-Levin spatial model of allelopathy

Nicolas Lanchier and Max Mercer Nicolas Lanchier was partially supported by
NSF grant CNS-2000792.

###### Abstract

Allelopathy refers to a type $0/-$ biological interaction that is neutral for
a so-called inhibitory species but detrimental for a so-called susceptible
species. To model this type of interaction in a spatially-structured
environment, Durrett and Levin introduced a variant of the multitype contact
process in which the death rate of the susceptible species is density-
dependent, increasing with the local density of the inhibitory species. Their
work combines mean-field analysis and simulations of the spatial model, and
our main objective is to give rigorous proofs of some of their conjectures. In
particular, we give a complete description of the behavior of the mean-field
model, including the global stability of the fixed points. Our main results
for the interacting particle system show the existence of two regimes
depending on the relative fitness of the individuals. When the inhibitory
species is the superior competitor, the inhibitory species always wins,
whereas when the susceptible species is the superior competitor, the
susceptible species wins if and only if the inhibitory effects do not exceed
some critical threshold. We also prove that, at least in dimensions $d\geq 3$,
the transition between these two regimes is continuous in the sense that, when
both species are equally fit, the inhibitory species wins even in the presence
of extremely weak inhibitory effects.

††footnotetext: AMS 2000 subject classifications: Primary 60K35 Keywords and
phrases: Interacting particle systems, allelopathy, coupling, block
construction, duality.

##  1 Introduction

Biological interactions among species can be beneficial $(+)$, neutral $(0)$
or detrimental $(-)$ for each of the interacting species. Biological
interactions that are neutral for one species and detrimental for another
species are referred to as amensalism. Allelopathy is a particular case of
amensalism called antibiosis amensalism in which one species produces toxic
biochemicals called allelochemicals that inhibit the growth and survival of
the other species. The species producing the allelochemicals is called the
inhibitory species while the other species is called the susceptible species.
Because the inhibitory species is immune to the allelochemicals it produces,
this mechanism provides a selective advantage to the inhibitory species. To
model allelopathy in a spatially-structured environment, Durrett and Levin [7]
introduced an interacting particle system in which each lattice point can be
empty or occupied by an individual of one of the two species. The system is
characterized by the natural birth and death rates of the species as well as
the strength of the inhibitory effects. The spatial component is included in
the form of local interactions assuming that the offspring are sent from the
parent’s location to nearby lattice points but also that the individuals of
the inhibitory species only affect nearby susceptible individuals. Their model
assumes in addition that offspring can only survive when sent to an empty
site, which also models competition for space.
More precisely, the Durrett-Levin model is a variant of the multitype contact
process [18] that models allelopathy by including a density-dependent death
rate for the susceptible species. Like in the multitype contact process, each
site of the $d$-dimensional lattice is in state

$\begin{array}[]{rcrcrcl}0&=&\hbox{empty}&&1&=&\hbox{occupied by a
type\leavevmode\nobreak\ 1 individual\leavevmode\nobreak\ (inhibitory
species)}\vspace*{4pt}\\\ &&\hbox{or}&&2&=&\hbox{occupied by a
type\leavevmode\nobreak\ 2 individual\leavevmode\nobreak\ (susceptible
species)}\end{array}$

so the state of the entire system at time $t$ is a spatial configuration

$\xi_{t}:\mathbb{Z}^{d}\longrightarrow\\{0,1,2\\}\quad\hbox{where}\quad\xi_{t}(x)=\hbox{state
at site\leavevmode\nobreak\ $x$ at time\leavevmode\nobreak\ $t$}.$

To describe the dynamics (local interactions), we let

$N_{x}=\\{y\in\mathbb{Z}^{d}:0<|\\!|x-y|\\!|\leq M\\}\quad\hbox{for all}\quad
x\in\mathbb{Z}^{d}$

be the neighborhood of site $x$, where $|\\!|\cdot|\\!|$ refers to the
Euclidean norm and $M$ is interpreted as a dispersal range. For every species
$i$, site $x$ and configuration $\xi$, we let

$f_{i}(x,\xi)=\frac{1}{\operatorname{card}(N_{x})}\ \sum_{y\in
N_{x}}\mathbf{1}\\{\xi(y)=i\\}$

be the fraction of neighbors of site $x$ that are occupied by an individual of
type $i$. Then, the spatial allelopathic model evolves according to the local
transitions

$\begin{array}[]{rclcrcl}0\to 1&\hbox{at rate}&\beta_{1}f_{1}(x,\xi)&&1\to
0&\hbox{at rate}&1\vspace*{4pt}\\\ 0\to 2&\hbox{at
rate}&\beta_{2}f_{2}(x,\xi)&&2\to 0&\hbox{at rate}&1+\gamma
f_{1}(x,\xi).\end{array}$ (1)

The two transitions on the left indicate that type $i$ individuals give birth
at rate $\beta_{i}$ to individuals of their own type. The offspring is sent to
a site chosen uniformly at random from the parent’s neighborhood, and takes
place in the system if and only if the target site is empty, which models
competition for space. The two transitions on the right indicate that the
individuals die naturally at rate one regardless of their type but the overall
death rate of the susceptible individuals is density-dependent, increasing
with respect to the local density of the inhibitory species, which models
allelopathy whose strength is measured by the parameter $\gamma$. The
particular case $\gamma=0$ corresponds to the multitype contact process which
was introduced by Neuhauser [18]. Another spatial model of allelopathy based
on the framework of interacting particle systems was studied more recently by
Lanchier [14] using a different modeling approach. His model is also a variant
of Neuhauser’s multitype contact process but, contrary to the Durrett-Levin
model, the death rates are both spontaneous. Instead, the model includes an
additional state interpreted as a frozen site, and allelopathy is modeled by
assuming that individuals of the inhibitory species produce metabolites at
their location, leaving after their death a frozen site that remains
inaccessible to the susceptible species (but not to the inhibitory species)
for a random amount of time.
Returning to the Durrett-Levin model (1), assuming that the population is
homogeneously mixing, and letting $u_{1}$ be the density of the inhibitory
species and $u_{2}$ be the density of the susceptible species, the process is
described by the following deterministic mean-field model:

$\begin{array}[]{rcl}u_{1}^{\prime}&=&F_{1}(u_{1},u_{2})=\beta_{1}u_{1}(1-u_{1}-u_{2})-u_{1}\vspace*{4pt}\\\
u_{2}^{\prime}&=&F_{2}(u_{1},u_{2})=\beta_{2}u_{2}(1-u_{1}-u_{2})-(1+\gamma
u_{1})u_{2}.\end{array}$ (2)

Durrett and Levin [7] proved that, whenever

$\beta_{1},\beta_{2}>1\quad\hbox{and}\quad\beta_{1}<\beta_{2}<(1+\gamma)\beta_{1}-\gamma,$
(3)

the system has two locally stable boundary fixed points, one with a positive
density of 1s and one with a positive density of 2s, as well as one interior
fixed point. Their numerical simulations also show that the system is bistable
in the sense that the interior fixed point is a saddle point and the system
converges to one of the two boundary fixed points, which is also suggested by
the solution curves in Figure 1.

$\beta_{1}=2,\ \beta_{2}=3,\ \gamma=1.2$$\beta_{1}=2,\ \beta_{2}=3,\ \gamma=3$

Figure 1:  Solution curves of the mean-field model for two different values of
$\gamma$ and the same birth rates. The boundary fixed points are the same but
the basin of attraction of the boundary fixed point in which only the
inhibitory/susceptible species is present increases/decreases with $\gamma$.

In particular, in the parameter region (3), none of the species can invade the
other species in its equilibrium, and the outcome of the competition depends
on the initial density of each species. In Section 2, we will identify all the
fixed points, study their local stability, and use the Bendixson–Dulac theorem
to give a rigorous proof of this result and a complete picture of the phase
structure of the mean-field model.
Our analysis together with numerical simulations shows that the spatial model
has a number of similarities with its mean-field approximation but also one
major difference: the parameter region where the mean-field model is bistable
is turned into a region where there is a strong type that wins even when
starting at low density in the spatial model. In particular, the outcome of
the competition depends on the parameters of the system but not on the initial
densities. To state our results and explore the phase structure, we say that
species $i$

$\begin{array}[]{rcrcl}\hbox{survives}&\
\hbox{when}&\liminf_{t\to\infty}P(\xi_{t}(x)=i)>0&\hbox{for
all}&x\in\mathbb{Z}^{d}\vspace*{4pt}\\\ \hbox{dies out}&\
\hbox{when}&\lim_{t\to\infty}P(\xi_{t}(x)=i)=0&\hbox{for
all}&x\in\mathbb{Z}^{d}\end{array}$

and wins when it survives while the other species dies out. In addition, we
say that both species coexist when they both survive. Letting $\beta_{c}$ be
the critical value of the contact process described below, the behavior of the
allelopathic model outside the parameter region
$\beta_{2}\geq\beta_{1}>\beta_{c}$ can be easily deduced from coupling
arguments to compare the spatial allelopathic model with Harris’ contact
process and Neuhauser’s multitype contact process. In the absence of one
species, the other species, say species 1, behaves like Harris’ contact
process [12], the interacting particle system in which sites can be either 0 =
empty or 1 = occupied, with local transitions

$\begin{array}[]{rclcrcl}0\to 1&\hbox{at
rate}&\beta_{1}f_{1}(x,\xi)&\quad\hbox{and}&1\to 0&\hbox{at
rate}&1.\end{array}$

As one of the most popular models of interacting particle systems, this
process was intensively studied, and we refer to Liggett [16, 17] for a review
of the main results. In particular, there exists a nondegenerate (positive and
finite) critical value $\beta_{c}$ that depends on both the spatial dimension
and the range of the interactions such that, starting with infinitely many 1s,

$\begin{array}[]{rcl}\beta_{1}\leq\beta_{c}&\Longrightarrow&\hbox{species 1
dies out}\vspace*{4pt}\\\ \beta_{1}>\beta_{c}&\Longrightarrow&\hbox{species 1
survives}.\end{array}$

The proof of extinction at the critical value $\beta_{c}$ is due to
Bezuidenhout and Grimmett [1]. To study the competition between both species,
we now assume that the process starts from a translation invariant measure
with a positive density of both species, so the initial configuration contains
infinitely many individuals of both types. In this case, the set of occupied
sites is dominated by a contact process with parameter
$\beta_{1}\vee\beta_{2}$ from which it follows that

$\begin{array}[]{rcl}\beta_{1}\leq\beta_{c}\quad\hbox{and}\quad\beta_{2}\leq\beta_{c}&\Longrightarrow&\hbox{both
species die out}.\end{array}$ (4)

Using another result of Bezuidenhout and Grimmett [2] about the exponential
decay of the subcritical contact process implies that, when a species is
subcritical, say species 2, it dies out fast enough that species 1 behaves
eventually as in the absence of species 2. In particular,

$\begin{array}[]{rcl}\hbox{(a)}\ \
\beta_{1}>\beta_{c}\quad\hbox{and}\quad\beta_{2}\leq\beta_{c}&\Longrightarrow&\hbox{species
1 wins}\vspace*{4pt}\\\ \hbox{(b)}\ \
\beta_{1}\leq\beta_{c}\quad\hbox{and}\quad\beta_{2}>\beta_{c}&\Longrightarrow&\hbox{species
2 wins}.\end{array}$ (5)

We now assume that the two birth rates are supercritical, in which case the
long-term behavior depends on the relative fitness of the two species and the
strength of allelopathy. Standard coupling arguments show that the set of 1s
is nondecreasing with respect to the birth rate $\beta_{1}$ and the parameter
$\gamma$ and nonincreasing with respect to the birth rate $\beta_{2}$, and
similarly but with the monotonicity reversed for the set of 2s. This implies
that, two of the parameters being fixed, there is at most one phase transition
from extinction to survival of a given species in the direction of the third
parameter. In addition, in the absence of inhibitory effects $\gamma=0$, the
process reduces to the multitype contact process, and a result of Neuhauser
[18] based on duality techniques shows that the species with the larger birth
rate wins.

$\beta_{1}=2.8$and $\gamma=0.45$$\beta_{1}=2.2$and
$\gamma=2.7$$\beta_{1}=2.2$and $\gamma=3.2$$\beta_{1}=2.8$and $\gamma=0.25$

Figure 2:  Snapshots of the process on the $300\times 300$ torus at time 200.
In all the pictures, the birth rate of the susceptible species is equal to
$\beta_{2}=3$, the black sites represent the inhibitory species and the white
sites the susceptible species. Note that, while decreasing $\beta_{1}$ and
increasing $\gamma$, the clusters of the inhibitory species are less dense but
the susceptible species cannot easily invade these clusters due to the
strength of allelopathy measured $\gamma$.

$\gamma$susceptiblespecies 2 wins$\beta_{c}$$\beta_{2}$$\beta_{1}$species 1
winsinhibitory(5.b)(8)(6)(5.b)(7)(6)(12)(10)(9)

Figure 3:  Phase structure of the process for a fixed $\beta_{2}>\beta_{c}$.

$\beta_{c}$$\beta_{1}$inhibitoryspecies 1 winssusceptiblespecies 2
wins$\beta_{c}$$\beta_{2}$extinction(4)(6)(5.b)(11)(5.b)(6)(6)(5.a)(5.a)(12)(10)

Figure 4:  Phase structure of the process for a fixed $\gamma>0$.

This, together with the monotonicity with respect to $\gamma$, implies that
the inhibitory species wins whenever it is the fittest:

$\begin{array}[]{rcl}\beta_{1}>\beta_{2}>\beta_{c}&\Longrightarrow&\hbox{species
1 wins}.\end{array}$ (6)

Combining (4)–(6) gives a complete description of the long-term behavior of
the spatial stochastic model outside the parameter region
$\beta_{2}\geq\beta_{1}>\beta_{c}$.
We now look at the behavior of the process in the parameter region
$\beta_{2}\geq\beta_{1}>\beta_{c}$, which is represented by the gray rectangle
in Figure 3 and the gray triangle in Figure 4. In this case, the long-term
behavior also depends on the strength of allelopathy: there is a critical
value for $\gamma$ below which the susceptible species wins and above which
the inhibitory species survives. Neuhauser’s results were later improved by
Durrett and Neuhauser [8] using a block construction to control the rate of
invasion of the superior competitor. Applying a standard perturbation argument
to their construction in the scenario where the susceptible species is the
superior competitor implies that the susceptible species still wins when the
inhibitory effects are sufficiently weak:

###### Theorem 1

For all $\beta_{2}>\beta_{1}>\beta_{c}$, there is $\gamma_{-}>0$ such that

$\begin{array}[]{rcl}\gamma<\gamma_{-}&\Longrightarrow&\hbox{species 2
wins}.\end{array}$ (7)

In contrast, using that the inhibitory species does not feel the presence of
the susceptible species in the limiting case $\gamma=\infty$, together with a
block construction and a perturbation argument, shows that the inhibitory
species survives when the inhibitory effects are sufficiently strong:

###### Theorem 2

For all $\beta_{2}>\beta_{1}>\beta_{c}$, there is $\gamma_{+}<\infty$ such
that

$\begin{array}[]{rcl}\gamma>\gamma_{+}&\Longrightarrow&\hbox{species 1
survives}.\end{array}$ (8)

Species 1 wins if in addition $M=d=1$.

Numerical simulations suggest that clusters quickly form and either the
susceptible species or the inhibitory species wins. In particular, we
conjecture that, like in the mean-field model, coexistence is not possible
except maybe for a parameter region with measure zero. This, together with the
previous two theorems and the monotonicity with respect to $\gamma$, implies
that for all $\beta_{2}>\beta_{1}>\beta_{c}$, there exists a nondegenerate
critical value $\gamma_{c}$ such that

$\begin{array}[]{rcl}\gamma<\gamma_{c}&\Longrightarrow&\hbox{species 2
wins}\vspace*{4pt}\\\ \gamma>\gamma_{c}&\Longrightarrow&\hbox{species 1
wins}.\end{array}$ (9)

Figure 2 shows realizations of the process near the phase transition, with
$\gamma$ slightly subcritical in the two pictures on the left where the
susceptible species wins and $\gamma$ slightly supercritical in the two
pictures on the right where the inhibitory species wins. Combining (6) and (9)
shows that when the susceptible species is the better competitor there is a
phase transition at some nondegenerate critical value $\gamma_{c}$ whereas
when the inhibitory species is the better competitor the critical value
$\gamma_{c}$ is degenerate, equal to zero. We conjecture that the transition
between these two regimes is continuous at point $\beta_{1}=\beta_{2}$ in the
sense that, when the species are equally fit, the inhibitory species wins even
in the presence of very weak inhibitory effects, meaning that the critical
value $\gamma_{c}$ is again degenerate in the symmetric case. Using duality
techniques inspired from Neuhauser [18] (tree structure of the dual process,
ancestor hierarchy, renewal points, etc.) and the transience of the first
ancestor in high dimensions, we were able to prove this conjecture when $d\geq
3$.

###### Theorem 3

Assume that $\beta_{1},\beta_{2}>\beta_{c}$. Then,

$\begin{array}[]{rcl}\beta_{1}=\beta_{2}\quad\hbox{and}\quad\gamma>0&\Longrightarrow&\hbox{species
1 wins in\leavevmode\nobreak\ $d\geq 3$}.\end{array}$ (10)

We now look at the effects of the birth rate of the susceptible species. In
the limiting case $\beta_{2}=\infty$, connected components of 2s with at least
two individuals can only expand because each time a type 2 individual dies the
resulting empty site is instantaneously reinvaded by the offspring of a nearby
type 2 individual. This implies that, even if the inhibitory effects are
strong, the susceptible species still wins provided it is a good enough
competitor:

###### Theorem 4

For all $\beta_{1},\gamma>0$, there is $\beta_{+}<\infty$ such that

$\begin{array}[]{rcl}\beta_{2}>\beta_{+}&\Longrightarrow&\hbox{species 2
wins}.\end{array}$ (11)

The theorem only gives the existence of a finite (possibly very large)
critical value $\beta_{+}$. Our last result shows that, in the presence of
long range interactions, the allelopathic model can be coupled with the grass-
bush-tree system [9, 10] to identify an explicit parameter region in which the
susceptible species survives. More precisely, we have the following.

###### Theorem 5

For all $\beta_{2}>\beta_{1}^{2}>1$ and $\gamma\leq\beta_{1}$, there is
$M_{0}<\infty$ such that

$\begin{array}[]{rcl}M>M_{0}&\Longrightarrow&\hbox{species 2
survives}.\end{array}$ (12)

The phase structure of the process obtained from (4)–(12) is depicted in
Figures 3 and 4. The gray rectangle in the first picture and the gray triangle
in the second picture represent the parameter region covered in our theorems.
The phase structure does not show any coexistence phase because this is what
our numerical simulations suggest. However, the absence of a phase of
coexistence in the spatial model is still an open problem.
The rest of this paper is devoted to the proofs and organized as follows.
Section 2 gives a complete description of the phase structure of the mean-
field model. In preparation for the analysis of the spatial model, Section 3
explains how to construct the process from a graphical representation and
deduce the monotonicity with respect to each of the parameters. The proofs of
Theorems 1, 2, 4 and 5 can be found in Sections 4–6 and rely on coupling
arguments and/or block constructions. Finally, the proof of Theorem 3, which
is more involved, will be carried out in Section 7.

##  2 Mean-field analysis

This section is devoted to the analysis of the mean-field model (2). In
particular, we give a complete description of the phase structure with the
limiting density if each type based on the parameters of the system. The key
parameter regions that appear in our analysis are

$\begin{array}[]{rcl}B_{0}&=&\\{(\beta_{1},\beta_{2},\gamma)\in\mathbb{R}_{+}^{3}:\beta_{1}<1\
\hbox{and}\ \beta_{2}<1\\}\vspace*{4pt}\\\
B_{1}&=&\\{(\beta_{1},\beta_{2},\gamma)\in\mathbb{R}_{+}^{3}:\beta_{1}>1\
\hbox{and}\ \beta_{2}<(1+\gamma)\beta_{1}-\gamma\\}\vspace*{4pt}\\\
B_{2}&=&\\{(\beta_{1},\beta_{2},\gamma)\in\mathbb{R}_{+}^{3}:\beta_{2}>1\
\hbox{and}\ \beta_{2}>\beta_{1}\\}.\end{array}$

Setting the right-hand side of (2) equal to zero gives the three trivial fixed
points

$p_{0}=(0,0),\quad p_{1}=\bigg{(}1-\frac{1}{\beta_{1}},0\bigg{)},\quad
p_{2}=\bigg{(}0,1-\frac{1}{\beta_{2}}\bigg{)}$

and the nontrivial fixed point $p_{12}=(\bar{u}_{1},\bar{u}_{2})$ where

$\bar{u}_{1}=\frac{1}{\gamma}\bigg{(}\frac{\beta_{2}}{\beta_{1}}-1\bigg{)}\quad\hbox{and}\quad\bar{u}_{2}=1-\frac{1}{\beta_{1}}-\bar{u}_{1}=1-\frac{1}{\beta_{1}}-\frac{1}{\gamma}\bigg{(}\frac{\beta_{2}}{\beta_{1}}-1\bigg{)}.$
(13)

The first step is to identify the sets of parameters for which each of the
four fixed points belongs to the biologically relevant two-dimensional simplex
$\Delta^{2}$. The fixed point $p_{0}$ is always in the two-dimensional
simplex, the fixed point $p_{1}$ belongs to the simplex if and only if
$\beta_{1}>1$, and the fixed point $p_{2}$ belongs to the simplex if and only
if $\beta_{2}>1$. In addition,

$\begin{array}[]{rclcl}p_{12}\in\Delta^{2}&\Longleftrightarrow&0<\bar{u}_{1},\bar{u}_{2}<1\vspace*{4pt}\\\
&\Longleftrightarrow&\beta_{1}<\beta_{2}<(1+\gamma)\beta_{1}-\gamma&\Longleftrightarrow&(\beta_{1},\beta_{2},\gamma)\in
B_{1}\cap B_{2}.\end{array}$ (14)

To study the local stability, note that the Jacobian matrix is given by

$J(u_{1},u_{2})=\left(\begin{array}[]{cc}\beta_{1}(1-2u_{1}-u_{2})-1&-\beta_{1}u_{1}\vspace*{2pt}\\\
-(\beta_{2}+\gamma)u_{2}&\beta_{2}(1-u_{1}-2u_{2})-1-\gamma
u_{1}\end{array}\right).$

In particular, the Jacobian matrices at $p_{0}$, $p_{1}$ and $p_{2}$ are
triangular, of the form

$J(p_{0})=\left(\begin{array}[]{cc}\beta_{1}-1&0\vspace*{2pt}\\\
0&\beta_{2}-1\end{array}\right)\quad
J(p_{1})=\left(\begin{array}[]{cc}1-\beta_{1}&\times\vspace*{2pt}\\\
0&-\gamma\bar{u}_{2}\end{array}\right)\quad
J(p_{2})=\left(\begin{array}[]{cc}\beta_{1}/\beta_{2}-1&0\vspace*{2pt}\\\
\times&1-\beta_{2}\end{array}\right).$

Using that $\bar{u}_{2}>0$ if and only if
$\beta_{2}<(1+\gamma)\beta_{1}-\gamma$ and looking at the sign of the two
eigenvalues on the diagonal of each of the Jacobian matrices, we deduce that

$\begin{array}[]{rcl}p_{0}\ \hbox{is locally
stable}&\Longleftrightarrow&\beta_{1}<1\ \hbox{and}\ \beta_{2}<1,\
\hbox{i.e.},\ (\beta_{1},\beta_{2},\gamma)\in B_{0}\vspace*{4pt}\\\ p_{1}\
\hbox{is locally stable}&\Longleftrightarrow&\beta_{1}>1\ \hbox{and}\
\beta_{2}<(1+\gamma)\beta_{1}-\gamma,\ \hbox{i.e.},\
(\beta_{1},\beta_{2},\gamma)\in B_{1}\vspace*{4pt}\\\ p_{2}\ \hbox{is locally
stable}&\Longleftrightarrow&\beta_{2}>1\ \hbox{and}\ \beta_{2}>\beta_{1},\
\hbox{i.e.},\ (\beta_{1},\beta_{2},\gamma)\in B_{2}.\end{array}$

Using numerical simulations, Durrett and Levin noticed that $p_{12}$ is a
saddle, which we now prove rigorously. First, notice that, because
$1-\bar{u}_{1}-\bar{u}_{2}=1/\beta_{1}$,

$\beta_{1}(1-2\bar{u}_{1}-\bar{u}_{2})-1=\beta_{1}(1-\bar{u}_{1}-\bar{u}_{2})-\beta_{1}\bar{u}_{1}-1=-\beta_{1}\bar{u}_{1}$

showing that the two coefficients on the first row of the Jacobian matrix at
the fixed point $p_{12}$ are in fact equal. It follows that the expression of
the determinant reduces to

$\begin{array}[]{rcl}\operatorname{Det}\,(J(p_{12}))&=&-\beta_{1}\bar{u}_{1}(\beta_{2}(1-\bar{u}_{1}-2\bar{u}_{2})-1-\gamma\bar{u}_{1}+(\beta_{2}+\gamma)\bar{u}_{2})\vspace*{4pt}\\\
&=&-\beta_{1}\bar{u}_{1}(\beta_{2}(1-\bar{u}_{1}-\bar{u}_{2})-1-\gamma(\bar{u}_{1}-\bar{u}_{2})\vspace*{4pt}\\\
&=&-\beta_{1}\bar{u}_{1}(\beta_{2}/\beta_{1}-\gamma(\bar{u}_{1}-\bar{u}_{2})-1).\end{array}$

Then, replacing $\bar{u}_{1}$ and $\bar{u}_{2}$ by their values in (13), we
get

$\operatorname{Det}\,(J(p_{12}))=-\bar{u}_{1}((1+\gamma)\beta_{1}-\gamma-\beta_{2})<0$

whenever the equivalent conditions in (14) hold. This shows that the two
eigenvalues have opposite signs therefore the interior fixed point is always a
saddle.

11$\beta_{1}$$\beta_{2}$slope 1slope $1+\gamma$$B_{1}\cap
B_{2}$$B_{1}$$B_{0}$extinction$u\to p_{0}$bistability$u\to p_{1}$or
$p_{2}$inhibitoryspecies 1 wins$u\to p_{1}$susceptiblespecies 2 wins$u\to
p_{2}$$B_{2}$$B_{2}$$B_{2}$$B_{1}$$B_{1}$$B_{1}$

Figure 5:  Phase structure of the mean-field model (2).

To deduce the limiting behavior of the system from the analysis of the local
stability of the fixed points, the last step is to apply the Bendixson–Dulac
theorem to exclude the existence of periodic orbits. More precisely, the
objective is to find a smooth function $\phi(u_{1},u_{2})$ such that the sign
of $\nabla\cdot(\phi F_{1},\phi F_{2})$ is constant almost everywhere in the
simplex. Using the function $\phi=1/(u_{1}u_{2})$, we get

$\begin{array}[]{l}\nabla\cdot(\phi F_{1},\phi
F_{2})=\displaystyle\frac{\partial(\phi F_{1})}{\partial
u_{1}}+\frac{\partial(\phi F_{2})}{\partial u_{2}}\vspace*{4pt}\\\ \hskip
20.0pt=\displaystyle\frac{\partial}{\partial
u_{1}}\bigg{(}\frac{\beta_{1}(1-u_{1}-u_{2})-1}{u_{2}}\bigg{)}+\displaystyle\frac{\partial}{\partial
u_{2}}\bigg{(}\frac{\beta_{2}(1-u_{1}-u_{2})-1-\gamma
u_{1}}{u_{1}}\bigg{)}=-\frac{\beta_{1}}{u_{2}}-\frac{\beta_{2}}{u_{1}}<0\end{array}$

for all $\beta_{1},\beta_{2}>0$ and all $u$ in the interior of the simplex.
This proves the absence of periodic orbits which, together with the local
stability of the fixed points, give the following limiting behavior. In the
parameter region $B_{0}$, the population goes extinct in the sense that $u\to
p_{0}$. In the parameter region $B_{1}\setminus B_{2}$, the inhibitory species
wins in the sense that $u\to p_{1}$ when starting with a positive density of
1s. Similarly, in the parameter region $B_{2}\setminus B_{1}$, the susceptible
species wins. Finally, in the parameter region $B_{1}\cap B_{2}$, the system
is bistable: for almost all initial conditions in the simplex, the densities
converge to either $p_{1}$ or $p_{2}$, indicating that the outcome of the
competition depends on the initial densities. Figure 5 shows a picture of the
phase structure.

##  3 Harris’ graphical representation

The starting point to study the spatial model and prove our results is to use
an idea of Harris [11] to construct the process graphically from a collection
of independent Poisson processes/exponential clocks. In the case of the
allelopathic model, we use four collections corresponding to the following
four updates: birth of a 1, birth of a 2, natural death, death of a 2 due to
inhibitory effects. To construct the process, it is convenient to think of the
lattice $\mathbb{Z}^{d}$ as the vertex set of a directed graph in which there
is a directed edge $\vec{xy}$ if and only if vertices $x$ and $y$ are
neighbors, and we let $N$ be the common size of the interaction neighborhoods.
The exponential clocks are attached to the vertices and directed edges of the
graph as follows.

  * •

Inhibition. Place an exponential clock with rate $\gamma/N$ along each
directed edge $\vec{xy}$. Each time the clock rings, say at time $t$, draw an
arrow $(x,t)\to(y,t)$ labeled with a 0 to indicate that if the tail of the
arrow is occupied by a 1 and the head of the arrow is occupied by a 2 then the
head of the arrow becomes empty.

  * •

Birth of a 1. Place an exponential clock with rate $\beta_{1}/N$ along each
directed edge $\vec{xy}$. Each time the clock rings, say at time $t$, draw an
arrow $(x,t)\to(y,t)$ labeled with a 1 to indicate that if the tail of the
arrow is occupied by a 1 and the head of the arrow is empty then the head of
the arrow becomes occupied by a 1.

  * •

Birth of a 2. Place an exponential clock with rate $\beta_{2}/N$ along each
directed edge $\vec{xy}$. Each time the clock rings, say at time $t$, draw an
arrow $(x,t)\to(y,t)$ labeled with a 2 to indicate that if the tail of the
arrow is occupied by a 2 and the head of the arrow is empty then the head of
the arrow becomes occupied by a 2.

  * •

Natural death. Place an exponential clock with rate one at each vertex $x$.
Each time the clock rings, say at time $t$, put a cross at $(x,t)$ to indicate
that site $x$ becomes empty.

Figure 6 shows an example of realization of this graphical representation and
how to construct the process starting from a given initial configuration.

43210$-1$$-2$$-3$$-4$12111111122222221200000

Figure 6:  Graphical representation of the allelopathic model. The black lines
represent the space-time region occupied by the inhibitory species and the
gray lines the region occupied by the susceptible species.

An argument of Harris [11] based on techniques from percolation theory shows
more generally that the allelopathic model on the infinite integer lattice
starting from any initial configuration is well-defined and can be constructed
using the collection of independent exponential clocks above.
The graphical representation can also be used together with the superposition
properties of the exponential distribution to couple processes with different
parameters or different dynamical structures, and deduce some monotonicity
results. For instance, fix $\gamma_{1}\leq\gamma_{2}$, and let $\xi_{t}^{i}$
be the process with birth rates $\beta_{1}$ and $\beta_{2}$, and parameter
$\gamma_{i}$ for $i=1,2$. Think of the first process as being generated from
the graphical representation described above with $\gamma=\gamma_{1}$ while,
according to the superposition property, the second process can be constructed
from the same graphical representation as the first process by also adding
type 0 arrows along the directed edges at rate $(\gamma_{2}-\gamma_{1})/N$,
which results in a joint construction (coupling) of the two processes. In
addition, starting both processes from the same initial configuration, one can
easily check that

$\begin{array}[]{rclcrcl}\xi_{t}^{1}(x)=1&\Longrightarrow&\xi_{t}^{2}(x)=1&\quad\hbox{and}&\xi_{t}^{2}(x)=2&\Longrightarrow&\xi_{t}^{1}(x)=2.\end{array}$

Using also the monotonicity of the expectation, we deduce that, the birth
rates being fixed, survival of the inhibitory species when $\gamma=\gamma_{1}$
implies survival of the inhibitory species when $\gamma=\gamma_{2}$, and
survival of the susceptible species when $\gamma=\gamma_{2}$ implies survival
of the susceptible species when $\gamma=\gamma_{1}$. The monotonicity with
respect to each of the birth rates mentioned in the introduction can be proved
similarly by coupling the birth arrows.
Recalling that the multitype contact process corresponds to the special case
$\gamma=0$, the previous monotonicity result also implies that if the 1s win
in the multitype contact process then they also win in the allelopathic model
with the same birth parameters for all $\gamma\geq 0$. This, together with
Neuhauser’s result [18, Theorem 1] which states that

$\begin{array}[]{rcl}\beta_{1}>\beta_{2}>\beta_{c}&\Longrightarrow&\hbox{species
1 wins}\end{array}$

for the multitype contact process, implies that the same result (6) also holds
for the allelopathic model. The graphical representation will also be used in
the next sections to couple the process with the grass-bush-tree system [9,
10], define good events in certain space-time regions when using block
constructions, and study the dual process of the model.

##  4 Proofs of Theorems 1 and 2

This section is devoted to the proofs of Theorem 1 and 2. Assuming that the
susceptible species is a better competitor than the inhibitory species, recall
that the theorems state that the susceptible species wins if the inhibitory
effects are sufficiently weak whereas the inhibitory species survives if the
inhibitory effects are sufficiently strong. The proofs of both theorems
consist in comparing the process in the two extreme cases $\gamma=0$ and
$\gamma=\infty$ with well-known interacting particle systems, and then using a
block construction along with a perturbation argument to prove the existence
of a phase transition at a nondegenerate critical value $\gamma_{c}$.
Throughout this section, $P_{\gamma}$ will refer to the probability for the
process with parameter $\gamma$.
The technique now known as the block construction appeared for the first time
in the work of Bramson and Durrett [3] and is reviewed in detail in Durrett’s
lecture notes [6]. The general idea of this approach is to couple the process
properly rescaled in space and time with oriented site percolation, which we
now briefly describe. To begin with, let

$\mathscr{L}=\\{(m,n)\in\mathbb{Z}^{d}\times\mathbb{Z}_{+}:m_{1}+\cdots+m_{d}+n\
\hbox{is even}\\}$

which we turn into a directed graph by placing arrows

$(m,n)\to(m^{\prime},n^{\prime})\quad\hbox{if and only
if}\quad|m_{1}-m_{1}^{\prime}|+\cdots+|m_{d}-m_{d}^{\prime}|=1\ \ \hbox{and}\
\ n^{\prime}=n+1.$

The sites in $\mathscr{L}$ are assumed to be open or closed with probability
$p$ and $1-p$, respectively, and whether a finite collection of sites are open
or closed are assumed to be independent events when these sites are
sufficiently far apart. We say that a site is wet if it can be reached from a
directed path of open sites starting at level $n=0$, we let $\mathscr{C}_{0}$
be the cluster of sites that can be reached from a directed path of open sites
starting at the origin, and we call the event that this cluster is infinite
the percolation event. Using a so-called contour argument, it can be proved
that there exists a nondegenerate critical value $p_{c}\in(0,1)$ for the
density of open sites such that

$\begin{array}[]{rcl}p>p_{c}&\Longrightarrow&P_{p}(|\mathscr{C}_{0}|=\infty)>0,\end{array}$

and we refer to Durrett [4, Section 10] for the details of the proof. A
consequence of this result is that, in the supercritical phase $p>p_{c}$ and
starting with infinitely many open sites at level $n=0$, the density of wet
sites at level $n$ converges to a positive limit as $n\to\infty$.
To prove Theorem 1, the basic idea is to observe that, under the assumptions
of the theorem, the allelopathic model consists of a small perturbation of a
multitype contact process in which type 2 individuals win. To turn our
intuition into a rigorous proof, we will apply a perturbation argument to the
block construction Durrett and Neuhauser [8, Section 3] used to study the
multitype contact process, the particular case $\gamma=0$. Changing a little
bit the notation in their construction, for each site $(m,n)\in\mathscr{L}$,
we define the space-time boxes

$\begin{array}[]{rcl}A_{m,n}&=&(mL,nT)+([-L,L]^{d}\times\\{T\\})\vspace*{4pt}\\\
B_{m,n}&=&(mL,nT)+([-L,L]^{d}\times[T,2T])\vspace*{4pt}\\\
C_{m,n}&=&(mL,nT)+([-3L,3L]^{d}\times[0,2T])\quad\hbox{where}\quad
T=L^{2}.\end{array}$

Note that $A_{m,n}$ is flat and represents the bottom of the space-time box
$B_{m,n}$. We partition $A_{m,n}$ into small cubes of size
$L^{0.1}\times\cdots\times L^{0.1}$, and define the events

$\begin{array}[]{rcl}E_{m,n}&=&\hbox{each of the small cubes
in\leavevmode\nobreak\ $A_{m,n}$ contains at least one
type\leavevmode\nobreak\ 2 individual}\vspace*{4pt}\\\ F_{m,n}&=&\hbox{the
space-time box\leavevmode\nobreak\ $B_{m,n}$ does not contain any
type\leavevmode\nobreak\ 1 individual}.\end{array}$

Durrett and Neuhauser’s proof, which relies on duality techniques along with a
repositioning algorithm, implies that, for the multitype contact process with
$\beta_{2}>\beta_{1}>\beta_{c}$ and for all $\epsilon>0$, there exists a
collection of good events $G_{m,n}$ that only depend on the graphical
representation in the larger space-time box $C_{m,n}$ such that, for all $L$
sufficiently large,

$\begin{array}[]{rl}\hbox{(a)}&P_{0}(G_{m,n})\geq 1-\epsilon/2\vspace*{4pt}\\\
\hbox{(b)}&E_{m,n}\cap G_{m,n}\ \Longrightarrow\ E_{m^{\prime},n^{\prime}}\cap
F_{m^{\prime},n^{\prime}}\ \ \hbox{for all}\ \
(m^{\prime},n^{\prime})\leftarrow(m,n).\end{array}$

In words, if the bottom of a space-time box contains many 2s then, with high
probability, not only the bottom of the boxes immediately above contains many
2s but also these boxes do not contain any 1s, as shown in Figure 7 in the
one-dimensional case.

$A_{m,n}$$B_{m-1,n+1}$$B_{m+1,n+1}$no 1s in thisspace-time boxno 1s in
thisspace-time boxone 2 in each small cubeno 1s and at leastone 2 in each
small cubeno 1s and at leastone 2 in each small cubeno 1s and at
least$B_{m,n}$

Figure 7:  Illustration of the block construction in Durrett and Neuhauser
[8].

In order to apply a perturbation argument, we now think of the allelopathic
model as being generated from the same graphical representation as the
multitype contact process along with type 0 arrows starting from each site at
rate $\gamma$, as shown in Figure 6. The scale parameter $L$ being fixed,
there exists $\gamma_{-}>0$ small such that the probability that the graphical
representation of the allelopathic model coincides with that of the multitype
contact process in the box $C_{m,n}$ is given by

$\begin{array}[]{l}P_{\gamma}(\hbox{no type\leavevmode\nobreak\ 0 arrows in
the space-time box}\ C_{m,n})\vspace*{4pt}\\\ \hskip
40.0pt=P(\operatorname{Poisson}(\gamma\operatorname{Vol}(C_{m,n}))=0)=\exp(-2L^{2}(6L+1)^{d}\,\gamma)\geq
1-\epsilon/2\end{array}$

for all $\gamma<\gamma_{-}$. In particular, for all $\gamma<\gamma_{-}$,

$P_{\gamma}(G_{m,n})=1-P_{\gamma}(G_{m,n}^{c})\geq
1-P_{0}(G_{m,n}^{c})-\epsilon/2\geq 1-\epsilon/2-\epsilon/2=1-\epsilon.$

Calling site $(m,n)\in\mathscr{L}$ a good site whenever $E_{m,n}\cap F_{m,n}$
occurs, [6, Theorem A.4] implies that there is a coupling of the process and
oriented site percolation with parameter $p=1-\epsilon$ such that the set of
good sites in the interacting particle system dominates the set of wet sites
in the percolation model. Choosing $\epsilon<1-p_{c}$ then implies that the
density of good sites converges to a positive limit as $n\to\infty$, which
proves survival of the susceptible species. This does not fully prove the
theorem because there is still a positive density $\epsilon>0$ of closed
sites, and the corresponding space-time box $B_{m,n}$ can potentially contain
1s. To prove extinction of the inhibitory species, the last step is to use a
result due to Durrett [5] which shows that, when $\epsilon$ is small enough,
not only the open sites percolate but also the closed sites do not percolate.
Because the 1s cannot appear spontaneously, the presence of an individual of
type 1 in box $B_{m,n}$ implies the existence of a directed path of closed
sites from level zero to site $(m,n)$, an event whose probability decays
exponentially fast with $n$ due to the lack of percolation of the closed
sites. This shows extinction of the inhibitory species and completes the proof
of Theorem 1.
To prove Theorem 2, we observe that, under the assumptions of the theorem, the
set of 1s in the allelopathic model consists of a small perturbation of a
supercritical contact process. The first work showing that the supercritical
contact process properly rescaled in space and time dominates oriented site
percolation with parameter arbitrarily close to one is the paper of
Bezuidenhout and Grimmett [1] where they used this coupling together with a
perturbation argument to prove extinction of the critical contact process.
Instead of explaining their construction, we simply reuse the result of
Durrett and Neuhauser [8] exchanging the roles of the two types of particles,
in the absence of 2s. Using the same space-time boxes but redefining the
events

$\begin{array}[]{rcl}E_{m,n}&=&\hbox{each of the small cubes
in\leavevmode\nobreak\ $A_{m,n}$ contains at least one
type\leavevmode\nobreak\ 1 individual},\end{array}$

their result implies that, for the (supercritical) contact process with
$\beta_{1}>\beta_{c}$ and for all $\epsilon>0$, there exists a collection of
good events $G_{m,n}$ that only depend on the graphical representation in the
larger space-time box $C_{m,n}$ such that, for all $L$ sufficiently large,

$\begin{array}[]{rl}\hbox{(a)}&P_{\infty}(G_{m,n})\geq
1-\epsilon/2\vspace*{4pt}\\\ \hbox{(b)}&E_{m,n}\cap G_{m,n}\ \Longrightarrow\
E_{m^{\prime},n^{\prime}}\ \ \hbox{for all}\ \
(m^{\prime},n^{\prime})\leftarrow(m,n).\end{array}$ (15)

In words, if the bottom of a space-time box $B_{m,n}$ contains many 1s then,
with high probability, the bottom of the boxes immediately above also contains
many 1s. Returning to the allelopathic model, observe that, in the limiting
case $\gamma=\infty$, the 2s in the neighborhood of a 1 are instantaneously
killed, which implies that the 1s only attempt to give birth onto empty sites
or sites already occupied by a type 1 particle. This shows that the 1s do not
feel the presence of the 2s and therefore evolve according to a supercritical
contact process. The next step is to apply a perturbation argument to prove
that the good events $G_{m,n}$ above still occur with probability arbitrarily
close to one when $\gamma$ is large but finite, which is done in the following
lemma.

###### Lemma 6

For all $\epsilon>0$, there exist good events $\bar{G}_{m,n}$ and large $L$
and $\gamma_{+}$ such that

$\begin{array}[]{rl}\hbox{(a)}&P_{\gamma}(\bar{G}_{m,n})\geq 1-\epsilon\
\hbox{for all}\ \ \gamma>\gamma_{+}\vspace*{4pt}\\\
\hbox{(b)}&E_{m,n}\cap\bar{G}_{m,n}\ \Longrightarrow\
E_{m^{\prime},n^{\prime}}\ \ \hbox{for all}\ \
(m^{\prime},n^{\prime})\leftarrow(m,n).\end{array}$

###### Proof.

Think of the allelopathic model as being generated from the same graphical
representation as the contact process along with type 0 arrows starting from
each site at rate $\gamma$ and type 2 arrows starting from each site at rate
$\beta_{2}$, as shown in Figure 6. Then, the set of 1s in the space-time box
$C_{m,n}$ evolves as if there were no 2s if each time a 2 gives birth onto a
site $x$ next to a 1, the offspring dies before any of the surrounding 1s
tries to send an offspring to $x$. This happens in particular if each time
there is a type 2 arrow pointing at $(x,t)\in C_{m,n}$, there are $N$ type 0
arrows starting from each site in $N_{x}$ and pointing at $x$ before any type
1 arrow pointing at $x$. Calling this event $F_{m,n}$, part (b) of the lemma
holds for the events

$\bar{G}_{m,n}=F_{m,n}\cap G_{m,n}\quad\hbox{for
all}\quad(m,n)\in\mathscr{L}.$

To estimate the probability of $\bar{G}_{m,n}$, define the events

$H_{m,n}=\hbox{there are less than\leavevmode\nobreak\
$2h_{L}=4\beta_{2}L^{2}(6L+1)^{d}$ type\leavevmode\nobreak\ 2 arrows pointing
at\leavevmode\nobreak\ $C_{m,n}$}.$

Because type 2 arrows occur at rate $\beta_{2}$ at each site,

$P(H_{m,n}^{c})=P(\operatorname{Poisson}(\beta_{2}\operatorname{Vol}(C_{m,n}))\geq
2h_{L})=P(\operatorname{Poisson}(h_{L})\geq 2h_{L}))\leq\epsilon/4$ (16)

for all $L$ sufficiently large. Now, let

$X_{1},X_{2},\ldots,X_{N}=\operatorname{Exponential}(\gamma/N)\quad\hbox{and}\quad
Y=\operatorname{Exponential}(\beta_{1})$

be independent. The parameter $L$ being fixed so that (15) and (16) hold,
recalling the rate of the type 0 arrows and type 1 arrows, and using the
superposition property, we get

$\begin{array}[]{rcl}P_{\gamma}(F_{m,n}^{c}\,|\,H_{m,n})&\leq&2h_{L}P_{\gamma}(\max(X_{1},X_{2},\ldots,X_{N})>Y)\vspace*{4pt}\\\
&\leq&2Nh_{L}P_{\gamma}(X_{1}>Y)=2Nh_{L}\beta_{1}/(\beta_{1}+\gamma/N)\leq\epsilon/4\end{array}$

for all $\gamma$ sufficiently large. In particular, there exists
$\gamma_{+}<\infty$ such that

$\begin{array}[]{rcl}P_{\gamma}(F_{m,n}^{c})&=&P_{\gamma}(F_{m,n}^{c}\,|\,H_{m,n})P(H_{m,n})+P_{\gamma}(F_{m,n}^{c}\,|\,H_{m,n}^{c})P(H_{m,n}^{c})\vspace*{4pt}\\\
&\leq&P_{\gamma}(F_{m,n}^{c}\,|\,H_{m,n})+P(H_{m,n}^{c})\leq\epsilon/4+\epsilon/4=\epsilon/2\end{array}$
(17)

for all $\gamma>\gamma_{+}$. Combining (15) and (17), we conclude that

$P_{\gamma}(\bar{G}_{m,n})\geq
1-P_{\gamma}(F_{m,n}^{c})-P_{\infty}(G_{m,n}^{c})\geq
1-\epsilon/2-\epsilon/2=1-\epsilon$

for all $\gamma>\gamma_{+}$, which proves part (a) of the lemma. ∎

As previously, calling site $(m,n)\in\mathscr{L}$ a good site when $E_{m,n}$
occurs, it follows from the lemma that there is a coupling with oriented site
percolation with parameter $p=1-\epsilon$ such that the set of good sites
dominates the set of wet sites in the percolation model. Choosing $\epsilon$
small enough implies that the density of good sites converges to a positive
limit as $n\to\infty$, which shows survival of the inhibitory species. The
stochastic domination also implies that there is a site (in fact infinitely
many sites) $(m,0)$ such that the cluster of good sites starting at $(m,0)$
expands linearly in all directions. Because invasion paths cannot jump over or
go around each other in the presence of one-dimensional nearest neighbor
interactions, we deduce that this cluster is void of 2s therefore species 1
wins when $M=d=1$. This completes the proof of Theorem 2.

##  5 Proof of Theorem 4

This section focuses on the effects of the competitiveness of the susceptible
species, measured by the birth rate $\beta_{2}$. Recall from (6) that the
inhibitory species wins whenever it is a better competitor than the
susceptible species. In contrast, Theorem 4 states that, even when the
inhibitory effects are strong, the susceptible species wins if it is highly
competitive. This and monotonicity imply the existence of a unique phase
transition from extinction to survival of the susceptible species in the
direction of the parameter $\beta_{2}$ as shown in Figure 4. To prove the
theorem, let

$\Lambda_{-}=\\{0,1\\}^{d}\quad\hbox{and}\quad\Lambda_{+}=\\{-1,0,1,2\\}^{d}$

and let $\bar{\xi}_{t}$ be the allelopathic model starting from

$\bar{\xi}_{0}(x)=2\ \ \hbox{for all}\ \
x\in\Lambda_{-}\quad\hbox{and}\quad\bar{\xi}_{0}(x)\neq 2\ \ \hbox{for all}\ \
x\notin\Lambda_{+}$

and modified so that births of 2s outside the larger cube $\Lambda_{+}$ are
suppressed. The next lemma shows that, in the limiting case
$\beta_{2}=\infty$, the set of 2s fully invades $\Lambda_{+}$ in a finite
deterministic time with probability arbitrarily close to one.

###### Lemma 7

Let $\beta_{2}=\infty$. Then, for all $\epsilon>0$, there exists $T<\infty$
such that

$P_{\infty}(\bar{\xi}_{T}(x)=2\ \hbox{for all}\ x\in\Lambda_{+})\geq
1-\epsilon/2.$

###### Proof.

Each time a 2 dies at some site $x\in\Lambda_{+}$, there is a 2 at some site
in $N_{x}$ that can send instantaneously an offspring to site $x$, which shows
that the set of 2s can only increase. To prove full invasion of the 2s, we
need to deal with two problems:

  * •

potential 1s in the set $\Lambda_{+}\setminus\Lambda_{-}$ that block the 2s,

  * •

the fact that particles in $\Lambda_{-}$ might not be able to send their
offspring to a corner of $\Lambda_{+}$, which is the case for instance in the
presence of nearest neighbor interactions.

However, regardless of the range, there exist $x_{1},x_{2},\ldots,x_{K}$ with
$K=4^{d}-2^{d}$ such that

$\Lambda_{+}\setminus\Lambda_{-}=\\{x_{1},x_{2},\ldots,x_{K}\\}\quad\hbox{and}\quad(\Lambda_{-}\cup\\{x_{1},\ldots,x_{i-1})\cap
N_{x_{i}}\neq\varnothing.$

In particular, the time it takes for the 2s to invade $\Lambda_{+}$ is smaller
than the time $S$ it takes to have one death mark at $x_{1}$ then one death
mark at $x_{2}$, and so on. By memoryless, this is the sum of $K$ independent
exponential random variables $X_{i}$ with mean one therefore, for all
$\epsilon>0$,

$\begin{array}[]{l}P_{\infty}(\bar{\xi}_{T}(x)=2\ \hbox{for all}\
x\in\Lambda_{+})\geq P(S<T)\vspace*{4pt}\\\ \hskip 40.0pt\geq P(X_{i}<T/K\
\hbox{for all}\ i=1,2,\ldots,K)=(1-e^{-T/K})^{K}\geq 1-\epsilon/2\end{array}$

for all $T$ sufficiently large. This completes the proof. ∎

Applying a perturbation argument like in the proof of Lemma 6, we deduce that
for all $\epsilon>0$, there exist sufficiently large $T<\infty$ and
$\beta_{+}<\infty$ such that

$P_{\beta_{2}}(\bar{\xi}_{T}(x)=2\ \hbox{for all}\ x\in\Lambda_{+})\geq
1-\epsilon\quad\hbox{for all}\quad\beta_{2}>\beta_{+}.$ (18)

Because the set of 2s in the modified process $\bar{\xi}_{t}$ is dominated by
its counterpart in the original allelopathic model with the same parameters
and because the evolution rules of the allelopathic model are translation
invariant in space, it follows from (18) that the set of 2s in the process
$\xi_{t}$ properly rescaled in space and time dominates supercritical oriented
site percolation. More precisely, call site $(m,n)\in\mathscr{L}$ a good site
whenever the following two events occur:

$\begin{array}[]{rcl}E_{m,n}&=&\hbox{the box $m+\Lambda_{-}$ is fully occupied
by 2s at time\leavevmode\nobreak\ $nT$}\vspace*{4pt}\\\ F_{m,n}&=&\hbox{the
space-time box\leavevmode\nobreak\ $(m,nT)+(\Lambda_{-}\times[0,T])$ does not
contain any 1s}.\end{array}$

Then, it follows from (18) that for all $\beta_{1}$ and $\gamma$, and for all
$\epsilon>0$, there exists a collection of good events $G_{m,n}$ such that,
for all $T$ and $\beta_{+}$ large,

$\begin{array}[]{rl}\hbox{(a)}&P_{\beta_{2}}(G_{m,n})\geq
1-\epsilon\vspace*{4pt}\\\ \hbox{(b)}&E_{m,n}\cap G_{m,n}\ \Longrightarrow\
E_{m^{\prime},n^{\prime}}\cap F_{m^{\prime},n^{\prime}}\ \ \hbox{for all}\ \
(m^{\prime},n^{\prime})\leftarrow(m,n)\end{array}$

for all $\beta_{2}>\beta_{+}$. In addition, because Lemma 7 applies to the
modified process $\bar{\xi}_{t}$, the event $G_{m,n}$ can be made measurable
with respect to the graphical representation of the process in

$(m,nT)+([-M-1,M+2]^{d}\times[0,2T]).$

As previously, this shows that the set of good sites dominates the set of wet
sites in an oriented site percolation process with parameter $p=1-\epsilon$.
Choosing $\epsilon>0$ small enough to ensure percolation of the open sites and
the lack of percolation of the closed sites, we conclude that the susceptible
species survives whereas the inhibitory species goes extinct.

##  6 Proof of Theorem 5

This section gives a coupling of the allelopathic model and the grass-bush-
tree system introduced by Durrett and Swindle [10] from which Theorem 5
follows easily. Changing their labeling of the states to facilitate the
comparison with the allelopathic model, the sites in the grass-bush-tree
system can be in state 0 = grass, 1 = tree, and 2 = bush. The process, denoted
by $\zeta_{t}$, is a variant of the multitype contact process modeling
ecological successions in which the tree is a superior competitor that can
give birth onto both grass and bushes, so the transition rates are

$\begin{array}[]{rclcrcl}0,2\to 1&\hbox{at rate}&\beta_{1}f_{1}(x,\zeta)&&1\to
0&\hbox{at rate}&1\vspace*{4pt}\\\ 0\to 2&\hbox{at
rate}&\beta_{2}f_{2}(x,\zeta)&&2\to 0&\hbox{at rate}&1.\end{array}$

Note that the dynamical structure of the process is the same as that of the
allelopathic model except that the death rate of the 2s is no longer density
dependent, which favors the 2s, but the 1s can give birth onto the 2s, which
favors the 1s. Assuming that $\gamma\leq\beta_{1}$, the two processes can be
constructed from the same graphical representation to define a coupling
$(\xi_{t},\zeta_{t})$ as follows.

  * •

Place an exponential clock with rate $\gamma/N$ along each directed edge
$\vec{xy}$. Each time the clock rings, say at time $t$, draw an arrow
$(x,t)\to(y,t)$ labeled with a 0. For the allelopathic model (first
coordinate), if the tail is occupied by a 1 and the head is in state 0 then
the 1 gives birth through the arrow while if the tail is occupied by a 1 and
the head is in state 2 then the 2 dies. For the grass-bush-tree system (second
coordinate), if the tail is occupied by a 1 and the head is in state 0 or 2
then the 1 gives birth through the arrow.

  * •

Place an exponential clock with rate $(\beta_{1}-\gamma)/N$ along each
directed edge $\vec{xy}$. Each time the clock rings, say at time $t$, draw an
arrow $(x,t)\to(y,t)$ labeled with a 1. For the allelopathic model, if the
tail is occupied by a 1 and the head is in state 0 then the 1 gives birth
through the arrow. For the grass-bush-tree system, if the tail is occupied by
a 1 and the head is in state 0 or 2 then the 1 gives birth through the arrow.

  * •

Place an exponential clock with rate $\beta_{2}/N$ along each directed edge
$\vec{xy}$. Each time the clock rings, say at time $t$, draw an arrow
$(x,t)\to(y,t)$ labeled with a 2. If the tail is occupied by a 2 and the head
is in state 0 then, for both the allelopathic model and the grass-bush-tree
system, the 2 gives birth through the arrow.

  * •

Finally, place an exponential clock with rate one at each vertex $x$. Each
time the clock rings, say at time $t$, put a cross at $(x,t)$ to indicate
that, for both the allelopathic model and the grass-bush-tree system, site $x$
becomes empty.

Note that type 0 arrows create more 0s in the allelopathic model and more 1s
in the grass-bush-tree system. In particular, starting both processes from the
same initial configuration and using the graphical representation above, one
expects that, for all $(x,t)\in\mathbb{Z}^{d}\times\mathbb{R}_{+}$,

$\xi_{t}(x)=1\ \Longrightarrow\
\zeta_{t}(x)=1\qquad\hbox{and}\qquad\xi_{t}(x)=2\ \Longleftarrow\
\zeta_{t}(x)=2.$ (19)

To prove this result, it suffices to prove that the set of states

$S=\\{(0,0),(0,1),(1,1),(2,0),(2,1),(2,2)\\}$

is closed under the dynamics of the coupling $(\xi_{t},\zeta_{t})$. Regardless
of the state of the coupling, a cross will turn this state into $(0,0)\in S$
so the crosses do not create new states not in $S$. Similarly, checking the
effect of each type of arrow on all possible configurations, we get

$\begin{array}[]{cccccccccccccc}&&\hbox{before}&\hbox{after}&&&&\hbox{before}&\hbox{after}&&&&\hbox{before}&\hbox{after}\\\
(0,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(0,0)&(0,1)&&(0,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(0,0)&(0,1)&&(2,0)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(0,0)&(2,0)\\\ (0,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,0)&(2,1)&&(0,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(2,0)&(2,1)&&(2,0)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(0,1)&(2,1)\\\ (0,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,2)&(2,1)&&(0,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(2,2)&(2,1)&&(2,1)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(0,0)&(2,0)\\\ (1,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(0,0)&(1,1)&&(1,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(0,0)&(1,1)&&(2,1)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(0,1)&(2,1)\\\ (1,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(0,1)&(1,1)&&(1,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(0,1)&(1,1)&&(2,2)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(0,0)&(2,2)\\\ (1,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,0)&(0,1)&&(1,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(2,0)&(2,1)&&(2,2)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(0,1)&(2,1)\\\ (1,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,1)&(0,1)&&(1,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(2,2)&(2,1)&&(2,2)&\overset{\hbox{\tiny
2}}{\longrightarrow}&(2,0)&(2,2)\\\ (1,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,2)&(0,1)&&(2,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(0,0)&(0,1)\\\ (2,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(0,0)&(0,1)&&(2,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(2,0)&(2,1)\\\ (2,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,0)&(2,1)&&(2,1)&\overset{\hbox{\tiny
1}}{\longrightarrow}&(2,2)&(2,1)\\\ (2,1)&\overset{\hbox{\tiny
0}}{\longrightarrow}&(2,2)&(2,1)\end{array}$

Because $\operatorname{card}(S)=6$, there are 36 possible pairs of states at
the tail and head of the arrows before an interaction but we have only listed
the pairs for which an update indeed happens, i.e., the states at the head of
the arrow before and after the interaction are different. Note that all the
states generated by the coupling remain in $S$ therefore (19) holds. In
addition, Durrett and Swindle [10] proved the following coexistence result for
the grass-bush-tree system:

$\begin{array}[]{rcl}\beta_{2}>\beta_{1}^{2}>1\quad\hbox{and}\quad M\
\hbox{large}&\Longrightarrow&\hbox{coexistence}.\end{array}$ (20)

In particular, Theorem 5 follows from (20) and the second implication in (19)
which states that the 2s in the allelopathic model dominate the 2s in the
grass-bush-tree system.

##  7 Proof of Theorem 3

This section is devoted to the proof of Theorem 3 which states that, at least
in $d\geq 3$, when both species are equally fit, even weak inhibitory effects
drive the susceptible species to extinction. The proof is mostly based on
duality techniques even though, strictly speaking, the allelopathic model does
not have a tractable dual process. The first step is to describe the dual
processes of the contact process and the symmetric multitype contact process.
To begin with, we think of the basic contact process with birth parameter
$\beta$ as being generated by the graphical representation that has unlabeled
birth arrows from each site to each of their neighbors at rate $\beta/N$ and
crosses/death marks at each site at rate one. Given a realization of this
graphical representation and two times $0<s<t$, we say that there is

a path $(y,t-s)\uparrow(x,t)$ or a dual path $(x,t)\downarrow(y,t-s)$

if we can go from one space-time point to the other one by moving
forward/backward in time in the graphical representation, crossing the birth
arrows in their direction/opposite of their direction, and avoiding the death
marks. More precisely, there are sites and times

$y=x_{1},x_{2},\ldots,x_{n}=x\quad\hbox{and}\quad
t-s=s_{0}<s_{1}<\cdots<s_{n}=t$

such that the following two conditions hold:

  * •

there is a birth arrow $(x_{i},s_{i})\to(x_{i+1},s_{i})$ for all
$i=1,2,\ldots,n-1$ and

  * •

the segments $\\{x_{i}\\}\times(s_{i-1},s_{i})$ are void of death marks
$\times$ for $i=1,2,\ldots,n$.

The dual process starting at $(x,t)$ is the set-valued process

$\hat{\xi}_{s}^{(x,t)}=\\{y\in\mathbb{Z}^{d}:\hbox{there is a dual path}\
(x,t)\downarrow(y,t-s)\\}\quad\hbox{for all}\quad 0\leq s\leq t.$ (21)

By construction of the contact process from the graphical representation, a
space-time point $(x,t)$ is occupied if and only if it can be reached from a
path starting at $(y,0)$ for some $y$ that is initially occupied. This happens
if and only if at least one of the dual paths starting at $(x,t)$ lands on an
occupied site, which gives the duality relationship

$\begin{array}[]{rcl}x\in\xi_{t}&\Longleftrightarrow&\hat{\xi}_{t}^{(x,t)}\cap\xi_{0}\neq\varnothing\end{array}$
(22)

where we have identified the configuration of the process with the set of
occupied sites. In other words, the dual process keeps track of the potential
ancestors of $(x,t)$, and we refer to the left-hand side of Figure 8 for a
picture. Note that the dual process exhibits a tree structure with branchings
along the birth arrows. In our example, $(x,t)$ is occupied if and only if at
least one of the four sites in black at the bottom of the picture is occupied
at time zero.
We now give a description of the dual process of Neuhauser’s multitype contact
process. In the symmetric case $\beta_{1}=\beta_{2}=\beta$, the process can be
constructed from the same graphical representation as the basic contact
process assuming that both types of individuals can give birth through the
arrows. In particular, the duality relationship (22) still holds: space-time
point $(x,t)$ is occupied if and only if the dual process starting at this
point intersects at least one occupied site in the initial configuration.

$(x,t)$$(x,t)$13211123212311243321243

Figure 8:  Dual process of the contact process on the left and dual process of
the neutral multitype contact process along with the ancestor hierarchy on the
right. In the right picture, the black line keeps track of the distinguished
particle while the numbers show the ancestor hierarchy at various times.

However, when the intersection contains particles of different types, the type
at point $(x,t)$, as opposed to its occupancy, is unclear. Neuhauser [18]
proved that the dual process of the symmetric multitype contact process
consists of the dual process of the basic contact process along with an
ancestor hierarchy in which the members can be arranged according to the order
they determine the type of $(x,t)$. We refer to Neuhauser [18] for a
description of this ancestor hierarchy, and to Lanchier and Neuhauser [15] for
a more formal definition using a labeling of the tree structure and the
lexicographic order. The right-hand side of Figure 8 shows the ancestor
hierarchy at various times. If the site labeled 1 at the bottom of the picture
is initially occupied then $(x,t)$ is of the same type as this site, if the
site labeled 1 is initially empty but the site labeled 2 is initially occupied
then $(x,t)$ is of the same type as this site, and so on. In general, $(x,t)$
is of the same type as the ancestor with the smallest index that is initially
occupied.
The first ancestor in the hierarchy plays a key role in the analysis of the
multitype contact process and the allelopathic model. This ancestor is also
called the distinguished particle, and its position at time $t-s$ = dual time
$s$ is denoted by $\hat{\xi}_{s}^{(x,t)}(1)$. Roughly speaking, the
distinguished particle starts at site $x$ and jumps to the lowest existing
branch of the tree structure each time it meets a death mark going backward in
time (see the black path in Figure 8). To study the evolution of the
distinguished particle, it is convenient to assume that the exponential clocks
in the graphical representation are defined for negative times so the process
(21) is well-defined for all $s>0$. Then, we say that $(x,t)$ lives forever if
the dual process starting at this point is nonempty for all times. Each time
the distinguished particle jumps onto a point that lives forever, this point
is called a renewal point. Because the distinguished particle is trapped in
the (infinite) subtree starting from a renewal point, its evolution before and
after a renewal point are determined by disjoint parts of the graphical
representation. In particular, the space-time displacements between
consecutive renewal points are independent and identically distributed.
Neuhauser also proved that these space-time displacements have exponentially
decaying tails from which she deduced that, like simple symmetric random
walks, the process that keeps track of the distinguished particle is recurrent
in one and two dimensions and transient in higher dimensions.
To study the allelopathic model when the two species are equally fit, we
construct the process using the graphical representation of the neutral
multitype contact process that has birth arrows and death marks, and also add
like in Figure 6 type 0 arrows, that we call kill arrows from now on, from
each site to each of their neighbors at rate $\gamma/N$. The key to proving
the theorem is to show that, when $(x,t)$ lives forever, there is a sequence
of distinguished particles that start at the tail of kill arrows pointing at
the first ancestor of $(x,t)$, live forever and never coalesce. To begin with,
we follow the path of the distinguished particle starting at $(x,t)$ going
backward in time, let $\sigma_{i}$ be the $i$th time we cross the head of a
birth arrow, let $x_{i}$ be the location of the tail of this arrow, and let
$y_{i}$ be the location of the head of this arrow. Then, we define the events

$\begin{array}[]{rcl}A_{i}&=&\hbox{there is one kill arrow\leavevmode\nobreak\
$x_{i}\to y_{i}$ in the time interval\leavevmode\nobreak\
$(\sigma_{i}-1,\sigma_{i})$ and no}\\\ &&\hbox{crosses at or other arrows
pointing at one of those two sites in this time interval}.\end{array}$

In case $A_{i}$ occurs, we let $\tau_{i}\in(\sigma_{i}-1,\sigma_{i})$ be the
time at which the kill arrow appears.

###### Lemma 8

Assume that $\gamma>0$. Then, $P(\limsup_{i\to\infty}A_{i})=1$.

###### Proof.

Looking at the rates of the exponential clocks in the graphical representation
and using that these exponential clocks are independent, we get

$\begin{array}[]{rcl}P(A_{i})&=&P(\operatorname{Poisson}(\gamma/N)=1)\times
P(\operatorname{Poisson}(2(\gamma+\beta+1)-\gamma/N)=0)\vspace*{4pt}\\\
&=&(\gamma/N)\,\exp(-2(\gamma+\beta+1))>0.\end{array}$ (23)

In addition, whenever $|\sigma_{j}-\sigma_{i}|>1$, the two events $A_{i}$ and
$A_{j}$ are independent, thus showing the existence of a subsequence of events
that are independent and have a positive probability (23).

0$\sigma_{i}$$\tau_{i}$$I_{3}$$I_{1}$$I_{2}$$I_{4}$$I_{5}$0$\zeta_{\sigma_{i}-2K-1}^{i}$$\zeta_{\sigma_{i}-2K-1}^{0}$$x_{i}$$y_{i}$

Figure 9:  Picture of the event $A_{i}$ on the left and of the event
$A_{i}\cap C_{i}$ on the right.

In particular, the result follows from the second Borel-Cantelli lemma. ∎

In view of Lemma 8, there exists a subsequence $A_{i_{k}}$ of events that all
occur. We again call this subsequence $A_{i}$ to avoid cumbersome notation.
Note that, whenever $(x_{i},\sigma_{i})$ is of type 1, this 1 must have killed
earlier any potential 2 at $(y_{i},\tau_{i})$ thanks to the presence of the
kill arrow therefore $(y_{i},\sigma_{i})$ is of type 1 as well. See the left-
hand side of Figure 9 for an illustration where as previously black means type
1 and gray means type 2. Because point $(y_{i},\sigma_{i})$ is on the path of
the distinguished particle starting at $(x,t)$, this implies that $(x,t)$ also
is of type 1. In particular, to prove the theorem, it suffices to show that,
whenever $d\geq 3$,

$\hbox{$(x_{i},\sigma_{i})$ is of type\leavevmode\nobreak\ 1}\quad\hbox{for
some}\quad i\in\mathbb{N}.$ (24)

Now, define the new collection of events

$\begin{array}[]{rcl}B_{i}&=&\hbox{the space-time point\leavevmode\nobreak\
$(x_{i},\sigma_{i})$ lives forever}.\end{array}$

###### Lemma 9

Assume that $\beta>\beta_{c}$. Then, $P(\limsup_{i\to\infty}B_{i})=1$.

###### Proof.

To begin with, note that, by self-duality of the contact process, the
probability of $B_{i}$ is equal to the probability of survival of the contact
process with parameter $\beta$, which is positive because $\beta>\beta_{c}$.
Now, assume that $B_{i}$ does not occur. Letting $j$ be the smallest integer
larger than $i$ such that the dual process starting at $(x_{i},\sigma_{i})$
dies out in less than $\sigma_{i}-\sigma_{j}$ units of time, $B_{j}$ is
determined by parts of the graphical representation that are disjoint from the
parts of the graphical representation that determine the dual process starting
at $(x_{i},\sigma_{i})$. Because disjoint parts of the graphical
representation are independent, we deduce that

$P(B_{j}\,|\,B_{i}^{c})=P(B_{j})=P(\hat{\xi}_{s}^{(x_{j},\sigma_{j})}\neq\varnothing\
\hbox{for all}\ s)>0.$

Using also that the events $B_{i}$ are positively correlated implies that,
with probability one, infinitely many of the events $B_{i}$ occur, which
proves the lemma. ∎

The lemma implies that there exists a subsequence $A_{i_{k}}\cap B_{i_{k}}$ of
events that all occur. As previously, we call this subsequence $A_{i}\cap
B_{i}$ to simplify the notation. Because the $(x_{i},\sigma_{i})$ live
forever, the distinguished particles starting from those points also live
forever, and the next step is to study how these particles interact. Because
these particles start at different times,

$\xi_{s}^{(x_{i},\sigma_{i})}(1)\quad\hbox{and}\quad\xi_{s}^{(x_{j},\sigma_{j})}(1)\quad\hbox{for}\quad
i\neq j$

do not refer to positions of different distinguished particles at the same
time so, to facilitate the comparison, we reindex the processes using the
actual time instead of the dual time. To also include the distinguished
particle starting at $(x,t)$ in the list, we let

$\zeta_{s}^{i}=\hat{\xi}_{\sigma_{i}-s}^{(x_{i},\sigma_{i})}(1)\quad\hbox{for
all}\quad i\geq 0\quad\hbox{where}\quad(x_{0},\sigma_{0})=(x,t).$

The process $\zeta_{s}^{i}$ is well-defined for all $i\leq\sigma_{i}$,
including negative $s$ since the graphical representation has been extended to
negative times. Now, observe that if the $i$th distinguished particle lands on
a 1 at time zero, meaning that $\xi_{0}(\zeta_{0}^{i})=1$, then space-time
point $(x_{i},\sigma_{i})$ must be of type 1. In particular, to prove (24), it
suffices to show that

$\xi_{0}(\zeta_{0}^{i})=1\quad\hbox{for some}\quad i\in\mathbb{N}.$ (25)

To prove this, the idea is to use transience in dimensions $d\geq 3$ to deduce
that an arbitrarily large number of distinguished particles do not coalesce.
More precisely, we will use the next result which follows from the proof of
Neuhauser [18, Lemma 5.5].

###### Lemma 10

Let $d\geq 3$. There exists $C>0$ such that, for all $K$ large,

$\begin{array}[]{rcl}|\\!|y-z|\\!|\geq
K&\Longrightarrow&P(|\\!|\hat{\xi}_{s}^{(y,t)}(1)-\hat{\xi}_{s}^{(z,t)}(1)|\\!|<s^{1/8}\
\hbox{for some}\ s)\leq CK^{-1/10}+2CK^{-3/32}.\end{array}$

Motivated by this lemma, we define the events

$\begin{array}[]{rcl}C_{i}&=&\hbox{the norm\leavevmode\nobreak\
$|\\!|\zeta_{\sigma_{i}-2K-1}^{i}-\zeta_{\sigma_{i}-2K-1}^{0}|\\!|$ is larger
than\leavevmode\nobreak\ $K$ but less than\leavevmode\nobreak\
$2K$}.\end{array}$

###### Lemma 11

For all $K>0$, we have $P(\limsup_{i\to\infty}C_{i})=1$.

###### Proof.

Subdividing $(\sigma_{i}-2K-1,\sigma_{i})$ into $2K+1$ intervals
$I_{1},I_{2},\ldots,I_{2K+1}$ of length one going backward in time, the
probability of $C_{i}$ is bounded from below by the probability that

  * •

the distinguished particle $\zeta_{s}^{0}$ does not jump between time
$\sigma_{i}$ and time $\sigma_{i}-2K-1$,

  * •

there are no crosses at and a single birth arrow pointing at $\zeta_{s}^{i}$
whose tail is further away from the distinguished particle in the intervals
$I_{2},I_{4},\ldots,I_{2K}$,

  * •

there are no birth arrows pointing at and one cross at $\zeta_{s}^{i}$ in the
intervals $I_{3},I_{5},\ldots,I_{2K+1}$.

See the right-hand side of Figure 9 for a picture of this more particular
event. A direct calculation using the rate of occurrence of the birth arrows
and crosses gives

$\begin{array}[]{rcl}P(C_{i})&\geq&P(\operatorname{Poisson}((2K+1)(\beta+1))=0)\vspace*{4pt}\\\
&&\hskip 10.0pt\times\,(P(\operatorname{Poisson}(\beta/N)=1)\times
P(\operatorname{Poisson}(\beta+1-\beta/N)=0))^{K}\vspace*{4pt}\\\ &&\hskip
10.0pt\times\,(P(\operatorname{Poisson}(1)=1)\times
P(\operatorname{Poisson}(\beta)=0))^{K}\vspace*{4pt}\\\
&=&(\beta/N)^{K}\,\exp(-(4K+1)(\beta+1))>0.\end{array}$ (26)

Note also that, when $|\sigma_{j}-\sigma_{i}|>2K+1$, the two events $A_{i}$
and $A_{j}$ are determined by disjoint parts of the graphical representation
and so are independent. In particular, there exists a subsequence of events
that are independent and have a positive probability (26) therefore, like in
the proof of Lemma 8, we can conclude using the second Borel-Cantelli lemma. ∎

Lemmas 8, 9 and 11 imply the existence of a sequence of space-time points,
that we again denote by $(x_{i},\sigma_{i})$ to simplify the notation, so that
the events $A_{i}\cap B_{i}\cap C_{i}$ occur. This, together with Lemma 10,
implies that the number of distinguished particles $\zeta_{s}^{i}$ that do not
coalesce can be made arbitrarily large. More precisely, we have the following
lemma.

###### Lemma 12

Let $d\geq 3$. Then, for all integers $n>0$,

$\begin{array}[]{l}\lim_{s\to-\infty}\inf_{0\leq k<l\leq
n}|\\!|\zeta_{s}^{i_{k}}-\zeta_{s}^{i_{l}}|\\!|=\infty\quad\hbox{for
some}\quad 0=i_{0}<i_{1}<i_{2}<\ldots<i_{n}.\end{array}$

###### Proof.

We prove the result by induction. Assuming that the statement of the lemma is
true, for all integers $K>0$, there exists a time $s_{K}>-\infty$ such that

$\begin{array}[]{l}|\\!|\zeta_{s}^{i_{k}}-\zeta_{s}^{i_{l}}|\\!|\geq
4K\quad\hbox{for all}\quad 0\leq k<l\leq n\quad\hbox{and}\quad
s<s_{K}.\end{array}$

In particular, for all $j$ large enough that $\sigma_{j}<s_{K}$, because the
event $C_{j}$ occurs,

$|\\!|\zeta_{\sigma_{j}-2K-1}^{j}-\zeta_{\sigma_{j}-2K-1}^{i_{k}}|\\!|\geq
K\quad\hbox{for all}\quad 0\leq k\leq n$

therefore it follows from Lemma 10 that the probability that the distinguished
particle $\zeta_{s}^{j}$ does not coalesce with any of the other $n+1$
distinguished particles is larger than

$\begin{array}[]{l}P(\lim_{s\to-\infty}|\\!|\zeta_{s}^{j}-\zeta_{s}^{i_{k}}|\\!|=\infty\
\hbox{for all}\ 0\leq k\leq n)\vspace*{4pt}\\\ \hskip
25.0pt=1-P(\lim_{s\to-\infty}|\\!|\zeta_{s}^{j}-\zeta_{s}^{i_{k}}|\\!|\neq\infty\
\hbox{for some}\ 0\leq k\leq n)\vspace*{4pt}\\\ \hskip 25.0pt\geq
1-(n+1)CK^{-1/10}-2(n+1)CK^{-3/32}\end{array}$

which can be made positive by choosing $K$ sufficiently large. In particular,
after a geometric number of trials, we find a particle $\zeta_{s}^{j}$ whose
distance from the other $n+1$ particles goes to infinity. This shows that the
statement of the lemma still holds replacing $n$ with $n+1$. ∎

To complete the proof of the theorem, we let

$\Theta_{t}=\\{\zeta_{0}^{i}:i\geq 0\ \hbox{and}\ \sigma_{i}>0\\}$

be the set of sites occupied by the distinguished particles of the
$(x_{i},\sigma_{i})$ that are contained in the positive time half-space.
According to Lemma 12, this set can be made arbitrarily large by choosing $t$
large enough. In particular, starting from a translation invariant
distribution with a positive density of particles of type 1, it follows from
Harris [13, Lemma 9.14] that

$\begin{array}[]{l}\lim_{t\to\infty}P(\Theta_{t}\cap\\{x:\xi_{0}(x)=1\\}\neq\varnothing)=1.\end{array}$

This shows that (25) occurs with probability one in the $t\to\infty$ limit
which, in turn, implies that (24) occurs with probability one in the limit,
and proves the theorem.

## References

  * [1] C. Bezuidenhout and G. Grimmett.  The critical contact process dies out.  Ann. Probab., 18(4):1462–1482, 1990.
  * [2] C. Bezuidenhout and G. Grimmett.  Exponential decay for subcritical contact and percolation processes.  Ann. Probab., 19(3):984–1009, 1991.
  * [3] M. Bramson and R. Durrett.  A simple proof of the stability criterion of Gray and Griffeath.  Probab. Theory Related Fields, 80(2):293–298, 1988.
  * [4] R. Durrett.  Oriented percolation in two dimensions.  Ann. Probab., 12(4):999–1040, 1984.
  * [5] R. Durrett.  Multicolor particle systems with large threshold and range.  J. Theoret. Probab., 5(1):127–152, 1992.
  * [6] R. Durrett.  Ten lectures on particle systems.  In Lectures on probability theory (Saint-Flour, 1993), volume 1608 of Lecture Notes in Math., pages 97–201. Springer, Berlin, 1995\.
  * [7] R. Durrett and S. A. Levin.  Allelopathy in spatially distributed populations.  Journal of theoretical biology, 185(2):165–171, 1997.
  * [8] R. Durrett and C. Neuhauser.  Coexistence results for some competition models.  Ann. Appl. Probab., 7(1):10–45, 1997.
  * [9] R. Durrett and R. Schinazi.  Asymptotic critical value for a competition model.  Ann. Appl. Probab., 3(4):1047–1066, 1993.
  * [10] R. Durrett and G. Swindle.  Are there bushes in a forest?  Stochastic Process. Appl., 37(1):19–31, 1991.
  * [11] T. E. Harris.  Nearest-neighbor Markov interaction processes on multidimensional lattices.  Advances in Math., 9:66–89, 1972.
  * [12] T. E. Harris.  Contact interactions on a lattice.  Ann. Probability, 2:969–988, 1974.
  * [13] T. E. Harris.  On a class of set-valued Markov processes.  Ann. Probability, 4(2):175–194, 1976.
  * [14] N. Lanchier.  A multitype contact process with frozen sites: a spatial model of allelopathy.  J. Appl. Probab., 42(4):1109–1119, 2005.
  * [15] N. Lanchier and C. Neuhauser.  A spatially explicit model for competition among specialists and generalists in a heterogeneous environment.  Ann. Appl. Probab., 16(3):1385–1410, 2006.
  * [16] T. M. Liggett.  Interacting particle systems, volume 276 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, New York, 1985.
  * [17] T. M. Liggett.  Stochastic interacting systems: contact, voter and exclusion processes, volume 324 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1999.
  * [18] C. Neuhauser.  Ergodic theorems for the multitype contact process.  Probab. Theory Related Fields, 91(3-4):467–506, 1992.


B_{k+1}^{\mathbf{x}})\cap\bar{B}_{\mathbf{x}}$. Otherwise, stop the steps.
Then, let $I^{\prime}$ denote the final step number. Herein, we see Equation
(158) holds for
$\mathcal{O}_{\mathbf{x}}=\\{B_{i}^{\mathbf{x}}\\}_{i=1}^{I^{\prime}}$ and
$\\{\mathbf{b}_{k}^{\mathbf{x}}\\}_{i=1}^{I^{\prime}-1}$ obtained through the
above steps.

Now, from (153) and (155), we have

$\displaystyle\left|e^{-T(\mathbf{x})}-e^{-T^{*}(\mathbf{x})}\right|$
$\displaystyle\leq$
$\displaystyle\left|e^{-T(\mathbf{x})}-e^{-T(\mathbf{X}_{(1)}(\mathbf{x}))}\right|+\left|e^{-T(\mathbf{X}_{(1)}(\mathbf{x}))}-e^{-T^{*}(\mathbf{x})}\right|$
$\displaystyle=$
$\displaystyle\left|e^{-T(\mathbf{x})}-e^{-T(\mathbf{X}_{(1)}(\mathbf{x}))}\right|+\left|e^{-T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))}-e^{-T^{*}(\mathbf{x})}\right|\qquad\qquad\therefore\text{(}\ref{Eq_Appendixtheoremsamplerequirement_2}\text{)}$
$\displaystyle\leq$
$\displaystyle\max\left\\{e^{-T(\mathbf{x})},e^{-T(\mathbf{X}_{(1)}(\mathbf{x}))}\right\\}\cdot\left|T(\mathbf{x})-T(\mathbf{X}_{(1)}(\mathbf{x}))\right|$
$\displaystyle\quad+\
\max\left\\{e^{-T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))}+e^{-T^{*}(\mathbf{x})}\right\\}\cdot\left|T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))-T^{*}(\mathbf{x})\right|\qquad\qquad\therefore\text{(}\ref{Eq_Appendixtheoremsamplerequirement_1}\text{)}$
$\displaystyle\leq$ $\displaystyle
C_{1}\cdot\left|T(\mathbf{x})-T(\mathbf{X}_{(1)}(\mathbf{x}))\right|$
$\displaystyle\quad+\
C_{2}\cdot\left|T^{*}(\mathbf{x})-T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))\right|$
$\displaystyle\leq$ $\displaystyle
C_{1}\cdot\left|T(\mathbf{x})-T(\mathbf{b}_{1}^{\mathbf{x}})\right|$
$\displaystyle\quad+\
C_{1}\cdot\sum_{i=1}^{I^{\prime}-2}\left|T(\mathbf{b}_{i}^{\mathbf{x}})-T(\mathbf{b}_{i+1}^{\mathbf{x}})\right|+C_{1}\cdot\left|T(\mathbf{b}_{I^{\prime}-1}^{\mathbf{x}})-T(\mathbf{X}_{(1)}(\mathbf{x}))\right|$
$\displaystyle\quad+\
C_{2}\cdot\left|T^{*}(\mathbf{x})-T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))\right|.$

Here, the equality in (B.5.2) is obtained because of the fact that
$\mathbf{X}_{(1)}(\mathbf{x})\in\mathbf{X}^{(K)}$ and $T(x)$ is the same as
$T^{*}(x)$ if $x\in\mathbf{X}^{(K)}$, which follows from (153).

Then, from the locally Lipschitz continuity of $T$ for each
$B_{k}^{\mathbf{x}}$ and the Lipschitz continuity of $T^{*}$, we obtain

$\displaystyle\left|e^{-T(\mathbf{x})}-e^{-T^{*}(\mathbf{x})}\right|$
$\displaystyle\leq$ $\displaystyle C_{1}\cdot
L_{1}\left|\mathbf{x}-\mathbf{b}_{1}^{\mathbf{x}}\right|$ (161)
$\displaystyle\quad+\ C_{1}\cdot
L_{1}\cdot\sum_{i=1}^{I^{\prime}-2}\left|\mathbf{b}_{i}^{\mathbf{x}}-\mathbf{b}_{i+1}^{\mathbf{x}}\right|+C_{1}\cdot
L_{1}\cdot\left|\mathbf{b}_{I^{\prime}-1}^{\mathbf{x}}-\mathbf{X}_{(1)}(\mathbf{x})\right|$
$\displaystyle\quad+\ C_{2}\cdot
L_{2}\cdot\left|\mathbf{x}-\mathbf{X}_{(1)}(\mathbf{x})\right|$
$\displaystyle\leq$ $\displaystyle C_{1}\cdot
L_{1}\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$
$\displaystyle\quad+\ C_{1}\cdot
L_{1}\cdot(I^{\prime}-2)\cdot\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|+C_{1}\cdot
L_{1}\cdot\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$
$\displaystyle\quad+\ C_{2}\cdot
L_{2}\cdot\left|\mathbf{x}-\mathbf{X}_{(1)}(\mathbf{x})\right|\quad\qquad\text{(}\therefore\\{\mathbf{b}_{i}^{\mathbf{x}}\\}_{i=1}^{I^{\prime}}\subset\bar{B}_{\mathbf{x}}\text{
\&
}\mathrm{diag}(\bar{B}_{\mathbf{x}})=\|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\|\text{)}$
$\displaystyle\leq$ $\displaystyle I^{\prime}\cdot C_{1}\cdot
L_{1}\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$
$\displaystyle\quad+\ C_{2}\cdot
L_{2}\cdot\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$
$\displaystyle=$ $\displaystyle\left(I^{\prime}\cdot C_{1}\cdot
L_{1}+C_{2}\cdot
L_{2}\right)\cdot\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$
$\displaystyle\leq$ $\displaystyle\left(I\cdot C_{1}\cdot L_{1}+C_{2}\cdot
L_{2}\right)\cdot\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|.$

By integrating both sides of (161) with $P$, we have

$\displaystyle E_{P}\left|e^{-T(\mathbf{x})}-e^{-T^{*}(\mathbf{x})}\right|$
$\displaystyle\leq$ $\displaystyle\left(I\cdot C_{1}\cdot L_{1}+C_{2}\cdot
L_{2}\right)\cdot E_{P}\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$
(162) $\displaystyle\leq$ $\displaystyle\left(I\cdot C_{1}\cdot
L_{1}+C_{2}\cdot
L_{2}\right)\cdot\mathrm{diag}(\Omega)\cdot\left(\frac{1}{K+1}\right)^{\frac{1}{d}}\qquad\qquad\therefore\
\text{Lemma}\ \ref{lemma_evaluating_nearestneibordistsq}$ $\displaystyle=$
$\displaystyle O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right).$

Here, we proved that the case ($\mathrm{i}$).

Next, we show the case ($\mathrm{ii}$). Let $T_{0}$ be a function such that
$T_{0}(\mathbf{x})$ is the same as $\log dQ/dP(\mathbf{X}_{i})$ where
$\mathbf{X}_{i}$ is the nearest variable to $\mathbf{x}$ in
$\mathbf{X}^{(K)}$. Specifically, let $V_{i}=\\{\mathbf{y}\in\Omega\ |\
\|\mathbf{X}_{i}-\mathbf{y}\|=\|\mathbf{X}_{(1)}(\mathbf{y})-\mathbf{y}\|\\}$.
Notably, $V_{i}\cap V_{j}=\phi$, and
$P(\Omega\setminus\sqcup_{i=1}^{K}V_{i})=0$ becasue $P$ is absolute continuous
to the Lebesgue measure.

Subsequently,

$\displaystyle
T_{0}(\mathbf{x})=\left\\{\begin{array}[]{ll}-\log\frac{dQ}{dP}(\mathbf{X}_{i})&\quad\text{if}\quad\mathbf{x}\in
V_{i},\\\ 0&\quad\text{otherwise.}\end{array}\right.$ (165)

Then, it holds that

$\displaystyle T_{0}(\mathbf{y})-T_{0}(\mathbf{x})=0\qquad\quad\text{for any}\
\ \ \mathbf{x},\mathbf{y}\in V_{i}.$ (166)

Thus, $T_{0}$ is locally Lipschitz continuous $P$-almost everywhere on $P$.

Let $L_{3}$ be a bi-Lipschitz constant for $T^{*}$:
$\frac{1}{L_{3}}\cdot\|\mathbf{x}-\mathbf{y}\|\leq\|T^{*}(\mathbf{x})-T^{*}(\mathbf{y})\|\leq
L_{3}\cdot\|\mathbf{x}-\mathbf{y}\|$. In addition, let $C_{3}>0$ be a positive
constant such that $e^{-T_{*}(\mathbf{x})}\geq C_{3}$ on $\Omega$. $C_{3}$
exists because $T^{*}(\mathbf{x})$ is Lipschitz continuous on the compact set
$\Omega$. Subsequently, we see

$\displaystyle\left|e^{-T_{0}(\mathbf{x})}-e^{-T^{*}(\mathbf{x})}\right|$
$\displaystyle\geq$
$\displaystyle\left|e^{-T_{0}(\mathbf{X}_{(1)}(\mathbf{x}))}-e^{-T^{*}(\mathbf{x})}\right|-\left|e^{-T_{0}(\mathbf{X}_{(1)}(\mathbf{x}))}-e^{-T_{0}(\mathbf{x})}\right|$
$\displaystyle=$
$\displaystyle\left|e^{-T_{0}(\mathbf{X}_{(1)}(\mathbf{x}))}-e^{-T^{*}(\mathbf{x})}\right|\qquad\qquad\therefore\text{(}\ref{Eq_Appendixtheoremsamplerequirement_5}\text{)}$
$\displaystyle=$
$\displaystyle\left|e^{-T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))}-e^{-T^{*}(\mathbf{x})}\right|\qquad\qquad\therefore\text{(}\ref{Eq_Appendixtheoremsamplerequirement_2}\text{)}$
$\displaystyle\geq$
$\displaystyle\min\left\\{e^{-T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))},e^{-T^{*}(\mathbf{x})}\right\\}\cdot\left|T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))-T^{*}(\mathbf{x})\right|\qquad\qquad\therefore\text{(}\ref{Eq_Appendixtheoremsamplerequirement_6}\text{)}$
$\displaystyle\geq$ $\displaystyle
C_{3}\cdot\left|T^{*}(\mathbf{X}_{(1)}(\mathbf{x}))-T^{*}(\mathbf{x})\right|$
$\displaystyle\geq$ $\displaystyle
C_{3}\cdot\frac{1}{L_{3}}\cdot\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|.$
(168)

Here, the equality in (B.5.2) is obtained because of the fact that
$\mathbf{X}_{(1)}(\mathbf{x})\in\mathbf{X}^{(K)}$ and $T_{0}(x)$ is the same
as $T^{*}(x)$ if $x\in\mathbf{X}^{(K)}$, which follows from (153).

By integrating (168) on both sides with $P$, we have

$\displaystyle E_{P}\left|e^{-T(\mathbf{x})}-e^{-T^{*}(\mathbf{x})}\right|$
$\displaystyle\geq$ $\displaystyle C_{3}\cdot\frac{1}{L_{3}}\cdot
E_{P}\left|\mathbf{X}_{(1)}(\mathbf{x})-\mathbf{x}\right|$ (169)
$\displaystyle=$ $\displaystyle
C_{3}\cdot\frac{1}{L_{3}}\cdot\mathrm{diag}(\Omega)\cdot
O\left(\frac{1}{K+1}\right)^{\frac{1}{d}}\qquad\qquad\therefore\ \text{Lemma}\
\ref{lemma_evaluating_nearestneibordistsq}$ $\displaystyle=$ $\displaystyle
O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right).$

From (162) and (169), we proved that the case ($\mathrm{ii}$).

Thus, the proof is complete. ∎

######  Corollary B.16.

Assuming the same assumption as in Theorem B.15. Then, the statement of
Theorem 7.1 still holds even if $T$ differs within $Op(1/\sqrt{K})$. That is,
($\mathrm{i}$) if $r(\mathbf{x})=p(\mathbf{x})/q(\mathbf{x})$ is Lipschitz
continuous on $\Omega$, then Equation (170) holds for any $T^{\prime}$ with
$T^{\prime}=T+Op(1/\sqrt{K})$ where $\exists
T\in\mathcal{T}_{\text{Lip}}^{(K)}$ such that

$E_{P}\left|\widehat{\frac{q(\mathbf{x})}{p(\mathbf{x})}}^{T^{\prime}}-\
\frac{q(\mathbf{x})}{p(\mathbf{x})}\right|\leq
O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right).$ (170)

In addition, ($\mathrm{ii}$) if $r(\mathbf{x})=p(\mathbf{x})/q(\mathbf{x})$ is
bi-Lipschitz continuous on $\Omega$, then there exists
$T_{0}\in\mathcal{T}_{\text{Lip}}^{(K)}$ such that for any $T^{\prime}$ with
$T^{\prime}=T_{0}+Op(1/\sqrt{K})$,

$E_{P}\left|\widehat{\frac{q(\mathbf{x})}{p(\mathbf{x})}}^{T^{\prime}}-\
\frac{q(\mathbf{x})}{p(\mathbf{x})}\right|=O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right).$
(171)

###### proof of Corollary B.16.

First, note that $e^{T+O_{p}(a_{n})}=e^{T}+O_{p}(a_{n})$ for any random
variable $T$, where $\\{a_{n}\\}_{i=1}^{\infty}$ is a sequence of positive
numbers with $\lim_{n\rightarrow\infty}a_{n}=0$. Thus,

$e^{-T^{\prime}}=e^{T+Op(1/\sqrt{K})}=e^{-T}+Op\left(\frac{1}{\sqrt{K}}\right).$
(172)

We now show the case ($\mathrm{i}$). Let $T^{*}=\log dQ/dP$. Then, from
Theorem 7.1, we see

$\displaystyle
E_{P}\left|\widehat{\frac{q(\mathbf{x})}{p(\mathbf{x})}}^{T^{\prime}}-\frac{q(\mathbf{x})}{p(\mathbf{x})}\right|$
$\displaystyle=$ $\displaystyle
E_{P}\left|e^{T(\mathbf{X})+O\left(1/\sqrt{K}\right)}-e^{-T^{*}(\mathbf{X})}\right|$
(173) $\displaystyle=$ $\displaystyle
E_{P}\left|e^{T(\mathbf{X})}+O_{p}\left(\frac{1}{\sqrt{K}}\right)-e^{-T^{*}(\mathbf{X})}\right|$
$\displaystyle=$ $\displaystyle
E_{P}\Bigg{|}e^{T(\mathbf{X})}-e^{-T^{*}(\mathbf{X})}\Bigg{|}+E_{P}\left|O_{p}\left(\frac{1}{\sqrt{K}}\right)\right|$
$\displaystyle\leq$ $\displaystyle
O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right)+O\left(\frac{1}{\sqrt{K}}\right)$
$\displaystyle=$ $\displaystyle
O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right).\qquad\qquad\qquad\therefore\
d\geq 2.$

Next, we show show the case ($\mathrm{ii}$).

$\displaystyle
E_{P}\left|\widehat{\frac{q(\mathbf{x})}{p(\mathbf{x})}}^{T^{\prime}}-\frac{q(\mathbf{x})}{p(\mathbf{x})}\right|$
$\displaystyle=$ $\displaystyle
E_{P}\left|e^{T_{0}(\mathbf{X})+O\left(1/\sqrt{K}\right)}-e^{-T^{*}(\mathbf{X})}\right|$
(174) $\displaystyle=$ $\displaystyle
E_{P}\left|e^{T_{0}(\mathbf{X})}+O_{p}\left(\frac{1}{\sqrt{K}}\right)-e^{-T^{*}(\mathbf{X})}\right|$
$\displaystyle\geq$ $\displaystyle
E_{P}\Bigg{|}e^{T_{0}(\mathbf{X})}-e^{-T^{*}(\mathbf{X})}\Bigg{|}-E_{P}\left|O_{p}\left(\frac{1}{\sqrt{K}}\right)\right|$
$\displaystyle=$ $\displaystyle
O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right)-O\left(\frac{1}{\sqrt{K}}\right)$
$\displaystyle=$ $\displaystyle
O\left(\left(\frac{1}{K}\right)^{\frac{1}{d}}\right).\qquad\qquad\qquad\therefore\
d\geq 2.$

Thus, the proof is complete. ∎

##  Appendix C Hyperparameter Settings used in the Experiments in Section 6

In this section, we provide the details of the hyperparameter settings used in
the experiments in Section 6.

###  C.1 Experimental Settings in Section 6.1

We used a 100×4 perceptron with ReLU activation. Adam algorithm in PyTorch and
NVIDIA T4 GPU were used for the training. For the hyperparameters in the
training, the learning rate was 0.001, bathSize was 2500, and the number of
epochs was 250.

###  C.2 Experimental Settings in Section 6.2

For KLIEP and uLSIF, the densratio library in R was used.
222https://cran.r-project.org/web/packages/densratio/index.html. The number of
kernels in KLIEP and uLSIF was 100, which was the default setting of the
methods in the R library. The other hyperparameters were tuned using the
methods in the R library with default settings.

For D3RE and $\alpha$-Div, we used 100×3 perceptrons with ReLU activation.
Adam algorithm in PyTorch and NVIDIA T4 GPU were used for the training.

For D3RE, the learning rate was 0.00005, bathSize 128, and the number of
epochs 250 for each dimension of the data, For $\alpha$-Div, the learning rate
was 0.0001, bathSize 128, and $\alpha$ was 0.5, for each dimension of the
data. When the data dimensions were 10, 20, 30 and 50, the numbers of epochs
for $\alpha$-Div was 250, and when the data dimension was 100, the number of
epochs was 300.