# Switching particle systems for foraging ants
showing phase transitions in path selections

Ayana Ezoe1, Saori Morimoto1, Yuya Tanaka1, Makoto Katori1 1111 Department of
Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-
ku, Tokyo 112-8551, Japan; e-mail<EMAIL_ADDRESS>
and Hiraku Nishimori2 2222 Meiji Institute for Advanced Study of Mathematical
Sciences, Meiji University, Nakano, Nakano-ku, Tokyo 164-8525, Japan; e-mail:
<EMAIL_ADDRESS>
(23 November 2023)

###### Abstract

Switching interacting particle systems studied in probability theory are the
stochastic processes of hopping particles on a lattice consisting of slow
particles and fast particles, where the switching between these two types of
particles happens randomly with a given transition rate. In the present paper,
we show that such stochastic processes of many particles are useful in
modeling group behaviors of ants. Recently the situation-depending switching
between two distinct types of primarily relied cues for ants in selecting
foraging paths has been experimentally studied by the research group of the
last author of this paper. We propose a discrete-time interacting random-walk
model on a square lattice, in which two kinds of hopping rules are included.
We report the numerical simulation results which exhibit the global changes in
selected homing paths from trailing paths of the ‘pheromone road’ to nearly-
optimal-paths depending on the switching parameters. By introducing two kinds
of order parameters characterizing the switching-parameter dependence of
homing-duration distributions, we discuss such global changes as phase
transitions realized in path selections of ants. Critical phenomena associated
with the continuous phase transitions are also studied.

Keywords: Switching interacting particle systems; Group behaviors of ants;
Switching of cues in foraging path selections; Interacting random-walk model;
Order parameters; Phase transitions and critical phenomena

##  1 Introduction

For the reaction-diffusion systems with double diffusivity, the interacting
particle systems in one dimension were introduced as the microscopic models,
in which we have two types of particles; fast particles and slow particles
[4]. The fast particle hops at rate 1 and the slow particles at rate
$\epsilon\in[0,1)$. The switching between these two types of particles happens
at rate $\gamma\in(0,\infty)$. An alternative view of the models is to
consider two layers of particles such that the hopping rate for the bottom
layer is 1, while that for the top layer is $\epsilon<1$, and to let particles
change layers at rate $\gamma$. It is interesting to know that a motivation to
study such switching particle systems came from population genetics [4].
Individuals live in colonies can behave as being either active or dormant.
Active individuals undergo clonal reproduction via resampling within a colony
and migrate between colonies by hopping around. They can become dormant and
then suspend resampling and migration until they become active again. Dormant
individuals reside in a seed bank. A useful review on the important roles of
dormancy in life sciences is given by [8]. In the switching interacting
particle systems the active (resp. dormant) individuals are represented by the
fast (resp. slow) particles. For other mathematical models and their analysis
for the population dynamics with seed-banks, see [2, 3, 5, 6].

The purpose of the present paper is to show the usefulness of the notion of
switching interacting particle systems for modeling the interesting group
behaviors of ants in foraging [7].

###  1.1 Switching particle systems modeling individual switching behaviors

Various species of ants make long foraging trails through a positive feedback
process of secretion and tracking of recruit pheromone, which enables them to
shuttle efficiently between the nest and food sources. A recent combinatorial
study of experiments and mathematical modeling for a species of garden ant
suggested, however, a situation-dependent switching of the primarily relied
cues from chemical ones (e.g. recruit pheromone) to other ones, mostly
probably visual cues [10].

In order to explain the sharp changes of path patterns of ant trails, which
were experimentally observed depending on geometrical setting of the nest and
the food source on a plane (see Section 1.2 below), Ogihara et al. proposed a
kind of multi-agent model [10]. Foraging ants are represented by biased random
walkers on a lattice, whose hopping probabilities to the nearest-neighboring
sites are weighted depending on the time-evolutionary field of pheromone. The
local intensity of pheromone field is increased by the secretion of pheromone
by exploring ants as well as by homing ants. (The pheromone field is decreased
by evaporation with a constant rate.) In addition to such a pheromone-
mediated-walk of ants, a visual-cues-mediated-walk is considered for the
homing ants, which tends to realize an optimal path to the nest. They
introduced a parameter $\alpha$ for each ant at each time, which measures the
conflict between the optimal path and the redundant path made by pheromone-
mediated-walk [10]. In order to make switching from the pheromone-mediated-
walk to the visual-cues-mediated-walk, they assumed a critical value
$\alpha_{\rm c}$, and when $\alpha>\alpha_{\rm c}$ the switching occurs and
the direction of the hopping is aligned toward the nest.

In the present paper, another mathematical model for foraging path selections
by ants is proposed following the idea of switching interacting particle
systems [4]. We consider a discrete-time interacting random-walk model of
hopping particles on a square lattice (a cellular automaton model) [1, 9]. The
exploring ants in the outward trip from the nest to the food source are
represented by random walkers in an exploring-period, while the homing ants in
the return trip from the food source to the nest are represented by random
walkers in a homing-period. We regard the exploring and homing ants performing
the pheromone-mediated-walk as the slow particles, and the homing ants
adopting the visual-cues-mediated-walk as the fast particles:

pheromone-mediated-walk $\displaystyle\iff\mbox{slow particle},$ visual-cues-
mediated-walk $\displaystyle\iff\mbox{fast particle}.$ (1.1)

In an exploring-period, all particles are slow, while in a homing-period,
switching transitions can occur from the slow particles to the fast particles,
which represent ‘pioneering ants’ establishing shorter paths to the nest. On
the other hand, the pioneering ants may have higher risk to encounter enemies
than the ants following the previous redundant path among many other ants. So
the fast particles should be switched back to the slow particles in order to
avoid such risk. The switching between the two types of particle behavior
occurs randomly at each time step as

$\displaystyle\mbox{slow particle $\Longrightarrow$ fast particle \quad with
probability $\gamma_{\rm sf}$},$ $\displaystyle\mbox{fast particle
$\Longrightarrow$ slow particle \quad with probability $\gamma_{\rm fs}$}.$
(1.2)

We will study the dependence of the global changes of homing-paths on the
parameters $\gamma_{\rm sf}$ and $\gamma_{\rm fs}$. Notice that, since here we
consider a discrete-time stochastic model, $\gamma_{\rm sf}$ and $\gamma_{\rm
fs}$ are not transition rates but transition probabilities taking values in
$[0,1]$.

###  1.2 Experiments suggesting phase transitions

Now we give a brief review of the experiments for the forage path selections
by ants reported by the research group of the last author of the present paper
(the Nishimori group). The following experiments showed the clear evidence of
the switching phenomena of primarily relied cues used by ants from the recruit
pheromone to the visual information [10].

They used Lasius Japonicus, a species of garden ant in Japan, raised in
plastic boxes of size $23\times 12\times 3.5$ cm equipped with special floors
and walls covered by plaster to keep the inner humidity. Each box contained
about 200–300 of ants and it is covered by a black plate to protect ants from
lights. From the remaining part of ants captured from the same natural colony,
recruit pheromone was extracted using a method of column chromatography.
Experimental processes were recorded by video camera. More detailed setting,
see the original paper [10].

In the box, they put a food source at a separated position from the nest in
the box. They realized a conflicting situation such that homing directions for
ants from the food source indicated by chemical cues and by visual cues
discord from each other as follows.

Figure 1: Three types of experimental settings and results of observed paths
of ants reported by the Nishimori group.

1\. At the beginning of each experiment, the preliminarily extracted recruit
pheromone was applied along one of the three kinds of polygonal lines
connecting the nest and the food source, which we will call the ‘pheromone
road’ in this paper. There one folding point was included in the lines with
the turning angles $\theta\in\\{45^{\circ},60^{\circ},90^{\circ}\\}$ (see
Fig.1). The preliminarily extracted pheromone has a sufficiently high density
to keep strong attraction of ants until the end of each experiment.

2\. It is considered from another experiment of the Nishimori group that
Lasius Japonicus can figure out the landmark at the distance of the order of
10 cm [11]. Hence ants in the present setup will be able to recognize the nest
direction from the food source. This direction along the optimal path from the
food source to the nest is different from the direction from the food sources
indicated by the ‘pheromone road’ as mentioned above.

After getting food at the food source, ants start homing walk, and a conflict
between the direction along the ‘pheromone road’ and the straight direction to
the nest required ants to make decision on which they rely chemical cues or
visual cues. The degree $\alpha$ of this conflict becomes larger as the angle
$\theta$ increases.

Figure 1 shows the results. In the upper pictures, the initially prepared
‘pheromone roads’ are indicated by light-gray lines, each of them have a
folding point with turning angle $\theta=45^{\circ},60^{\circ}$, and
$90^{\circ}$, respectively. Lower pictures show trajectories of foraging ants
recorded during the middle time period (about 10 min) of each experiment with
duration 60 min. Only in the case of $\theta=90^{\circ}$, a direct path
between the nest and the food source was established, whereas in other cases,
no direct path was established until the end of experiment at time $t=60$ min.

It was claimed [10] that the change of the dominant path from trailing paths
of the ‘pheromone road’ to the direct path did not associate with a continuous
geometrical change of path, and instead, the direct path spontaneously emerged
only in the case of $\theta=90^{\circ}$.

The above observation in experiments suggests that, if we introduce suitable
parameters which control individual motion of ants, there are critical values
of parameters and the change of dominant path can occur only in the
‘supercritical regime’ of parameters. In the present paper, we introduce two
kinds of order parameters characterizing global changes of path patterns of
foraging ants in homing walks from the food source to the nest. Roughly
speaking, they are given by

$\displaystyle M$ $\displaystyle=\frac{\sharp\\{\mbox{pioneering ants and
their followers}\\}}{\sharp\\{\mbox{homing ants}\\}},$
$\displaystyle\widetilde{M}$ $\displaystyle=\frac{\sharp\\{\mbox{ants trailing
the `pheromone road'}\\}}{\sharp\\{\mbox{homing ants}\\}},$ (1.3)

where $\sharp\\{\mbox{elements}\\}$ denotes the total number of elements in a
set $\\{\mbox{elements}\\}$. The precise definition of $M$ and $\widetilde{M}$
will be given in Section 3. We will discuss the possibility to regard the
global changes of path patterns as continuous phase transitions associated
with critical phenomena [1, 9, 12].

##  2 Discrete-time Stochastic Model on a Square Lattice

###  2.1 Switching random walks interacting through pheromone field

[General setting] Consider an $(L-1)\times(L-1)$ square region on a square
lattice. The set of vertices (sites) is given by

$V_{L}:=\\{v=(x,y):x\in\\{0,1,\dots,L-1\\},y\in\\{0,1,\dots,L-1\\}\\}.$ (2.1)

We write the Euclidean distance between two vertices $v_{1},v_{2}\in V_{L}$ as
$|v_{1}-v_{2}|$.

With discrete time

$t\in\mathbb{N}_{0}:=\\{0,1,\dots\\},$ (2.2)

we consider a set of random walks of a given number of particles,
$N\in\mathbb{N}:=\\{1,2,\dots\\}$, on $V_{L}$. They are interacting with each
other through a time-dependent pheromone field as explained below. We do not
impose, however, any exclusive interaction between particles. Hence each
vertex $v\in V_{L}$ can be occupied by any number of particles at each time.

The nest and food source are assumed to be located at the origin $v_{\rm
n}:=(0,0)$ and at the most upper-right vertex $v_{\rm f}:=(L-1,L-1)$,
respectively.

All particles start from $v_{\rm n}$ at $t=0$ and perform drifted walk toward
$v_{\rm f}$. After arriving at $v_{\rm f}$, each particle immediately starts
drifted walk from $v_{\rm f}$ toward $v_{\rm n}$ and then returns to $v_{\rm
n}$. In this way, particles shuttle between $v_{\rm n}$ and $v_{\rm f}$ for a
given time period $T$. When the particle is in a time period for walking from
$v_{\rm n}$ to $v_{\rm f}$, we say that it is in an exploring-period, and when
it is in a time period for walking from $v_{\rm f}$ to $v_{\rm n}$, it is said
to be in a homing-period.

We will denote the two types of hopping rules, which represent the slow mode
and the fast mode of the drifted random walk, respectively. The particles are
labeled by $j=1,\dots,N$. At each time $t\in\mathbb{N}_{0}$, the state of a
particle is specified by

$\displaystyle\mbox{the location}:\,\,v\in V_{L},$ $\displaystyle\mbox{and the
types of hopping : the slow mode (s) or the fast mode (f)}.$

Hence, at each time $t\in\mathbb{N}_{0}$, the $N$-particle configuration is
given by a set

$X(t):=\\{X_{j}(t):j=1,\dots,N\\}$ (2.3)

of pairs of random variables:

$X_{j}(t)=(v_{j}(t),\sigma_{j}(t)),\quad v_{j}(t)\in
V_{L},\quad\sigma_{j}(t)\in\\{{\rm s},{\rm f}\\},$ (2.4)

$j=1,\dots,N$. We will define a discrete-time stochastic process
$(X(t))_{t\in\mathbb{N}_{0}}$.

[Time-dependent pheromone field]

The pheromone is put on vertices and its intensity is expressed by a
nonnegative integer,

$f(v,t)\in\mathbb{N}_{0},\quad v\in V_{L},\quad t\in\mathbb{N}_{0}.$ (2.5)

That is, $f(v,t)=f\in\mathbb{N}_{0}$ means that there are $f$ units of
pheromone on the vertex $v\in V_{L}$ at time $t\in\mathbb{N}_{0}$. Define the
$\Gamma$-shaped subset of $V_{L}$,

$\displaystyle\Gamma_{L,\ell}=$ $\displaystyle\\{(x,y)\in V_{L}:0\leq
x\leq\ell-1,0\leq y\leq L-1\\}$ $\displaystyle\cup\\{(x,y)\in V_{L}:0\leq
x\leq L-1,L-\ell\leq y\leq L-1\\}.$ (2.6)

We will call $\Gamma_{L,\ell}$ the $\Gamma$-region, which represents the
‘pheromone road’. The variable $\ell$ indicates the width of the
$\Gamma$-region. See Fig.2.

Figure 2: Setting of the model on the square region $V_{L}$ is illustrated.
Paths from the food source $v_{\rm f}$ to the nest $v_{\rm n}$ in the homing-
period are classified. The path $\widetilde{\pi}$ shown by a red line is an
example of $\Gamma$-path which is completely included in the $\Gamma$-region,
$\Gamma_{L,\ell}$. Two examples of nearly-optimal-paths are given by $\pi_{1}$
and $\pi_{2}$ drawn by blue lines. Some parts of paths are outside of
$\Gamma_{L,\ell}$ and each homing duration $A$ satisfies $2(L-1)\leq A\leq
2(L-1)+\langle\Delta A_{0}\rangle/2$.

With a parameter $f_{0}\in\mathbb{N}$, the initial value of the pheromone
field is given by

$f(v,0)=\begin{cases}f_{0},&\mbox{if $v\in\Gamma_{L,\ell}$},\cr 0,&\mbox{if
$v\in V_{L}\setminus\Gamma_{L,\ell}$},\end{cases}$ (2.7)

where $V_{L}\setminus\Gamma_{L,\ell}$ denotes the complement of
$\Gamma_{L,\ell}$ in $V_{L}$. With parameters $f_{1},f_{2}\in\mathbb{N}_{0}$,
for each arrival of a particle at each vertex $v\in V_{L}$, secretion of
pheromone by an ant is represented by the following increment of the
intensity,

$f\to\begin{cases}f+f_{1},&\mbox{in an exploring-period},\cr f+f_{2},&\mbox{in
a homing-period}.\end{cases}$ (2.8)

If $v\in V_{L}\setminus\Gamma_{L,\ell}$, each unit of pheromone evaporates
with probability $q$ independently at each time step $t\to t+1$. That is, at
the time step $t\to t+1$,

$f\to f-n\quad\mbox{with probability}\quad\binom{f}{n}q^{n}(1-q)^{f-n},\quad
n=0,1,\dots,f.$ (2.9)

If $v\in\Gamma_{L,\ell}$, however, only excess of $f$ over $f_{0}$ evaporates.
That is, if and only if $f>f_{0}$,

$f\to f-n\quad\mbox{with
probability}\quad\binom{f-f_{0}}{n}q^{n}(1-q)^{f-f_{0}-n},\quad
n=0,1,\dots,f-f_{0}.$ (2.10)

[Hopping rule in an exploring-period] When ants are in an exploring-period,
they are all in the slow mode; $\sigma_{j}(t)\equiv{\rm s}$. For
$X_{j}(t)=(v_{j}(t),{\rm s})$, we define a set of the nearest-neighboring
vertices of $v_{j}(t)$ as

$\displaystyle\Lambda^{\rm s}_{j}(t)$ $\displaystyle:=\begin{cases}\\{v\in
V_{L}:|v-v_{j}(t)|=1\\}=\\{(1,0),(0,1)\\},&\mbox{if $v_{j}(t)=v_{\rm
n}$},\cr\\{v\in V_{L}:|v-v_{j}(t)|=1,v\not=v_{j}(t-1)\\},&\mbox{if
$v_{j}(t)\in V_{L}\setminus\\{v_{\rm n}\\}$}.\end{cases}$ (2.11)

Then at each time step $t\to t+1$,

$v_{j}(t)\to v\quad\mbox{with
probability}\quad\frac{f(v,t)}{\sum_{w\in\Lambda^{\rm
s}_{j}(t)}f(w,t)},\quad\mbox{if and only if $v\in\Lambda^{\rm s}_{j}(t)$},$
(2.12)

provided that $\sum_{w\in\Lambda^{\rm s}_{j}(t)}f(w,t)>0$. If
$\sum_{w\in\Lambda^{\rm s}_{j}(t)}f(w,t)=0$; that is, $f(v,t)=0$ for all
$v\in\Lambda^{\rm s}_{j}(t)$, then

$v_{j}(t)\to v\quad\mbox{with probability}\quad\frac{1}{|\Lambda_{j}^{\rm
s}(t)|},\quad\mbox{if and only if $v\in{\Lambda}^{\rm s}_{j}(t)$},$ (2.13)

where $|\Lambda_{j}^{\rm s}(t)|$ denotes the total number of vertices in
$\Lambda^{\rm s}_{j}(t)$. Notice that the previous position $v_{j}(t-1)$ is
not included in $\Lambda^{\rm s}_{j}(t)$ in order to forbid backward walking.

(Prohibition against cyclic motions) In an exploring-period, if
$v_{j}(t+1)-v_{j}(t)\in\\{(-1,0),(0,-1)\\}$, that is, the step is leftward or
downward, the hopping is regarded as a retrograde motion. We prohibit any
succession of such retrograde motions in order to avoid cyclic motions of
walkers. When the particle is in the domain $\Gamma_{L,\ell}$, this
prohibition rule is strengthened so that all downward steps are forbidden, if
$0\leq x\leq\ell-1$, and all leftward steps are forbidden, if $L-\ell\leq
y\leq L-1$.

[Hopping rules in a homing-period]

(Slow mode) For $X_{j}(t)=(v_{j}(t),{\rm s})$ in a homing-period, we define a
set of the nearest-neighboring vertices of $v_{j}(t)$ as

$\displaystyle\widetilde{\Lambda}^{\rm s}_{j}(t)$
$\displaystyle:=\begin{cases}\\{v\in
V_{L}:|v-v_{j}(t)|=1,v\not=v_{j}(t-1)\\},\cr\hskip 142.26378pt\mbox{if
$v_{j}(t)\in V_{L}\setminus\\{v_{\rm f}\\}$},\cr\\{v\in
V_{L}:|v-v_{j}(t)|=1\\}=\\{(L-2,L-1),(L-1,L-2)\\},\cr\hskip
142.26378pt\mbox{if $v_{j}(t)=v_{\rm f}$}.\end{cases}$ (2.14)

Notice that the previous position $v_{j}(t-1)$ is not included in
$\widetilde{\Lambda}^{\rm s}_{j}(t)$ also in order to forbid backward walking.
Then at each time step $t\to t+1$,

$v_{j}(t)\to v\quad\mbox{with
probability}\quad\frac{f(v,t)}{\sum_{w\in\widetilde{\Lambda}^{\rm
s}_{j}(t)}f(w,t)}.\quad\mbox{if and only if $v\in\widetilde{\Lambda}^{\rm
s}_{j}(t)$},$ (2.15)

provided that $\sum_{w\in\widetilde{\Lambda}^{\rm s}_{j}(t)}f(w,t)>0$. If
$\sum_{w\in\widetilde{\Lambda}^{\rm s}_{j}(t)}f(w,t)=0$; that is, $f(v,t)=0$
for all $v\in\widetilde{\Lambda}^{\rm s}_{j}(t)$, then

$v_{j}(t)\to v\quad\mbox{with
probability}\quad\frac{1}{|\widetilde{\Lambda}_{j}^{\rm s}(t)|},\quad\mbox{if
and only if $v\in\widetilde{\Lambda}^{\rm s}_{j}(t)$}.$ (2.16)

(Prohibition against cyclic motions) In a homing-period, if
$v_{j}(t+1)-v_{j}(t)\in\\{(1,0),(0,1)\\}$, that is, the step is rightward or
upward, the hopping is regarded as a retrograde motion. We prohibit any
succession of such retrograde motions in order to avoid cyclic motions of
walkers. When the particle is in the domain $\Gamma_{L,\ell}$, this
prohibition rule is strengthened so that all rightward steps are forbidden, if
$L-\ell\leq y\leq L-1$, and all upward steps are forbidden, if $0\leq
x\leq\ell-1$.

(Fast mode) For $X_{j}(t)=(v_{j}(t),{\rm f})$ in a homing-period, we define a
subset of the nearest-neighboring vertices of $v_{j}(t)$ as

$\widetilde{\Lambda}^{\rm f}_{j}(t):=\\{v\in
V_{L}:|v-v_{j}(t)|=1,|v|<|v_{j}(t)|\\}.$ (2.17)

Then in the time step $t\to t+1$,

$v_{j}(t)\to v\quad\mbox{with
probability}\quad\frac{1}{|\widetilde{\Lambda}_{j}^{\rm f}(t)|},\quad\mbox{if
and only if $v\in\widetilde{\Lambda}^{\rm f}_{j}(t)$}.$ (2.18)

Note that since we have considered the system in a subset $V_{L}$ of the
square lattice, $|\widetilde{\Lambda}^{\rm f}_{j}(t)|\in\\{1,2\\}$. If $x=0$
or $y=0$ in $v_{j}(t)=(x,y)$, $|\widetilde{\Lambda}^{\rm f}_{j}(t)|=1$, and
hence, the hopping of the fast mode is deterministic.

(Switching) We introduce parameters $\gamma_{\rm sf},\gamma_{\rm fs}\in[0,1]$.
Then in the time step $t\to t+1$ in a homing-period, we change the mode of
hopping as

$\displaystyle\sigma_{j}(t)$ $\displaystyle={\rm s}\to{\rm f}\quad\mbox{with
probability $\gamma_{\rm sf}$},$ $\displaystyle\sigma_{j}(t)$
$\displaystyle={\rm f}\to{\rm s}\quad\mbox{with probability $\gamma_{\rm
fs}$}.$ (2.19)

It means that the mode is not changed with probability $1-\gamma_{\rm sf}$
(resp. $1-\gamma_{\rm fs}$), when the particle is in the slow (resp. fast)
mode at each time.

[Initial condition, update rule, random switching at food source, and time
duration]

(i) We start from the initial configuration such that all $N$ particles are in
the slow mode and at the origin,

$X_{j}(0)=(0,{\rm s}),\quad j=1,\dots,N.$ (2.20)

(ii) We perform hopping of particles and switching of the mode in a homing-
period sequentially according to the numbering of particles $j=1,\dots,N$. The
update of the pheromone field is done at each time step.

(iii) When $v_{j}(t)=v_{\rm f}$, that is, an ant arrives at the food source,
switching between the slow mode and the fast mode is randomly occurs:

$\sigma_{j}(t+1)=\begin{cases}{\rm s},&\mbox{with probability 1/2},\cr{\rm
f},&\mbox{with probability 1/2}.\end{cases}$ (2.21)

(iv) Let $\tau(n)$ be the time duration so that the total number of particles
which have returned to the nest $v_{\rm n}$ at least once is $n$. We perform
each numerical simulation up to time step

$T=\tau(\kappa N),$ (2.22)

where $\kappa$ is a parameter. That is, during $T$, totally $\kappa N$
particles shuttle between the nest and the food source.

###  2.2 Parameter setting

We set the parameters concerning the time-dependent pheromone field as

$\displaystyle f_{0}=1000,\quad f_{1}=0,\quad f_{2}=10,\quad q=0.01.$ (2.23)

We assume that the density of particles is fixed to be

$\rho:=\frac{N}{L^{2}}=\frac{1}{8}=0.125,$ (2.24)

the ratio of the width of the $\Gamma$-region defined by (2.6) with respect to
$L$ is given by

$\frac{\ell}{L}=\frac{1}{20},$ (2.25)

and each simulation is done for the time duration (2.22) with

$\kappa=4.$ (2.26)

We have performed the numerical simulation of the present stochastic model on
the square lattices with sizes $L=40$, $60$, and $80$. In the present paper,
we will report the dependence of the simulation results on the switching
parameters $\gamma_{\rm sf}$ and $\gamma_{\rm fs}$ which are changed in the
interval $[0,1]$. We have confirmed that the numerical results given below are
not changed qualitatively by changing the parameter setting (2.23)–(2.26). In
particular, the setting (2.26) for the time duration of each simulation is
sufficient to observe stable distributions of homing duration, which will be
reported in Section 3. The validity of the $L$-dependence of parameters (2.24)
and (2.25) will be discussed in Section 4.1. Systematic study of quantitative
dependence of the model on the other parameters than $L,\ell,N$ and
$\gamma_{\rm sf}$ (with the relation $\gamma_{\rm fs}=1-\gamma_{\rm sf}$) is
considered as one of the future problems as discussed in the item (v) in
Section 5. Table 1 summarizes the setting, changing, and fixed parameters of
the present model.

Table 1: List of the setting, changing, and fixed parameters in the present discrete-time stochastic model on a square lattice setting parameters |  | setting
---|---|---
$L$ | system size | 40, 60, 80
$\ell$ |  width of the $\Gamma$-region | $L/20$
$N$ | number of particles | $\rho L^{2}=L^{2}/8$
changing parameters |  | range of value
$\gamma_{\rm sf}$ | switching probability from slow mode to fast mode | $[0,1]$
$\gamma_{\rm fs}$ | switching probability from fast mode to slow mode |  $[0,1]$ or $1-\gamma_{\rm sf}$
fixed parameters |  | value
$f_{0}$ |  initial intensity of pheromone in the $\Gamma$-region | 1000
$f_{1}$ | amount of pheromone dropped by an exploring particle | 0
$f_{2}$ | amount of pheromone dropped by a homing particle | 10
$q$ | evaporation rate of pheromone | 0.01
$\kappa$ |  time duration of simulation $T=\tau(\kappa N)$ | 4

##  3 Distributions of Homing Duration and Order Parameters

###  3.1 $\Gamma$-paths and order parameter $\widetilde{M}$

Figure 3:  Histogram of the homing duration $A_{0}$ of particles in one sample
of simulation, when $(\gamma_{\rm sf},\gamma_{\rm fs})=(0,1)$, in the system
with size $L=60$. All particles are in the slow mode and trail $\Gamma$-paths.
In this observed distribution, mean of $A_{0}$ is $188.9$, the maximum and
minimum values are 210 and 166, respectively, and then $\Delta A_{0}=44$.

For each particle in the homing-period, we trace the path from the food source
$v_{\rm f}$ to the nest $v_{\rm n}$,

$\pi=\\{v_{\rm f}=v_{0}\to v_{1}\to v_{2}\to\cdots\to v_{A}=v_{\rm n}\\}.$
(3.1)

Here the number of steps in $\pi$ indicating the path length is expressed by
$A$. In the present paper, we call this random variable $A$ the homing
duration.

First we consider the extreme case with $\gamma_{\rm sf}=0$ and $\gamma_{\rm
fs}=1$. In this case all particles are in the slow mode following the
‘pheromone road’. Hence all paths in the homing-period are completely included
in the $\Gamma$-region, $\Gamma_{L,\ell}$, defined by (2.6). In general, if
the path in the homing-period is completely included in the $\Gamma$-region,
then we say that the path is a $\Gamma$-path. (See Fig.2.) In this extreme
case, we write the random variable of homing duration as $A_{0}$ by putting
subscript 0. Figure 3 shows a typical distribution of homing duration $A_{0}$
in one sample of simulation for the system with size $L=60$. We notice that
the minimal value of the homing duration is $2(L-1)=118$, which is realized by
a $\Gamma$-shaped homing walk along the uppermost edges and then the leftmost
edges of the square lattice. The observed distribution of $A_{0}$ in this
simulation is, however, centered at a larger value $\simeq 189$. It means that
the slow mode paths are far from optimal and include many redundant walks
which are enhanced by the pheromone field. In the observed distribution, we
measured the width $\Delta A_{0}$ defined by the difference between the
observed maximum and minimum values of $A_{0}$. We calculated the averaged
value $\langle\Delta A_{0}\rangle$ over several samples of numerical
simulation. The results are the following:

$\displaystyle\langle\Delta A_{0}\rangle$ $\displaystyle=26.6\pm
1.8,\quad\mbox{for $L=40$}$ $\displaystyle\langle\Delta A_{0}\rangle$
$\displaystyle=41.6\pm 2.7,\quad\mbox{for $L=60$}$ $\displaystyle\langle\Delta
A_{0}\rangle$ $\displaystyle=54.4\pm 3.2,\quad\mbox{for $L=80$},$ (3.2)

where the errors are given by the standard deviations over ten samples of
numerical simulation for the systems with sizes $L=40$ and $60$, and five
samples for the system with size $L=80$. The above results show a linear
dependence of $\langle\Delta A_{0}\rangle$ on size $L$,

$\langle\Delta A_{0}\rangle\simeq 0.7L-0.83,$ (3.3)

as shown by Fig.4.

Figure 4:  Dependence of $\langle\Delta A_{0}\rangle$ on $L$ is well described
by a linear fitting; $\langle\Delta A_{0}\rangle\simeq 0.7L-0.83$. The errors
are given by the standard deviations over ten samples of numerical simulation
for the systems with sizes $L=40$ and $60$, and five samples for the system
with size $L=80$.

We define $\widetilde{M}$ by the ratio of the number of observed particles
following $\Gamma$-paths to the total number of homing particles in
simulation; i.e. $\kappa N=\kappa L^{2}/8$ ;

$\widetilde{M}:=\frac{\sharp\\{\mbox{homing particle following a
$\Gamma$-path}\\}}{\kappa N}.$ (3.4)

As mentioned above, $\widetilde{M}=1$, if $(\gamma_{\rm sf},\gamma_{\rm
fs})=(0,1)$.

From now on we assume the relation

$\gamma_{\rm sf}+\gamma_{\rm fs}=1.$ (3.5)

As $\gamma_{\rm sf}$ increases from 0, $\widetilde{M}$ will decrease from 1,
since particles in the fast mode appear and they make deviation from the
‘pheromone road’. In other words, $\widetilde{M}$ will play a role of the
order parameter for the path selection which persists the original ‘pheromone
road’ in the homing-period.

###  3.2 Nearly-optimal-paths and order parameter $M$

(a) Dependence on $\gamma_{\rm sf}$ with the relation $\gamma_{\rm
fs}=1-\gamma_{\rm sf}$ of the distributions of homing duration $A$ of
$\Gamma$-paths (shown by red curves) and of non-$\Gamma$-paths (by blue
curves) for the system with $L=60$. Three distributions of $A$ are shown for
the $\Gamma$-paths with $\gamma_{\rm sf}=0$, $0.1$, and $0.2$, and ten for the
non-$\Gamma$-paths with $\gamma_{\rm sf}=0.1$, $0.2$, $0.3$, $0.4$, $0.5$,
$0.6$, $0.7$, $0.8$, $0.9$, and $1$. In both cases, the right edge (the point
at which the right tail of distribution becomes zero) decreases monotonically
as $\gamma_{\rm sf}$ increases.

(b) Means of the distribution of $A$ for the non-$\Gamma$-paths (shown by the
blue curves in (a)) are plotted versus $\gamma_{\rm sf}\in(0,1]$. The standard
deviations of distribution of $A$ are shown by the error bars.

Figure 5: Distributions of homing duration $A$. The shaded region indicates
the range of $A$, $2(L-1)\leq A\leq 2(L-1)+\langle\Delta A_{0}\rangle/2$, for
the nearly-optimal-paths. For $L=60$, $2(L-1)=118$ and $\langle\Delta
A_{0}\rangle/2=20.8$ by (3.2).

For the system with size $L=60$, Fig. 5 (a) shows the dependence of the
distribution of homing duration $A$ on $\gamma_{\rm sf}$ with the relation
(3.5). When $\gamma_{\rm sf}\gtrsim 0.3$, $\Gamma$-path is not observed any
more and observed paths are dominated by non-$\Gamma$-paths. As $\gamma_{\rm
sf}$ increases, the observed support of distribution of homing duration for
the non-$\Gamma$-paths shifts to the lower-valued region of $A$. It means that
as $\gamma_{\rm sf}$ increases, more pioneer walkers appear who build shorter
paths from the food source to the nest, since particles can be in the fast
mode more frequently. In addition to the shift to the left, we also see in
Fig. 5 (a) that the bell-shaped distribution of $A$ for the the
non-$\Gamma$-paths becomes sharper as $\gamma_{\rm sf}$ increases. Monotonic
decrease of the mean $\langle A\rangle$ and the standard deviation $\sigma$ of
the distribution of $A$ as the increment of $\gamma_{\rm sf}$ is shown by Fig.
5 (b) for the non-$\Gamma$-paths. This implies that the shorter paths are
strengthened by followers of pioneer walkers via the interaction through
pheromone field. As $\gamma_{\rm sf}$ approaches to 1, the distribution
quickly sharpens and seems to be condensed to the shortest value of $A$, i.e.
the length of the optimal path, $2(L-1)=118$. In other words, as $\gamma_{\rm
sf}$ approaches to 1, a large amount of particles tend to trail the paths
which are very near to the optimal path. We want to call such paths the
nearly-optimal-paths.

In order to define the nearly-optimal-paths, we use the value $\langle\Delta
A_{0}\rangle$ determined by the homing-duration distribution when
$(\gamma_{\rm sf},\gamma_{\rm fs})=(1,0)$ in Section 3.1. That is, if the
homing duration $A$ of the path satisfies the condition

$2(L-1)\leq A\leq 2(L-1)+\frac{1}{2}\langle\Delta A_{0}\rangle,$ (3.6)

then we say that it is a nearly-optimal-path. This range (3.6) of $A$ is
indicated by a shaded region in Figs. 5 (a) and (b). Then we define another
order parameter $M$ in addition to $\widetilde{M}$ by the ratio of the number
of particles following nearly-optimal-paths in a homing-period to the total
number of homing particles in simulation, $\kappa N$,

$M:=\frac{\sharp\\{\mbox{homing particle following a nearly-optimal-
path}\\}}{\kappa N}.$ (3.7)

Figure 2 illustrates an example of $\Gamma$-path which contributes to the
order parameter $\widetilde{M}$ and two examples of nearly-optimal-paths which
contribute to the order parameter $M$. We will compare the dependence on
$\gamma_{\rm sf}$ of these two kinds of order parameters $M$ and
$\widetilde{M}$ with changing the system size $L$. The choice of
$\langle\Delta A_{0}\rangle/2$ as a width in the condition (3.6) which defines
the nearly-optimal-paths is to make the dependence of $M$ on $L$ be comparable
with that of $\widetilde{M}$. The definitions (3.6) and (3.7) work well as
shown below.

##  4 Phase Transitions and Critical Phenomena in Path Selections

###  4.1 Almost data-collapse with respect to system size

(a) Numerical results of $M_{L}$ and $\widetilde{M}_{L}$ for the finite
systems with sizes $L=40$, $60$, and $80$.

(b) $M_{\infty}$ and $\widetilde{M}_{\infty}$ for the infinite system
estimated by the $1/L$-plots as explained in Section 4.2. The critical values
$\gamma_{\rm sf}^{\rm c}\simeq 0.63$ and $\widetilde{\gamma_{\rm sf}^{\rm
c}}\simeq 0.08$ estimated there are indicated by arrows.

Figure 6: Order parameters $M$ and $\widetilde{M}$ are plotted versus
$\gamma_{\rm sf}\in[0,1]$, where $\gamma_{\rm fs}=1-\gamma_{\rm sf}$.

With the relation between parameters (3.5), Fig.6 (a) shows the $\gamma_{\rm
sf}$-dependence of the two kinds of order parameters, $M$ and $\widetilde{M}$,
for the systems with different sizes, $L=40$, $60$, and $80$. For the systems
with sizes $L=40$, $60$, (resp. $80$) we have obtained 10 samples (resp. 3
samples) for each parameter $\gamma_{\rm sf}\in[0,1]$. The lines interpolate
the averaged values of these samples. We have confirmed that the standard
deviations are very small, e.g. $3.0\times 10^{-3}$ for $M$ at $\gamma_{\rm
sf}=0.6$ and $1.5\times 10^{-2}$ for $\widetilde{M}$ at $\gamma_{\rm sf}=0.1$
for $L=60$, and hence error bars are omitted in Fig.6 (a). By definition,
$\widetilde{M}\to 1$ as $\gamma_{\rm sf}\to 0$, and $M\to 1$ as $\gamma_{\rm
sf}\to 1$. We see that when $\gamma_{\rm sf}$ is small, $M=0$ and
$\widetilde{M}>0$, while when $\gamma_{\rm sf}$ is large, $M>0$ and
$\widetilde{M}=0$. The rapid decrease of $\widetilde{M}$ around $\gamma_{\rm
sf}\simeq 0.1$, and the rapid increase of $M$ around $\gamma_{\rm sf}\simeq
0.6$ indicate the global switching phenomena of path selections between the
$\Gamma$-paths and the nearly-optimal-paths.

Figure 6 (a) shows that the differences of the values of $M$ and
$\widetilde{M}$ caused by enlarging the system size $L$ are relatively small.
Roughly speaking, this fact will be regarded as a kind of data collapse, which
shows independence of the global switching phenomena in path selections on the
size $L$ in the present model. In other words, our settings of the number of
particles, $N=\rho L^{2}\propto L^{2}$, given by (2.24), the width of the
$\Gamma$-region (‘pheromone road’), $\ell=L/20\propto L$, given by (2.25), and
the condition (3.6) for the nearly-optimal-paths using the quantity,
$\langle\Delta A_{0}\rangle\simeq aL+b$, (see (3.3)) are very simple but they
provide a proper scaling property of the system with respect to the change of
the system size $L$.

###  4.2 Evaluations of critical values and critical exponents

On the other hand, in the vicinity of the values of $\gamma_{\rm sf}$ at which
$\widetilde{M}$ approaches zero ($\gamma_{\rm sf}\simeq 0.1$) and at which $M$
emerges ($\gamma_{\rm sf}\simeq 0.6$), the changes show systematic dependence
on $L$. This observation suggests that we will have critical phenomena in the
$L\to\infty$ limit and the present global switching phenomena in path
selections can be considered as continuous phase transitions.

In order to explain our method to estimate the order parameters in the
$L\to\infty$ limit, here we write the order parameters as $M_{L}$ and
$\widetilde{M}_{L}$ for the system with size $L$. We assume the following
simple dependence of the order parameters on the system size $L$,

$M_{\infty}=M_{L}+\frac{c_{1}}{L},\quad\widetilde{M}_{\infty}=\widetilde{M}_{L}+\frac{\widetilde{c_{1}}}{L},\quad\mbox{as
$L\to\infty$},$ (4.1)

where $c_{1}$ and $\widetilde{c_{1}}$ are fitting parameters, and $M_{\infty}$
and $\widetilde{M}_{\infty}$ are the estimated values of the order parameters
for the infinite system. (See the item (ii) in Section 5 for a possible
improvement by introducing a critical exponent $\nu$.) Then if we plot the
values of $M_{L}$ (resp. $\widetilde{M}_{L}$) with respect to $1/L$, the
$L\to\infty$ limit of order parameter $M_{\infty}$ (resp.
$\widetilde{M}_{\infty}$) will be evaluated by the ‘$y$-intercept’. The
estimated curves of $M_{\infty}$ and $\widetilde{M}_{\infty}$ are shown for
$\gamma_{\rm sf}\in[0,1]$ in Fig. 6 (b).

(a) Fitting with $\gamma_{\rm sf}^{\rm c}=0.63$ and $\beta=1.59$ for
$M_{\infty}$.

(b) Fitting with $\widetilde{\gamma_{\rm sf}^{\rm c}}=0.08$ and
$\widetilde{\beta}=1.51$ for $\widetilde{M}_{\infty}$.

Figure 7: Log-log plots of the order parameters $M_{\infty}$ and
$\widetilde{M}_{\infty}$ with respect to the deviations of the parameter
$\gamma_{\rm sf}$ from its critical values, $\gamma_{\rm sf}-\gamma_{\rm
sf}^{\rm c}$ and $\widetilde{\gamma_{\rm sf}^{\rm c}}-\gamma_{\rm sf}$,
respectively. The slopes give the critical exponents $\beta$ and
$\widetilde{\beta}$.

The results suggest that we can define the two kinds of critical values of
$\gamma_{\rm sf}$ according to the two kinds of order parameters, $M_{\infty}$
and $\widetilde{M}_{\infty}$, as

$\gamma_{\rm sf}^{\rm c}:=\inf\\{\gamma_{\rm
sf}:M_{\infty}>0\\},\quad\widetilde{\gamma_{\rm sf}^{\rm
c}}:=\sup\\{\gamma_{\rm sf}:\widetilde{M}_{\infty}>0\\}.$ (4.2)

From the $1/L$-plots, we have evaluated the critical values as

$\gamma_{\rm sf}^{\rm c}\simeq 0.63,\quad\widetilde{\gamma_{\rm sf}^{\rm
c}}\simeq 0.08.$ (4.3)

See Fig. 6 (b).

The continuous changes of $M$ and $\widetilde{M}$ at these critical points
suggest that the phase transitions are continuous. Then the critical phenomena
will be described by the following power laws as the functions of the
parameter $\gamma_{\rm sf}$,

$\displaystyle M_{\infty}(\gamma_{\rm sf})$ $\displaystyle\simeq
c_{2}(\gamma_{\rm sf}-\gamma_{\rm sf}^{\rm c})^{\beta},\quad\gamma_{\rm
sf}\gtrsim\gamma_{\rm sf}^{\rm c},$
$\displaystyle\widetilde{M}_{\infty}(\gamma_{\rm sf})$
$\displaystyle\simeq\widetilde{c_{2}}(\widetilde{\gamma_{\rm sf}^{\rm
c}}-\gamma_{\rm sf})^{\widetilde{\beta}},\quad\gamma_{\rm
sf}\lesssim\widetilde{\gamma_{\rm sf}^{\rm c}},$ (4.4)

where $\beta$ and $\widetilde{\beta}$ are the critical exponents of order
parameters $M$ and $\widetilde{M}$, respectively [1, 9, 12]. As shown by
Fig.7, the power laws are observed and the log-log plots give the values

$\beta\simeq 1.6,\quad\widetilde{\beta}\simeq 1.5.$ (4.5)

##  5 Concluding Remarks and Future Problems

At the end of this paper, we list out the concluding remarks and possible
future problems.

(i) The interactions between ants are given through the pheromone field, which
evolves in time affected by the previous behaviors of other ants. In the
present study, we have studied such an interacting particle system by
numerically simulating a cellular automaton model. What kind of theory is
relevant to describe such kind of temporally inhomogeneous interaction with
memory effects?

(ii) In the finite-size scaling theory [1, 9, 12], the $1/L$-fitting (4.1)
should be refined by introducing exponents $\nu$ and $\widetilde{\nu}$ as

$M_{\infty}=M_{L}+\frac{c_{1}}{L^{\nu}},\quad\widetilde{M}_{\infty}=\widetilde{M}_{L}+\frac{\widetilde{c_{1}}}{L^{\widetilde{\nu}}},\quad\mbox{as
$L\to\infty$}.$ (5.1)

In the present study, we have obtained numerical results for only three sizes
$L=40$, $60$, and $80$. Hence it is rather difficult to evaluate the values of
$\nu$ and $\widetilde{\nu}$ by numerical fitting. We have assumed the mean-
field value $\nu=\widetilde{\nu}=1$ in the fitting (4.1). The obtained
numerical results (4.5) suggest $\beta=\widetilde{\beta}=3/2$. Systematic
study of finite-size scaling based on larger scale simulation will be an
important future problem. The correlation functions and the response functions
of the present model for the group behavior of ants should be carefully
considered in order to understand the critical phenomena described by the
critical exponents $\nu$, $\delta$, and $\gamma$.

(iii) Any effective theory is desired to describe the phase transitions and
critical phenomena in the path selections for foraging ants. As mentioned in
the item (ii) above, first we will try to establish the mean-field theory.
Fractional values of critical exponents suggest the emergence of fractal
structures in paths of ants at the moment when the nearly-optimal-paths are
first established, as well as at the moment when the ‘pheromone road’ vanishes
in the path selections of ants.

(iv) By the definition (2.19), the parameters $\gamma_{\rm sf}$ and
$\gamma_{\rm fs}$ can take any values in $[0,1]$ independently. We have
assumed in the present paper, however, the special relation (3.5) between
these two parameters. The present study on the global switching of path
selections shall be generalized to the whole parameter space $(\gamma_{\rm
sf},\gamma_{\rm fs})\in[0,1]^{2}$. As a result, we will obtain the phase
diagram showing three types of phases; (a) the phase in which homing particles
select the $\Gamma$-paths (i.e. trailing paths of the ‘pheromone road’)
indicated by $\widetilde{M}>0$ and $M=0$, (b) the phase in which homing
particles select the nearly-optimal-paths indicated by $\widetilde{M}=0$ and
$M>0$, and (c) the intermediate phase.

(v) As shown by Table 1, we have fixed many parameters except $L,\ell$, $N$
and $\gamma_{\rm sf}$ with the relation (3.5). More general and systematic
study of the present model will be desired by investigating the dependence of
the results on other parameters. For example, if we change the parameter
$\kappa$, the time duration of numerical observation is changed. Then the
transient or relaxation phenomena in the initial time period and longer-term
behavior of the present processes will be clarified. It will be interesting
research subject to study precise dependence of the results on the evaporation
rate $q$ of pheromone, since this parameter will control the spatio-temporal
correlations between particles via the pheromone field.

(vi) The present study was motivated by the experiments reported by the group
of the last author (HN) of this paper as briefly explained in Section 1.2
[10]. In our modeling, we adopted the setting with $\theta=90^{\circ}$ for the
turning angle of the ‘pheromone road. (See Figs. 1 and 2 again.) The reason is
that any global switching phenomena in path selections was not observed in
other setting with $\theta=45^{\circ}$ and $\theta=60^{\circ}$. In the present
model, we have introduced other parameters $\gamma_{\rm sf}$ and $\gamma_{\rm
fs}$ (the switching parameters between the fast and slow modes) and showed
that the global switching phenomena in path selections are realized as
continuous phase transitions associated with the critical phenomena at the
critical values $\gamma_{\rm sf}^{\rm c}$ and $\widetilde{\gamma_{\rm sf}^{\rm
c}}$. This result suggests that even if we adopt the setting with
$\theta=45^{\circ}$ or $\theta=60^{\circ}$, we will be able to observe such
phase transitions provided that we properly choose the parameters $\gamma_{\rm
sf}$ and $\gamma_{\rm fs}$. In order to simulate the models with the setting
with $\theta=45^{\circ}$ or $\theta=60^{\circ}$ efficiently, we shall change
the lattice from the present square lattice to the triangular or the honeycomb
lattices. We expect that the critical phenomena associated with the phase
transitions representing the global switching phenomena in path selections are
universal and do not depend on any details of the lattice structure. If so,
the present study on lattice models will provide the universal statements to
the path selection phenomena observed in real foraging ants.

Acknowledgements The present authors would like to thank Mitsugu Matsushita
and Hirohiko Shimada for useful discussion on the present work. This work was
supported by the Research Institute for Mathematical Sciences, an
International Joint Usage/Research Center located in Kyoto University. MK is
supported by the Grant-in-Aid for Scientific Research (C) (No.19K03674), (B)
(No.23H01077), (A) (No.21H04432), and the Grant-in-Aid for Transformative
Research Areas (A) (No.22H05105) of the Japan Society for the Promotion of
Science (JSPS). HN is supported by the Grant-in-Aid for Scientific Research
(B) (No.20H01871), and the Grant-in-Aid for Transformative Research Areas (A)
(No.21H05293) and (A) (No.21H05297) of JSPS.

## References

  * [1] B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems, Cambridge University Press, Cambridge, UK, 1998.
  * [2] F. den Hollander, S. Nandan, Spatially inhomogeneous populations with seed-banks: I. Duality, existence and clustering, J. Theor. Probab. 35 (2022) 1792–1841.
  * [3] F. den Hollander, S. Nandan, Spatially inhomogeneous populations with seed-banks: II. Clustering regime, Stochastic Process. Appl. 150 (2022) 116–146.
  * [4] S. Floreani, C. Giardinà, F. den Hollander, S. Nandan, F. Redig, Switching interacting particle systems: Scaling limits, uphill diffusion and boundary layer, J. Stat. Phys. 186 (2022) 33 (45 pages).
  * [5] A. Greven, F. den Hollander, M. Oomen, Spatial populations with seed-bank: Well-posedness, duality and equilibrium, Electon, J. Probab. 27 (18) (2022) 1–88.
  * [6] A. Greven, F. den Hollander, M. Oomen, Spatial populations with seed-bank: Renormalisation on the hierarchical group, to appear in Memoirs of AMS. arXiv:math.PR/2110.02714
  * [7] B. Hölldobler, E. O. Wilson, The Ants, Harvard University Press, Cambridge, MA, 1990.
  * [8] J. T. Lennon, F. den Hollander, M. Wilke-Berenguer, J. Blath, Principle of seed banks and the emergence of complexity from dormancy, Nat. Commun. 12 (2021) 4807 (16 pages).
  * [9] J. Marro, R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Cambridge University Press, Cambridge, UK, 1999.
  * [10] Y. Ogihara, O. Yamanaka, T. Akino, S. Izumi, A. Awazu, H. Nishimori, Switching of primarily relied information by ants: A combinatorial study of experiment and modeling, in: (eds.) T. Ohira, T. Uzawa (Eds.), Mathematical Approaches to Biological Systems –Networks, Oscillations, and Collective Motions, Springer-Verlag, 2015, pp.119–137.
  * [11] S. Shinoda, Analysis of the influence of visual cues for foraging ants (in Japanese). Graduate Thesis, Department of Mathematics, Hiroshima University, 2013.
  * [12] T. Vicsek, A. Zafeiris, Collective motion. Phys. Rep. 517 (2012) 71–140.