Physica A xx (xxxx) xxx–xxx

Contents lists available at ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

Stochastic patterns in a 1D Rock–Paper–Scissors model withmutation

Q1 Claudia Cianci a, Timoteo Carletti b,∗a Dipartimento di Sistemi e Informatica and INFN, University of Florence, Via S. Marta 3, 50139 Florence, Italyb naXys, Namur Center for Complex Systems, University of Namur, rempart de la Vierge 8, B 5000 Namur, Belgium

h i g h l i g h t s

• Characterisation of stochastic patterns in Rock–Paper–Scissors model with mutation.• Determination of the Master Equation for Rock–Paper–Scissors model with mutation.• Study of the Fokker–Planck equation for Rock–Paper–Scissors model with mutation.• Analytical computation of fluctuations power spectrum.• Numerical study of the parameters dependence of the fluctuations.

a r t i c l e i n f o

Article history:Received 17 January 2014Received in revised form 2 April 2014Available online xxxx

Keywords:Stochastic processesNonlinear dynamicsSpatio-temporal patternsStochastic patternsStochastic simulations

a b s t r a c t

In the framework of a 1D cyclic competitionmodel, the Rock–Paper–Scissorsmodel, wherethree kinds of generic agents are allowed to mutate, to interact and to move in space, westudy the formation of stochastic patterns, where all the agents do coexist. We modelledthe problem using an individual-based setting and we used the system size van Kampenexpansion to deal with the Master Equation. We have hence been able to characterise thespatio-temporal patterns using the power spectrum of the fluctuations. We proved thatsuch patterns are robust against the intrinsic noise and they can be found for parametervalues beyond the ones fixed by the deterministic approach (mean field approximation).We complement such analytical results with numerical simulations based on the Gillespiealgorithm.

© 2014 Published by Elsevier B.V.

0. Introduction 1

Understanding the formation of spatio-temporal patterns is a long lasting problem, transversal to several research do- 2

mains, ranging from physics (fractures [1]), passing through biology (spots and stripes in animals [2]) and chemistry (the 3

celebrated Belousov–Zhabotinsky reaction [3–5]), to social sciences (spatial games [6–9]), just to name few examples. The 4

underlying phenomenon is thus the emergence of complex non-equilibrium structures that self-organise and persist as the 5

result of the interplay of noise and deterministic dynamics. 6

Since the pioneeringwork of A. Turing [10], it iswell known that spatio-temporal self-organised patterns can spontaneously 7

emerge in a reaction–diffusion system: a small perturbation of a homogeneous stable equilibrium can be amplified, through 8

the presence of the diffusion, and eventually drive the system into a non homogeneous spatial solution. Several of the above 9

∗ Corresponding author. Tel.: +32 0 81724903; fax: +32 0 81724914.E-mail address: timoteo.carletti@fundp.ac.be (T. Carletti).

http://dx.doi.org/10.1016/j.physa.2014.05.0010378-4371/© 2014 Published by Elsevier B.V.

2 C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx

mentioned examples can be explained by the Turing mechanism despite the very different shapes exhibited: waves, spirals,1

spots, both stationary or time dependent.2

In the classical setting, the emergence of Turing instability needs two diffusing interacting species, the activator and the3

inhibitor one; systems of three [11] simultaneously diffusing species have also been considered and shown to display a4

rich zoology of possible patterns and instabilities. Patterns can also develop if only one species is allowed to diffuse in the5

embedding medium [12].6

Besides such deterministic models where the concentrations of the interacting species are assumed to take continuous7

values, one can develop an alternative stochastic individual based description, that accounts for the discrete nature of the8

involved species and where the stochastic contributions arise from the finite size corrections.9

In a series of recent publications, it has been shown that the intrinsic noise is able to create stochastic patterns for10

parameter values for which the deterministic dynamics predicts a stable homogeneous state; the stochastic effects can11

be amplified via a resonant mechanism and thus give rise to stochastic Turing patterns [13–18].12

There is now a well established analytical framework where such stochastic Turing patterns can be studied, that is the13

van Kampen system size expansion. This method allows us to expand the Master Equation in the system size; at the first14

order one recovers the deterministic mean-field model while at the second order, one can obtain a Fokker–Planck equation15

describing the stochastic fluctuations. Instead of explicitly solving the Fokker–Planck equation, one can infer the presence16

of the stochastic patterns by studying the power spectrum of the fluctuations. Such analytical results can be complemented17

by numerical simulations based on the Gillespie algorithm.18

Let us observe that the tools developed in such framework can be used beyond their very first framework of definition.19

Thus motivated by this literature, we decided to apply these techniques in a different setting; in fact as we will explain20

later on, the patterns we found would not be strictly speaking due to a Turing mechanism, requiring that some eigenvalues21

should change the sign of their real parts and thus destabilise the homogeneous solution, they will be purely stochastic22

ones; nevertheless the Fokker–Planck equation would still provide the right framework where to analyse such patterns. In23

our opinion, other studies available in the literature that exhibit such stochastic patterns, see for instance Refs. [19–22],24

could be analysed using such tools.25

The aim of this paper is to study, along the above lines, the existence of stochastic patterns in a Rock–Paper–Scissors26

model with mutation, where three kinds of agents diffuse and interact. Such a model has been introduced firstly in Ref. [19]27

and more recently generalised by Refs. [23,24], allowing for mutation and spatial diffusion. In the latter papers, authors28

proved the existence of spiral waves, in both the deterministic and stochastic model, for small enough mutation rate and29

characterised the temporal behaviour of such spirals in terms of the Hopf frequency of the limit cycle present in the aspatial30

model.31

For a sake of claritywe hereby restrict our analysis to a 1D spatialmodel, however our findings could be straightforwardly32

extended to the 2D case as well. Because the spatio-temporal spectrum of the patterns we found, is very close to the one33

determined in Refs. [23,24], we decided to name such patterns 1D spiral waves. Ourmain result will be to prove the existence34

of stochastic spiral waves for parameter values beyond the ones provided for the mean field approximation by Refs. [23,24].35

Let us finally stress that the hereby proposed approach is different from the Complex Ginzburg–Landau equation used in36

[25,26] and the multi-scale method used in Ref. [24]. To conclude we also performed dedicated stochastic simulations using37

the Gillespie algorithm and confirmed a posteriori the adequacy of the predictions obtained from the theoretical power38

spectrum.39

The paper is organised as follows. In Section 1 we will introduce the model that will be studied in Section 2 using the40

Master Equation and the van Kampen system size expansion and then in the mean-field approximation in Section 3. Finally,41

Section 4 will be devoted to the derivation of the Fokker–Planck equation and of its use to study the intrinsic stochastic42

fluctuations.43

1. The model44

For a sake of completeness let us briefly present the Rock–Paper–Scissors model with mutation; we refer the interested45

reader to Refs. [23,24] for a more complete description. Three populations of agents, say bacteria [27], hereby named S1,46

S2 and S3 are considered; each bacterium can move, reproduce itself and interact-fight with bacteria of the other species.47

The competition is metaphorically described by a Rock–Paper–Scissors game, RPS for short, that is, S1 overcomes S2, S248

overcomes S3 that in turn overcomes S1.49

The non-spatial RPS model possesses an unstable coexistence equilibrium and three unstable equilibria where only one50

species survives, while the generic orbit accumulates to a heteroclinic cycle, that is for longer and longer interval of times51

the amount of two populations of bacteria is almost 0 and the third one almost 1, then the system suddenly jumps to another52

configurationwhere two other species are almost extinguished and so on in a cyclic way [19]. Introducing themutation, one53

can prove [24] that the coexistence equilibrium can become stable if themutation rate is large enough, while if themutation54

rate decreases the system undergoes through a Hopf bifurcation and a limit cycle is created. The 2D spatial extension of the55

model is characterised by a coexistence of species and by the development of spatio-temporal patterns, more precisely,56

spiral waves [23,24], that thus enhance and sustain the biodiversity.57

Let us now introduce the 1D individual based description of the above presented model. The three species of bacteria58

evolve on a linear chain composed by Ω cells with periodic boundary conditions. Each cell has a finite carrying capacity, say59

C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx 3

N and hereby assumed to be the same for all the cells. The number of bacteria of species S1, respectively S2 and S3, in the 1

jth cell, will be denoted by S j1, respectively S j2 and S j3. We also considered the effect of excluded volume, denoting by E j the 2

number of empty spaces available in the jth cell, we finally got: 3

S j1 + S j2 + S j3 + E j= N ∀j. (1) 4

Summarising, the superscript denotes the space location, while the subscript the bacterium type. 5

A bacterium canmove from one cell to one of its two neighbouring cells if enough space is available [28], i.e. the number 6

of vacancies in the destination cell is strictly positive. Assuming all bacteria to have the same diffusivity coefficient, hereby 7

named δ, we obtain: 8

S ji + Ekδ

GGGGGGA Ski + E j, ∀j and k ∈ j, i ∈ 1, 2, 3, (2) 9

where we borrowed from Ref. [29] the notation k ∈ j to denote that k is any of the neighbouring cell of the jth cell. 10

A bacterium can reproduce itself and the resulting offspring will occupy an available space in the same cell, if enough 11

space is at its disposal. We will assume all the bacteria to have the same reproductivity coefficient hereby named β: 12

S ji + E jβ

GGGGGGGA S ji + S ji ∀j, i ∈ 1, 2, 3. (3) 13

In the spirit of the cyclic interaction of the RPS model, we assume that bacteria of species Si dominates over Si+1 while 14

being dominated by Si−1, where we define S3+1 ≡ S1 and S0 ≡ S3. To simplify we further assume that the competition rate 15

is the same for all the bacteria and it will be denoted by σ . Assuming such interactions to hold only among bacteria living 16

in the same cell, we get: 17

S ji + S ji+1

σGGGGGGGA S ji + E j

∀j, i ∈ 1, 2, 3. (4) 18

Let us also consider the presence of a process of dominance-replacement, with rate ζ : 19

S ji + S ji+1

ζGGGGGGA S ji + S ji ∀j, i ∈ 1, 2, 3. (5) 20

Finally themutation introduces the possibility that a bacteriumof one species can transform into one of the other species, 21

that is: 22

S jiµ

GGGGGGGA S ji+1 and S jiµ

GGGGGGGA (6) 23

The time evolution of the above model is completely described by the Master Equation, governing the evolution of the 24

probability to have, at any given time, an amount of bacteria S j1, Sj2 and S j3 in any cells, j ∈ 1, . . . , Ω. As already stated, the 25

van Kampen expansion will provide, at first order, the mean field description of the system, we thus decide to postpone a 26

detailed analysis of the dynamics of such model to Section 3, after having introduced the Master Equation. 27

2. The Master Equation and the van Kampen expansion 28

The state of the system at any time t is completely determined by the amount of bacteria of each species in each cell, thus 29

because of the constraint (1), it will be enough to have: n(t) = [S11(t), S12(t), S

13(t), . . . , S

Ω1 (t), SΩ

2 (t), SΩ3 (t)]. The goal of 30

this section is to introduce a framework where the system evolution can be studied, that is the so-called (chemical) Master 31

Equation. 32

Starting from the ‘‘chemical reactions’’ (2)–(6) it is possible to compute the transition probabilities, Tn′

|n, i.e. the 33

probability for the system to jump from state n(t) to a new compatible one n′(t ′), in small time interval, |t − t ′| ≪ 1. The 34

Master Equation is thus obtained by taking into account all the possible ways by which the system can leave a given state, 35

n, and reach a new state n′: 36

dPdt

(n, t) =

n′=n

T (n|n′)P(n′, t) − T (n′

|n)P(n, t). (7) 37

More precisely the probability that a bacterium S ji , i ∈ 1, 2, 3, moves from the jth cell to a compatible kth one, is given 38

by [30]: 39

TS ji − 1, Ski + 1, E j

+ 1, Ek− 1|S ji , S

ki , E

j, Ek

=δ

zΩS jiE

k

N2where k ∈ j. (8) 40

4 C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx

The factor z stands for the number of nearest neighbour cells, being all the movements equally probable, in the following1

1D case with nearest neighbours we will set z = 2. Let us observe that the above formula is based on the assumption that2

the transition probabilities are proportional to the concentration of each species involved in the reaction and to the rate of3

success of the reaction, that is we assume that in each cell the bacteria are well stirred. Let us also stress that we hereby4

make use of the intensive transition probabilities per unit of time; this approach has been recently largely used (see for5

instance Ref. [31]) and proved to be equivalent to the classical method presented by van Kampen [32] once we rescale the6

time variable (see below).7

The reproduction process (3) of one Si bacterium in the jth cell corresponds to the transition probability:8

TS ji + 1, E j

− 1|S ji , Ej

=β

Ω

S jiEj

N2, (9)9

whereas to the selection mechanism (4), where a S ji bacterium in the jth cell, fights against and destroys an S ji+1 bacterium10

we associate:11

TS ji+1 − 1, E j

+ 1|S ji , Sji+1

=

σ

Ω

S ji+1Sji

N2. (10)12

The remaining cases, Eqs. (5) and (6), can be handled similarly. Let us however observe the different normalisation for the13

mutation Eq. (6), being a ‘‘mono-molecular’’ reaction:14

TS ji±1 + 1|S ji

=

µ

Ω

S jiN

. (11)15

To simplify the notations and to prepare the set up for the following van Kampen expansion, we introduce the step16

operator:17

ϵ±

ij f (. . . , Sji , . . .) = f (. . . , S ji ± 1, . . .), (12)18

where f represents a generic function, the subscript i denotes the different species and j the spatial location. In this way we19

can rewrite the Master Equation (7) as follows:20

dPdt

(n, t) =

3i=1

Ωj=1

k∈j

(ϵ+

i j ϵ−

i k − 1)T (S ji − 1, Ski + 1, E j+ 1, Ek

− 1|S ji , Ski , E

j, Ek)21

+ (ϵ−

i j ϵ+

i k − 1)T (S ji + 1, Ski − 1, E j− 1, Ek

+ 1|S ji , Ski , E

j, Ek)

P(n, t)22

+

3i=1

Ωj=1

(ϵ+

ij ϵ−

i+1 j − 1)T (S ji − 1, S ji+1 + 1, . . . |S ji , Sji+1 . . .)23

+ (ϵ+

i j ϵ−

i−1 j − 1)T (S ji − 1, S ji−1 + 1, . . . |S ji , Sji−1, . . .)24

+ (ϵ−

i j − 1)T (S ji + 1, . . . |S ji , . . .) + (ϵ+

i+1 j − 1)T (S ji+1 − 1, . . . |S ji+1, . . .)25

+ (ϵ+

i+1 jϵ−

i j − 1)T (S ji+1 − 1, S ji + 1, . . . |S ji+1, Sji , . . .)

P(n, t). (13)26

Such an equation is difficult to handle analytically and one has to resort to approximation techniques to progress in the27

study; a possibility is to use the celebrated van Kampen system size expansion [32], a perturbative calculation that recovers28

the mean-field system at the first order and a Fokker–Planck equation describing the fluctuations, at the second order.29

The starting point is the following ansatz, the number of bacteria in each cell is given by a ‘‘regular’’ function plus a30

stochastic contribution, vanishing in the limit of large system size:31

S jiN

= φji +

ξji

√N

, (14)32

more precisely, φji will denote the deterministic concentration of the species i in cell j, in the limit N → ∞, while ξ

ji is a33

stochastic variable that quantifies the intrinsic fluctuations that perturb the idealised mean field deterministic solution φji .34

The amplitude factor 1/√N encodes the finite size of the system and it is the small parameter in the following perturbative35

analysis.36

C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx 5

Putting the van Kampen ansatz into the Master Equation, developing the step operators, collecting together the terms 1

with the same power of√N and rescaling time by t/(NΩ), one recovers at the first order: 2

dφj1

dt(t) = φ

j1[β(1 − r j) − σφ

j3 + ζ (φ

j2 − φ

j3)] + µ(φ

j3 + φ

j2 − 2φj

1) + δ1φj1 + δ(φ

j11r j − r j1φ

j1)

dφj2

dt(t) = φ

j2[β(1 − r j) − σφ

j1 + ζ (φ

j3 − φ

j1)] + µ(φ

j3 + φ

j1 − 2φj

2) + δ1φj2 + δ(φ

j21r j − r j1φ

j2)

dφj3

dt(t) = φ

j3[β(1 − r j) − σφ

j2 + ζ (φ

j1 − φ

j2)] + µ(φ

j1 + φ

j2 − 2φj

3) + δ1φj3 + δ(φ

j31r j − r j1φ

j3),

(15) 3

where we introduced r j =3

i=1 φji and the discrete Laplacian 1fj :=

2z

k∈j

fk − fj

. Let us remember that in the present 4

case of 1D system with nearest neighbours z = 2 and k ∈ j − 1, j + 1. 5

The effect of the finite carrying capacity reflects in the abovemean-field equations through the non-linear cross diffusion 6

terms (φji1r j − r j1φ

ji) which appear to modify the conventional Fickean behaviour. These are second order contributions Q2 7

in the concentrations and are therefore important in the regime of high densities [33,15,17]. 8

The expansion to the next leading order will determine a Fokker–Planck equation describing the probability distribution 9

of the fluctuation, Π(ξ , t), that will be introduced and analysed in Section 4. 10

3. Analysis of the mean-field system 11

Let us start our analysis by considering the spatially homogeneous solutions of the previous system (15), namely we 12

assume the following limit does exist and it is independent from the spatial location j: 13

limN→∞

S jiN

= φi, (16) 14

hence (15) rewrites: 15

dφi

dt(t) = φi[β(1 − (φi + φi+1 + φi−1)) − σφi−1 + ζ (φi+1 − φi−1)] + µ(φi−1 + φi+1 − 2φi) i ∈ 1, 2, 3, (17) 16

where we used once again the notation φ3+1 ≡ φ1 and φ0 ≡ φ3. 17

A straightforward analysis [23] of the above system shows that it admits the equilibrium point: 18

S∗=

β

3β + σ,

β

3β + σ,

β

3β + σ

, (18) 19

and the system behaviour can be summarised by: 20

• for µ > µH , where µH =βσ

6(3β+σ), S∗ is stable focus and the trajectories generically converge to S∗; 21

• at µ = µH there is a supercritical Hopf bifurcation and the associated frequency is ωH =

√3β(σ+2ζ )

2(3β+σ); 22

• µ < µH , S∗ is an unstable focus and a stable limit cycle emerges from the Hopf bifurcation. 23

Because of the cyclic competition, the Jacobianmatrix of the system (17) is a circulating matrix, that evaluated at S∗ reduces 24

to: 25

JS∗ =

a0 a2 a1a1 a0 a2a2 a1 a0

, (19) 26

where: 27

a0 =−β2

− 2µ(3β + σ)

3β + σa1 =

13β + σ

−β2

− σβ + µ(3β + σ) − ζβ

a2 =1

3β + σ

−β2

+ ζβ + µ(3β + σ).

(20) 28

The eigenvalues are thus easily obtained: 29

λ0 = −β, 30

λ1,2 =1

2(3β + σ)

(−6µ(3β + σ) + σβ) ± i

√3 (σβ + 2ζβ)

. (21) 31

6 C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx

0.2

0.15

0.1

0.05

00 0.5 1 1.5 2 2.5 3

Fig. 1. Parameters plane (ζ , µ). We fix β = 1 and σ = 1 and we delimited two zones corresponding to two different dynamical behaviours: (zone I)existence of a stable limit cycle, (zone II) presence of a stable fixed point. The zones are separated by the lineµ = µH where the system undergoes througha Hopf bifurcation.

In Fig. 1 we summarise the dynamics of the homogeneous system as a function of two parametersµ and ζ once we fixed1

the remaining two β and σ . Let us observe that the plane (ζ , µ) is divided in two zones, in the first one (zone I) the system2

presents a stable limit cycle and an unstable fixed point, while in the second one (zone II) there is a stable fixed point. The3

line separating the two zones, µ = µH , corresponds to the supercritical Hopf bifurcation.4

We are now able to recover the space dependence and interested in identifying conditions yielding to a spontaneous5

amplification of the perturbation and eventually translate in the emergence of stochastic patterns. To this end, and following6

the standard approach, we consider the linear stability analysis of the full system (15) close to the homogeneous solution7

φi = S∗ for i = 1, 2 and 3. To better understand the system behaviour we analyse the linearised system in the Fourier space,8

where the Jacobian of the non-homogeneous system, J∗NH, reads:9

J∗NH = JS∗ + D∗∆,10

∆ is the Fourier transform of the Laplacian and D∗ is the diffusion matrix evaluated at S∗:11

D∗= δ

b0 b1 b1b1 b0 b1b1 b1 b0

, (22)12

where:13

b0 = (β + σ)/(3β + σ) and b1 = β/(3β + σ). (23)14

The eigenvalues of the linearised system, in Fourier space, are (once we approximate ∆ with −k2):Q315

ρ0 = −β − δk2, (24)16

ρ1,2 =1

2(3β + σ)

−6µ(3β + σ) + σβ − 2δσk2

±

i√3

2(3β + σ)[σβ + 2ζβ] . (25)17

We can observe that ρ0 is always negative and thus corresponds to a stable direction also for the spatial system. The18

interesting dynamics is hence reduced to study the other two eigenvalues. The imaginary parts of these eigenvalues do19

not depend on k2 and thus they are the same as the homogeneous case. The real parts differ for the new term −2δσk2. So20

we can conclude that the mode k = 0 has the same behaviour as the aspatial system; moreover because the mode k = 021

dominates the dynamics induced by the other modes, even if the real parts of ρ1,2 can change their signs with respect to the22

aspatial case, they cannot introduce any new dynamical behaviour. So the patterns we eventually find would not due to a23

Turing like mechanism because no eigenvalue will change its real part and thus destabilise the homogeneous solution, they24

will be purely stochastic ones.25

The solutions of the system (15) for parameters in zone II converge to the spatial homogeneous solution, on the other26

hand once parameters are fixed in the zone I one can obtain stable patterns, i.e. spatially organised and time synchronised27

structures, as reported in Fig. 2, where we represent on the left panel the results of a numerical integration of the 1D system28

and on the right panel a snapshot of the numerical integration of the 2D model where spiral waves can be observed as29

already reported by Ref. [23].30

A more complete understanding of the patterns presented in Fig. 2 can be obtained by analysing the Fourier spectrum,31

both spatial and temporal one (see Fig. 3). In Fig. 4 we plot the power spectrum of the temporal Fourier transform of the32

patterns shown in Fig. 2, we can observe that in both 1D and 2D cases, the spectra behave in a similar way with a clear33

peak at a frequency that is close to the Hopf frequency; also the spatial power spectra (data not shown) exhibit a similar34

C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx 7

a b

Fig. 2. Deterministic patterns. Panel (a): numerical integration of 1D system, we report the time evolution of the concentrations of species S1 , being theones for S2 and S3 similar because of the cyclic dominance. Panel (b): generic time snapshot of the spiral waves for the 2D model, still for species S1 . Bothsimulations refer to the following parameter values: ζ = 0.6, β = 1, σ = 1,µ = 0.02 < µH = 1/24, δ = 1/L2 and the spatial domain has been discretisedinto L = 128 identical cells.

Fig. 3. Numerical power spectrum in the spatio-temporal domain for the 1D model for the same parameters used for Fig. 2.

a b

Fig. 4. Numerical power spectrum in the temporal domain. Panel (a) represents the logarithm of the power spectrum for the 1Dmodel for a generic spatialposition. Panel (b) represents the logarithm of the power spectrum of the 2D model for a fixed generic spatial coordinates. Simulations have been doneusing the same set of parameters used for Fig. 2.

8 C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx

behaviour with a decrease of the spectrum as a function of the spatial modes. The similarity of such behaviours allows us to1

term the pattern observed in the panel (a) of Fig. 2 one dimensional spiral waves [34].2

In previous studies [23,24] authors studied the robustness of the spiralwaves against the noise and of the systemparame-3

ters and concluded that stochastic spirals waves exist for parameters in zone I, that is in the same range as for themean-field4

approximation. Our goal is to prove that stochastic spiral waves do exist in a larger parameters domain, covering part of the5

zone II, where the mean-field solutions converge to the homogeneous one. To achieve our goal we will characterise such6

patterns by analysing the power spectrum of the fluctuations. To this endwe need to introduce and study the Fokker–Planck7

equation that governs the evolution of the fluctuations. Let us observe that our approach is based on methods similar to the8

ones developed in Refs. [35,14] and thus it is completely different from the Complex Ginzburg–Landau equation used in9

[25,26] and the multi-scale method used in Ref. [24].10

4. Fokker–Planck equation and fluctuations power spectrum11

As already stated, the next to the leading order in the van Kampen system size expansion allows us to characterise the12

distribution of the fluctuations, Π(ξ , t), in fact a cumbersome computation allows us to derive a Fokker–Planck equation13

for Π(ξ , t):14

dΠ

dt(ξ , t) = −

3i=1

Ωj=1

∂

∂ξji

Q ji (ξ)Π(ξ , t)

+

12

3i=1

Ωj=1

p∈j

3h=1

Rih∂2Π(ξ , t)

∂ξji ∂ξ

ph

, (26)15

wherewedenoted by ξ = (ξ 11 , ξ 1

2 , ξ 13 , . . . , ξΩ

1 , ξΩ2 , ξΩ

3 ). In the above expressionQ ji can be expressed in terms of the Jacobian16

M of themean-field system (17) evaluated at the fixed point S∗. For a sake of claritywe splitM into two parts: the one named17

M(d), depending on the diffusion part of the mean-field equation, i.e. involving the Laplacian, and the remaining one, M(r),18

i.e. associated to the reaction terms. More explicitly we get:19

M(r)i i−1 = µ − (β + σ + ζ )S∗

20

M(r)i i+1 = µ − (β + ζ )S∗

21

M(r)ii = −2µ + β

1 − 4S∗

− σ S∗

22

M(r)ih = 0 in all the remaining cases. (27)23

On the other hand the spatial contribution is:24

M(d)ii = δ[1 + 2S∗

]25

M(d)ih = δS∗ for all h = i. (28)26

Finally:27

Mih = M(r)ih + M(d)

ih . (29)28

We can thus write:29

Q ji (ξ) =

3h=1

M(r)

ih ξjh + M(d)

ih 1ξjh

, (30)30

where ∆ is the discrete Laplacian.31

The same splitting can be applied to the matrix R, still evaluated at the fixed point S∗:32

R(r)ii = βS∗(1 − 3S∗) + 4µS∗

+ 2ζ S∗+ σ(S∗)233

R(r)i i−1 = −µS∗

34

R(r)i i+1 = −µS∗

+ ζ (S∗)235

R(r)ih = 0 in all the remaining cases, (31)36

and37

R(d)ii = −2δS∗

[1 − 3S∗]38

R(d)ih = 0 h = i, (32)39

and thus we get:40

Rjpih =

R(r)ih + R(d)

ih ∆jp

, (33)41

C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx 9

where ∆jp denotes the discrete Laplacian associated to the linear lattice with periodic boundary conditions: 1

∆jp = Wjp − 2δjp, 2

being δjp the Kronecker delta and Wjp the matrix given by: 31 if j = p ± 10 otherwise. 4

To handle the Fokker–Planck is not so straightforward we thus prefer to pass to an equivalent Langevin equation [36], 5

where the noise term intrinsically depends on the system fluctuations: 6

dξ ji

dt= Q j

i (ξ) + ηji(t), (34) 7

and the stochastic contribution satisfies the following relations: 8

⟨ηji(t)⟩ = 0 and ⟨η

ji(t)η

ph(t

′)⟩ = Rjpihδ(t − t ′). (35) 9

Recalling Eq. (30), introducing spatial and temporal Fourier variables Eq. (34) reads (denoting by f k(ω) the spatio-temporal 10

Fourier transformation of a function f (x, t)): 11

−iωξik(ω) =

3h=1

[Mrihξ

kh + Md

ih1ξkh] + ηk

i (ω), (36) 12

where, 13

⟨ηki (ω)ηk′

h (ω′)⟩ = Ω Rkihδk,−k′δω,−ω′ . (37) 14

Eq. (36) can be rewritten as: 15h

−iωδih − [Mr

ihξkh + Md

ih1ξkh]

ξkh = ηk

i (ω), (38) 16

whose solution is: 17

ξkh =

i

Φk

ih(ω)−1

ηki (ω), (39) 18

being 19

Φkih(ω) = −iωδih − [Mr

ihξkh + Md

ih1ξkh]. (40) 20

We are now able to analytically compute the power spectrum Pi(k, ω) of the fluctuations for each species i = 1, 2, 3: 21

Pi(k, ω) = ⟨|ξki (ω)|2⟩ = Ω

3j=1

3u=1

[Φk(ω)]−1ij Rk

ju[ΦkĎ(ω)]−1

ui , (41) 22

where ΦĎ= ΦT . 23

Using the above formula for the power spectrum of the fluctuations we are able to study the microscopic system for 24

parameter values outside the regions of deterministic order, that is part of zone II, with the goal of looking for the signatures 25

of a spatio-temporal organisation. Because themean-field deterministic model will not display the same patterns, the latter 26

should ultimately reflect the discreteness of the investigated stochastic model. To complement our analytical results, we 27

will compare the power spectrum (41) with the numerical one obtained through a spatio-temporal FFT of the solutions of 28

the microscopic system got using the Gillespie algorithm. 29

Results reported in Fig. 5 allow to conclude that the system has a spatio-temporal organisation also in (part of) zone II, 30

in fact both power spectra present a clear peak in the ω variable, whose value is close to the Hopf frequency ωH , and rapidly 31

decrease as k2 increase. To quantify such similarities we compute the maximum error, maxω,k |Pnum(ω, k) − Pth(ω, k)|, that 32

results to be 0.24. To better appreciate the similarities we show in Fig. 6a comparison of two sections of the two spectra, Q4 33

ω = ωmax (being the value at which the spectrum has amaximum) and k = kmin (being the smallest non zero spatial mode), 34

we can clearly see that the main discrepancy arises in the width of the peak. The agreement between the two spectra, gives 35

a confirmation a posteriori of the validity of our analytical formula and the assumptions so far used. 36

In Fig. 7 we report a numerical simulation of the Gillespie algorithm of the 1D individual based model, for the same 37

parameters in zone II used to obtain the power spectrum presented in Fig. 5, where we can observe the spatio-temporal 38

patterns, distorted by the noise. 39

Let us stress that the two spectra reported in Fig. 5 present also some differences, the main ones being the larger width 40

of the numerical spectrum in the ω variable close to the maximum, that is not present in the theoretical one (see left 41

10 C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx

P(ω

,k)

1

0.5

0

P(ω

,k)

1

0.5

03

21

32

10

0

0.2

0.4

0.6

k k

ω ω0.2

0.4

0.6

a b

Fig. 5. Power spectrum of the fluctuations. Panel (a): we report the analytical power spectrum given by formula (41). Panel (b): the numerical powerspectrum of the fluctuations for species S1 is reproduced (N = 5000 and Ω = 32). Both panels refer to parameter values in zone II; more precisely, wehave set µ = 0.043 > µH , ζ = 0.6, β = 1, σ = 1 and δ = 1/(2 · Ω2). The numerical power spectrum is obtained by averaging over 150 independentrealisations.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 2.5 3

a b

Fig. 6. Bidimensional sections of the power spectra of the fluctuations for species S1 presented in Fig. 5. Panel (a): we compare the section k = const forboth the numerical spectrum – solid blue (on line) curve – and the analytical spectrum – black dashed curve. Panel (b): we compare the section ω = constof the above spectra, solid blue (on line) curve for the numerical one and black dashed curve for the theoretical one. Both panels refer to parameter valuesin zone II; more precisely, we have set µ = 0.043 > µH , ζ = 0.6, β = 1, σ = 1 and δ = 1/(2 · 322). The numerical power spectrum is obtained byaveraging over 150 independent realisations and N = 5000.

Fig. 7. Stochastic patterns. We report the time evolution of the concentration of S1 obtained through a numerical integration of 1D system using theGillespie algorithm to be compared with the one of Fig. 2. The parameter values are the same used to get the results presented in Fig. 5.

C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx 11

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.05

0.1

0.15

0.35 0.4 0.45

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25

0.2

0.15

0.1

0.05

00.35 0.4 0.45

106

105

104

103

102

102 103 104

a b c

Fig. 8. Impact of the number of averages and of the finite carrying capacity on the numerical spectrum. Panel (a): using the bidimensional section k = constwe report the numerical spectrum obtained using several independent repetitions uponwhichwe compute the averages (black dot-dashed curve n = 100,green (on line) dotted curve n = 200, blue (on line) solid curve n = 10, red (on line) dashed curve n = 50), the system size has been set to N = 5000. Insetof panel (a): a zoom to emphasise the queue of the power spectrum. Panel (b): using the bidimensional section k = constwe report the numerical spectrumcomputed for different system sizes (solid blue (on line) curve N = 1000, dot-dashed black curve N = 5000, dotted green (on line) curve N = 510, dashedred (on line) curve N = 210), the number of independent realisations upon which to compute the averages has been set to n = 200. Inset of panel (b): azoom to emphasise the fluctuations in the queue of the power spectrum. Panel (c): we report the variation of the spectrum as a function of the system sizeN , the solid line represents a best linear fit performed retaining only the five largest values of N: log10 y = a log10 N + b with a = −1.22, 95% confidenceinterval (−1.35, −1.10), and b = 7.06, 95% confidence interval (6.62, 7.49). All panels refer to parameter values in zone II; more precisely, we have setµ = 0.043 > µH , ζ = 0.6, β = 1, σ = 1 and δ = 1/(2 · 322). (For interpretation of the references to colour in this figure legend, the reader is referred tothe web version of this article.)

panel of Fig. 6), and the long tail for all values of k and ω ∼ ωmax exhibited by the numerical spectrum and absent in 1

the theoretical one that can be seen also in the right panel of Fig. 6. Such phenomena are direct consequences of the intrinsic 2

stochasticity of the model: the carrying capacity N induces finite size corrections that cannot be removed with a finite 3

number of independent realisations upon which to compute the averages, the theoretical spectrum thus cannot reproduce 4

such traits. We performed a dedicated set of numerical simulations to test such hypotheses and the results reported in Fig. 8 5

confirm our previous statement: both the number of realisations used to compute the averages and the system size do play 6

a role in the computation of the numerical spectrum, the larger such quantities the smoother the spectrum. In particular in 7

panel (c) of Fig. 8 we reported the variation [37] of the numerical spectrum, namely a proxy for the roughness, as a function 8

of the system size, we can observe that the spectrum becomes smoother and smoother as N increases. 9

In conclusion, besides the previous numerical phenomena, we can observe that both numerical and theoretical spectra 10

presented in Fig. 5 are very similar to the one reported in Fig. 3 for parameter values in the zone I where the system exhibits 11

deterministic spiral waves. We can thus conclude that the results presented in Ref. [23] hold in a larger domain, for instance 12

for µH < µ < µ∗, for some positive µ∗. To determine such value µ∗ and more generally the parameters set, beyond which 13

the noise will completely destroy the patterns, is surely an interesting question to which we will devote a forthcoming 14

analysis, we hereby limit ourselves to present a preliminary numerical result by showing the ‘‘degradation’’ of the power 15

spectrum as the parameter µ increases. Inspecting Fig. 9 we can observe that the roughness become larger and larger as 16

well as the spectrumwidth. Taking as reference the shape of the spectrum computed for µ = 0.043 we could, qualitatively, 17

state that the patterns persist up to µ ∼ 0.051, beyond this value the spectrum deteriorates too much. 18

Let us finally conclude that such stochastic patterns do persist for large values ofµwell beyondµH as can be numerically 19

inspected in Fig. 10, where we report numerical simulations of the individual based model and the corresponding tempo- 20

spatial Fourier transform for µ = 0.05 ≫ µH . 21

5. Conclusion 22

Spatio-temporal patterns are widely spread and encompass several research fields; in this scenario the Turing instability 23

is one of the mechanisms that can be used to understand the emergence of such ordered patterns in reaction–diffusion 24

models. Recent results have shown that such spatio-temporal patterns are robust against intrinsic noise and they can persist 25

for parameter values well beyond the ones fixed by the deterministic setting, e.g. the mean-field approximation. 26

Motivated by such results, in this paper we have considered a 1D version of the Rock–Paper–Scissors model with 27

mutation [24], able to describe the coexistence of different species in a square lattice. We formulated the model as an 28

individual based one, taking into account for the finite carrying capacity of each lattice cell. A preliminary analysis of the 29

mean-field approximation allows us to recover the results by Ref. [23], proving the existence of spatio-temporal ‘‘wave-like 30

spirals’’ once the homogeneous model admits a limit cycle because of a Hopf bifurcation, µ < µH . 31

However, being the proposed model inherently stochastic, we computed the Master Equation, describing the evolution 32

of probability of being in a given state, and thus we performed a van Kampen system size expansion. Studying the first two 33

terms of such approximation, we have been able to prove that such patterns are also present in (part of) the µ > µH zone 34

where the mean-field approximation predicts the existence of a stable homogeneous solution; hence such structures are 35

12 C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00.80.70.60.50.40.30.20.1

0.043

0.047

0.051

0.055

0.059

0.063

Fig. 9. Numerical power spectrum as a function of µ. Using the bidimensional section k = const we report the numerical spectrum computed for severalvalues of µ > µH . All spectra refer to parameter values in zone II; more precisely, we have set N = 5000, ζ = 0.6, β = 1, σ = 1, δ = 1/(2 · 322) and eachspectrum has been computed averaging 200 independent realisations of the Gillespie algorithm.

P(ω

,k)

1

0.8

0.6

0.4

0.2

03

2.52

1.5 1 0.50.2

0.40.6

0.8

ba

Fig. 10. Stochastic patterns for large µ. Left panel, we report the time evolution of the concentration of S1 obtained through a numerical integration of 1Dsystem using the Gillespie algorithmwith parametersµ = 0.05, ζ = 0.6, β = 1, σ = 1, δ = 1/(2Ω2),N = 50 andΩ = 128. Right panel, power spectrumin the spatio-temporal domain for the 1D model for the same parameters used in left panel. The numerical power spectrum is obtained by averaging over150 independent realisations.

entirely driven by the intrinsic noise. As already stated our method is complementary to the ones used previously in the1

literature, the Complex Ginzburg–Landau equation and the multi-scale method.2

Acknowledgements3

The authors would like to warmly thank Duccio Fanelli for useful comments and discussions.4

This research used computational resources of the ‘‘Plateforme Technologique de Calcul Intensif (PTCI)’’ located at the5

University of Namur, Belgium, which is supported by the F.R.S.-FNRS.6

This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization),7

funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientificQ58

responsibility rests with its author(s).9

References10

[1] A. Ord, B.E. Hobbs, Fracture pattern formation in frictional, cohesive, granular material, Phil. Trans. R. Soc. A 368 (2010) 95.11

[2] J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, third ed., Springer-Verlag, 2003.12

C. Cianci, T. Carletti / Physica A xx (xxxx) xxx–xxx 13

[3] B.P. Belousov, Periodically acting reaction and its mechanism, Coll. Abstr. Radiat. Med. 145 (1959) 147.Q6 1

[4] S. Strogatz, Non Linear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, Perseus Book Group, 2001. 2

[5] L. Yang, M. Dolnik, A.M. Zhabotinsky, E.R. Epstein, Pattern formation arising from interaction between turing and waves instability, J. Chem. Phys. 117(2002) 7259.

3

[6] M.A. Novak, R.M. May, Evolutionary games and spatial chaos, Nature 359 (1992) 827. 4

[7] M.A. Novak, S. Bonhoeffer, R.M. May, More spatial games, Int. J. Bifurcation Chaos 4 (1993) 33. 5

[8] J.Y. Wakano, Ch. Hauert, Pattern formation and chaos in spatial ecological public goods games, J. Theoret. Biol. 268 (2011) 30. 6

[9] R. deForest, A. Belmonte, Spatial pattern dynamics due to the fitness gradient flux in evolutionary games, Phys. Rev. E 87 (2013) 062138. 7

[10] A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B 237 (1952) 37–72. 8

[11] R.A. Satnoianu, M. Menzinger, P.K. Maini, Multispecies reaction diffusion models and the turing instability revisited, Math. Biol. 41 (2000) 493–512. 9

[12] B. Ermentrout, M. Lewis, Pattern formation in systems with one spatially distributed species, Bull. Math. Biol. 59 (1997) 533–549. 10

[13] T. Butler, N. Goldenfeld, Fluctuation-driven turing patterns, Phys. Rev. E 84 (2011) 011112. 11

[14] T. Biancalani, D. Fanelli, F. Di Patti, Stochastic turing patterns in the brusselator model, Phys. Rev. E 81 (2010) 046215. 12

[15] D. Fanelli, C. Cianci, F. Di Patti, Multispecies reaction diffusion models and the turing instability revisited, Eur. Phys. J. B 86 (2013) 142. 13

[16] E.A. Gaffney, T.E. Woolley, R.E. Baker, P.K. Maini, Stochastic reaction and diffusion on growing domains: understanding the breakdown of robustpattern formation, Phys. Rev. E 84 (2011) 046216.

14

[17] C. Cianci, D. Fanelli, Stochastic patterns and the role of crowding, Discontinuity, Nonlinearity, Complexity 2 (2013) 301–319. 15

[18] L. Cantini, C. Cianci, D. Fanelli, E. Massi, L. Barletti, Stochastic turing patterns for systems with one diffusing species, J. Math. Biol. (2014). 16

[19] R.M. May, W.J. Leonard, Non linear aspects of competition between three species, SIAM J. Appl. Math. 29 (2) (1975) 243–253. 17

[20] M. Kness, D. Barkley, L.S. Tuckerman, Spiral-wave dynamics in a simple model of excitable media: the transition from simple to compound rotation,Phys. Rev. A 42 (1990) 2489–2492.

18

[21] D. Barkley, Linear stability analysis of spiral waves in excitable media, Phys. Rev. Lett. 68 (1992) 2090–2093. 19

[22] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys. 17 (1955) 257–278. 20

[23] T. Reichenbach, M. Mobilia, E. Frey, Mobility promotes and jeopardizes biodiversity in rockpaperscissors games, Nature 30 (2007) 448. 21

[24] B. Szczesny, M. Mobilia, A.M. Rucklidgez, When does cyclic dominance lead to stable spiral waves? Europhys. Lett. (2013) 28012. 22

[25] E. Frey, Evolutionary game theory: theoretical concepts and applications to microbial communities, Physica A 389 (2010) 4265–4298. 23

[26] T. Reichenbach, M. Mobilia, E. Frey, Noise and correlations in a spatial population model with cyclic competition, Phys. Rev. Lett. 99 (2007) 238105. 24

[27] Let us observe that the analogy is pertinent because the RPS behaviour has been found in experiment with Escherichia coli bacteria [38]. 25

[28] We could also have consider the possibility for two bacteria to hop, that is exchange their places in two neighbouring cells. Because this new action 26

will not have introduced any new phenomenon, we decided to not consider it and to have a model as simple as possible. 27

[29] D. Fanelli, T. Dauxois, F. Di Patti, A.J. McKane, Enhanced stochastic oscillations in autocatalytic reactions, Phys. Rev. E 79 (2009) 036112. 28

[30] To lighten the notations, we hereby indicate only the variables whose values change because of the transition. 29

[31] A.J. McKane, T.J. Newman, Stochastic models in population biology and their deterministic analogs, Phys. Rev. E 70 (2004) 041902. 30

[32] N.G. van Kampen, Stochastic Preocesses in Physics and Chemistry, North-Holland, Amsterdam, 1992. 31

[33] D. Fanelli, A.J. McKane, Diffusion in a crowded environment, Phys. Rev. E 82 (2010) 021113. 32

[34] Let us observe that this spectrum is also very similar to the one obtained applying a FFT to pictures of spirals. 33

[35] C.A. Lugo, A.J. McKane, Quasi-cycles in a spatial predator–prey model, Phys. Rev. E 78 (2008) 051911. 34

[36] C.W. Gardiner, Handbook of Stochastic Methods, second ed., Springer, 1985. 35

[37] Given a differentiable function f (x) its variation on the interval [a, b] is given by ba ∥f ′(x)∥ dx, in the present case we approximate the first derivative 36

with the forward finite difference. 37

[38] B. Kerr, M.A. Riley, M.W. Feldman, B.J. Bohannan, Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors, Nature 418 (2002)172.

38