We analyze the spatio-temporal patterns of two competing species in the
presence of two white noise sources: an additive noise acting on the
interaction parameter and a multiplicative noise which affects directly the
dynamics of the species densities. We use a coupled map lattice (CML) with
uniform initial conditions. We find a nonmonotonic behavior both of the pattern
formation and the density correlation as a function of the multiplicative noise
intensity.Comment: 10 pages, 7 figures. accepted for publication in Acta Phys. Pol. 

Fiasconaro, A.

Spagnolo, B.

Valenti, D.

English

arXiv

We analyze the spatio-temporal patterns of two competing species in the presence of two white noise sources: an additive noise acting on the interaction parameter and a multiplicative noise which aects directly the dynamics of the species densities. We use a coupled map lattice (CML) with uniform initial conditions. We nd a nonmonotonic behavior both of the pattern formation and the density correlation as a function of the multiplicative noise intensity

VALENTI D

FIASCONARO A

SPAGNOLO B

Archivio istituzionale della ricerca - Università di Palermo

Vol. 35 (2004) ACTA PHYSICA POLONICA B No 4PATTERN FORMATION AND SPATIALCORRELATION INDUCED BY THE NOISEIN TWO COMPETING SPECIES∗D. Valenti, A. Fiasconaro and B. SpagnoloINFM - Unità di Palermo, Group of Interdisciplinary Physics andDipartimento di Fisica e Tecnologie Relative, Università di PalermoViale delle Scienze, I-90128 Palermo, Italy(Received December 4, 2003)We analyze the spatio-temporal patterns of two competing species inthe presence of two white noise sources: an additive noise acting on theinteraction parameter and a multiplicative noise which affects directly thedynamics of the species densities. We use a coupled map lattice (CML)with uniform initial conditions. We find a nonmonotonic behavior bothof the pattern formation and the density correlation as a function of themultiplicative noise intensity.PACS numbers: 05.40.–a, 87.23.Cc, 89.75.Kd, 87.23.–n1. IntroductionWe present a stochastic model for spatial distribution of two compet-ing species. Our theoretical model could be useful to describe biologicalsystems, where the presence of fluctuations, such as random variability oftemperature, can modify strongly the dynamics of an ecosystem [1, 2]. Wefocus on the role played by the noise on the transient dynamics of the spatialdistributions of two competing species belonging to an ecosystem describedby generalized Lotka–Volterra equations [3] in the presence of multiplicativenoise. We find nonmonotonic behaviors for the pattern formation and thedensity correlation of the species as a function of the multiplicative noise in-tensity. The theoretical results could contribute to select environmental andperiodical driving forces using the proper space and time scales to developphysical models of population dynamics useful to interpret spatial patternsin the abundance of the species [4, 5].∗ Presented at the XVI Marian Smoluchowski Symposium on Statistical Physics,Zakopane, Poland, September 6–11, 2003.(1481)1482 D. Valenti, A. Fiasconaro, B. Spagnolo2. The modelTo study the spatial effects due to the presence of noise sources we con-sider a discrete time evolution model, which is the discrete version of theLotka–Volterra equations with diffusive terms, namely a coupled map lat-tice [6].The time evolution of the spatial distribution for the two species is givenby the following equationsxn+1i,j = µxni,j(1 − xni,j − βnyni,j) +√σxxni,jXni,j + D∑γ(xnγ − xni,j) , (1)yn+1i,j = µyni,j(1 − yni,j − βnxni,j) +√σyyni,jYni,j + D∑γ(ynγ − yni,j) , (2)where xni,j and yni,j denote respectively the densities of prey 1 and prey 2 inthe site (i, j) at the time step n, µ is proportional to the growth rate, D isthe diffusion constant,∑γ indicates the sum over the four nearest neighbors.The random terms are modeled by independent Gaussian variables de-noted by Xni,j , Yni,j with zero mean and variance unit and βn takes intoaccount for the interaction between the species. In Eqs. (1) and (2), σx andσy are the intensities of the multiplicative noise which models the interactionbetween the species and the environment.2.1. Stochastic resonanceIt is known that for β < 1 a coexistence regime takes place, that isboth species survives, while for β > 1 an exclusion regime is established,that is one of the two species vanishes after a certain time [7, 8]. Coexis-tence and exclusion of one of the two species correspond to stable states ofthe Lotka–Volterra’s deterministic model [3]. Real ecosystems are immersedin a noisy nonstationary environment, so also the interaction parameter isaffected by the noise and some other deterministic periodical driving suchas the temperature. The change in the competition rate between exclusionand coexistence occurs randomly because of the coupling between the lim-iting resources and the noisy environment. A random variation of limitingresources produces a random competition between the species. The noise,therefore, together with the periodic force determines the crossing from adynamical regime (β < 1, coexistence) to the other one (β > 1, exclusion).To describe this continuous and noisy behavior of the interaction parame-ter β(t) we consider an Ito stochastic differential equation with a bistablepotential and a periodical driving forcedβ(t)dt= −dU(β)dβ+ γ cos(ω0t) + ξβ(t) , (3)Pattern Formation and Spatial Correlation . . . 1483where U(β) is the bistable potential (see Fig. 1)U(β) = h(β − (1 + ρ))4/η4 − 2h(β − (1 + ρ))2/η2 , (4)the periodical driving mimics the climatic temperature oscillations, and ξβ(t)is a white Gaussian noise with 〈ξβ(t)〉 = 0 and 〈ξβ(t)ξβ(t′)〉 = σβδ(t − t′).Since the dynamics of the species strongly depends on the value of the in-teraction parameter, we initially consider the time evolution of β for σβ = 0(see Fig. 2).0.90 0.93 0.96 0.99 1.02 1.05 1.08U-0.002-0.0010.0000.0010.002hFig. 1. The bistable potential U(β) of the interaction parameter β(t). The potentialU(β) is centered on β = 0.99. The parameters of the potential are h = 6.25×10−3,η = 0.05, ρ = −0.01.t0 100 2000.900.951.001.051.10Fig. 2. Time evolution of the interaction parameter β for σβ = 0 and initial valueβ(0) = 0.94. The values of the parameters are: γ = 1.5 · 10−1, ω0/(2pi) = 10−2. Inthe absence of noise β(t) oscillates below the critical value βc = 1. The transientbehavior near the initial point β(0) = 0.94 is due to the choice of cosine functionin Eq. (3) as deterministic periodical driving.1484 D. Valenti, A. Fiasconaro, B. SpagnoloWe note that in the absence of the additive noise ξβ(t), β(t) shows aperiodical evolution but its values always remain below β = 1, i.e. in thecoexistence regime. The noise can synchronize with the periodical drivingforce. In this case a Stochastic Resonance (SR) [9, 10] effect appears whichaffects strongly the dynamics of the system [7, 11, 12]. Therefore we fix theadditive noise intensity at the value σβ = 2.65 × 10−3 corresponding to acompetition regime with β periodically switching from coexistence to exclu-sion regions [12] (see Fig. 3). The SR effect in the dynamics of interactionparameter β induces SR phenomenon in the dynamics of two competingspecies. This produces noise-induced anticorrelated periodic oscillations intime evolution of the two species densities (see Ref. [12]).t0 100 2000.900.951.001.051.10Fig. 3. Time evolution of the interaction parameter β(t) for σβ = 2.65 × 10−3.The values of the parameters and the initial value β(0) are the same as in Fig. 2.For this level of noise a synchronization appears (Stochastic Resonance) and β(t)oscillates quasi-periodically below and above the critical value βc = 1.2.2. Spatial distributionsWe consider the time evolution of the spatial distribution of the ecosys-tem, described by Eqs. (1) and (2), in the SR dynamical regime. Theinteraction parameter βn of Eqs. (1), (2) corresponds to the value of contin-uous β(t) of Eq. (3) taken at the step n. So we fix the additive noise at thevalue σβ = 2.65 × 10−3 and we vary both the intensities of multiplicativenoise. In Figs. 4 and 5 we report the spatio-temporal patterns of the twospecies for different values of the multiplicative noise intensity σ = σx = σy,namely σ = 10−12, 10−8, 10−4, 10−1 with µ = 2, D = 0.05, γ = 1.5 × 10−1,ω0/(2pi) = 10−2, β(0) = 0.94 and x0i,j = y0i,j = 0.5 at all sites (i, j). We seethat for very low noise intensity (see Fig. 4(a)) an average correlation onthe considered lattice (N = 100× 100) between the species is observed. ForPattern Formation and Spatial Correlation . . . 1485(a) (b)Fig. 4. Spatial distributions at different times for (a) σ = 10−12 and (b) σ = 10−8.The value of the additive noise is fixed at σβ = 2.65 × 10−3. The values of theparameters are: µ = 2, D = 0.05, γ = 1.5× 10−1, ω0/(2pi) = 10−2, N = 100× 100.The initial values are x0i,j = y0i,j = 0.5 for all sites (i, j) and β(0) = 0.94.(a) (b)Fig. 5. Spatial distributions at different times for (a) σ = 10−4 and (b) σ = 10−1.The additive noise intensity, the values of the parameters and the initial conditionsare the same as in Fig. 4.higher noise intensities (see Fig. 4(b), 5(a)) an anticorrelation between thetwo species is observed: the two species tend to occupy different positions.The anticorrelation is more evident in Fig. 5(a). By increasing the multi-plicative noise the anticorrelation is strongly reduced (see Fig. 5(b)). Further1486 D. Valenti, A. Fiasconaro, B. Spagnoloincrease of the noise (σ = 10+3) causes the anticorrelation to disappear andthe two species densities become uncorrelated. We note also that the averagesize of the patterns increases with the noise intensity non-monotonically: atvery low noise intensity (σ = 10−14) the spatial distribution is almost uni-form, by increasing the noise intensity spatial patterns arise (see Figs. 4(b),5(a)) and a further increase of the noise intensity reduces the average sizeof the patterns (see Fig. 5(b)).2.3. Spatial correlationIn order to evaluate the spatial correlation between the two species for thenoise intensities considered we calculate, at the time step n, the correlationcoefficient 〈cn〉 defined on the lattice as〈cn〉 = covnxysnxsny(5)withcovnxy =∑i,j(xni,j − x¯n)(yni,j − y¯n)N, (6)where x¯n, snx, y¯n, sny are the mean value and the root mean square, respec-tively, of species 1 and species 2, obtained over the whole spatial grid at thetime step n, covnxy is the corresponding covariance and N = 100 × 100 thenumber of sites which compose the grid. The behavior of the correlationcoefficient 〈cn〉 as a function of the time for different levels of the multi-plicative noise has been reported in Fig. 6. We observe a nonmonotonicbehavior of 〈cn〉 as a function of the multiplicative noise intensity. In factfor low noise intensities σ = 10−12, 〈cn〉 shows weak oscillation around 1,that is strong correlation between the two species. For higher levels of thenoise σ = 10−10, 〈cn〉 is affected by fluctuations and its values vary stronglyas a function of the time. A further increase of the multiplicative noise, i.e.σ = 10−8 and σ = 10−4, determines an oscillation of 〈cn〉 around a negativevalue, that is anticorrelation between the two species, with the frequency ofthe periodical forcing. For higher intensities of the noise σ = 10−1, the valueof the correlation coefficient 〈cn〉 increases and it vanishes for σ = 10+3. Fi-nally to show clearly the nonmonotonic behavior of 〈cn〉, we calculate thetime average of the correlation coefficient 〈cn〉t and we report it as a functionof the multiplicative noise intensity in Fig. 7. A clear minimum is shown,which corresponds to the anticorrelated oscillations shown in the time evo-lution of two competing species in each point of our spatial grid [12]. Wenote therefore the different role of the two noise sources in the ecosystemdynamics. The additive noise determines the conditions of the dynamicalPattern Formation and Spatial Correlation . . . 1487t0 20 40 60 80 100 120 140 160 180 200 220<cn>-1.5-1.0-0.50.00.51.01.5Fig. 6. Correlation coefficient 〈cn〉 as a function of time. For low levels of themultiplicative noise (σ = 10−12) the species are strongly correlated and 〈cn〉 isapproximately constant. By increasing the intensity of the multiplicative noise(σ = 10−10) 〈cn〉 shows big fluctuations. A further increase of the noise (σ =10−8, σ = 10−4) causes strong anticorrelation between the two species with 〈cn〉oscillating at the frequency of the periodical forcing. For very high levels of noise,the anticorrelation is reduced (σ = 10−1) and finally it disappears (σ = 10+3), thatis the species are totally uncorrelated.-11 -9 -7 -5 -3 -1 1 3<cn>t-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2Fig. 7. Time average of the correlation coefficient 〈cn〉t as a function of the multi-plicative noise in semilog scale.regime, the multiplicative noise produces a coherent response of the system(see Ref. [12]), which is responsible for the appearance of anticorrelationbehavior in the spatial patterns of the species.1488 D. Valenti, A. Fiasconaro, B. Spagnolo3. ConclusionsWe present a study on the role of the noise in the spatial distributionsof two interacting species. The main result is that noise plays a very im-portant role in population dynamics and cannot be neglected. Noise canhave a constructive role and it is responsible for the enhancement of theresponse of the system to a driving force producing stochastic resonance.By using a discrete time evolution model, which is the discrete version ofthe Lotka–Volterra equations with diffusive terms, in the presence of a mul-tiplicative noise and with a random interaction parameter, we analyze thetemporal behaviors of the spatial distributions for an ecosystem consistingof two species. The noise induces spatio-temporal behaviors which are ab-sent in the deterministic dynamics, i.e. pattern formation with the sameperiodicity of the deterministic force. Moreover, appearance of temporaloscillation is observed in the correlation coefficient between the two species.We find a nonmonotonic behavior of the time average correlation coefficientas a function of the multiplicative noise. In fact at low levels of the noiseintensity the species densities are almost uniform and a strong correlationappears in the spatial distributions. By increasing the noise intensity weobserve pattern formation and anticorrelated behavior, which exhibit thesame periodicity of the deterministic driving force and correspond to theminimum of Fig. 7. For higher values of the noise intensity no patterns orcorrelations appear. Our model could be useful to explain spatio-temporalbehaviors of populations, whose dynamics is strongly affected by the noiseand by the environmental physical variables.This work has been supported by INTAS Grant 01-450, by INFM andMIUR.REFERENCES[1] C. Zimmer, Science 284, 83 (1999); see also the special section on ComplexSystems, Science 284, 79 (1999); O.N. Bjornstad, B.T. Grenfell, Science 293,638 (2001).[2] S. Ciuchi, F. de Pasquale, B. Spagnolo, Phys. Rev. E53, 706 (1996);F. de Pasquale, B. 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Pattern formation and spatial correlation induced by the noise in two competing species

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