A coupled map lattice of generalized Lotka-Volterra equations in the presence
of colored multiplicative noise is used to analyze the spatiotemporal evolution
of three interacting species: one predator and two preys symmetrically
competing each other. The correlation of the species concentration over the
grid as a function of time and of the noise intensity is investigated. The
presence of noise induces pattern formation, whose dimensions show a
nonmonotonic behavior as a function of the noise intensity. The colored noise
induces a greater dimension of the patterns with respect to the white noise
case and a shift of the maximum of its area towards higher values of the noise
intensity.Comment: 6 pages, 3 figure

Fiasconaro, A.

Spagnolo, B.

Valenti, D.

English

arXiv

A coupled map lattice of generalized Lotka-Volterra equations in the presence of colored multiplicative noise is used to analyze the spatiotemporal evolution of three interacting species: one predator and two preys symmetrically competing each other. The correlation of the species concentration over the grid as a function of time and of the noise intensity is investigated. The presence of noise induces pattern formation, whose dimensions show a nonmonotonic behavior as a function of the noise intensity. The colored noise induces a greater dimension of the patterns with respect to the white noise case and a shift of the maximum of its area towards higher values of the noise intensity

FIASCONARO, Alessandro

VALENTI, Davide

SPAGNOLO, Bernardo

Archivio istituzionale della ricerca - Università di Palermo

July 29, 2005 15:39 WSPC/167-FNL 00269Fluctuation and Noise LettersVol. 5, No. 2 (2005) L305–L311c© World Scientific Publishing CompanyFluctuation and Noise LettersVol. 5, No. 2 (2005) 000–000c© World Scientific Publishing CompanyNONMONOTONIC PATTERN FORMATION IN THREE SPECIESLOTKA–VOLTERRA SYSTEM WITH COLORED NOISEA. FIASCONARO∗, D. VALENTI and B. SPAGNOLODipartimento di Fisica e Tecnologie Relative and INFM, Group of Interdisciplinary Physics†Universita` di Palermo, Viale delle Scienze pad. 18, I–90128 Palermo, Italy∗afiasconaro@gip.dft.unipa.itReceived 14 January 2005Revised 19 March 2005Accepted 10 April 2005Communicated by Werner Ebeling and Bernardo SpagnoloA coupled map lattice of generalized Lotka–Volterra equations in the presence of coloredmultiplicative noise is used to analyze the spatiotemporal evolution of three interactingspecies: one predator and two preys symmetrically competing each other. The correlationof the species concentration over the grid as a function of time and of the noise intensityis investigated. The presence of noise induces pattern formation, whose dimensions showa nonmonotonic behavior as a function of the noise intensity. The colored noise inducesa greater dimension of the patterns with respect to the white noise case and a shift ofthe maximum of its area towards higher values of the noise intensity.Keywords: Statistical mechanics; population dynamics; noise induced effects; Lotka–Volterra equations.1. IntroductionThe addition of noise in mathematical models of population dynamics can be usefulto describe the observed phenomenology in a realistic and relatively simple form.This noise contribution can give rise to non trivial effects, modifying sometimes in anunexpected way the deterministic dynamics. Examples of noise induced phenomenaare stochastic resonance, noise delayed extinction, temporal oscillations and noise-induced pattern formation [1–5]. Biological complex systems can be modelled asopen systems in which interactions between the components are nonlinear and anoisy interaction with the environment is present [6]. Recently it has been foundthat nonlinear interaction and the presence of multiplicative noise can give rise topattern formation in population dynamics of spatially extended systems [7–9]. Thereal noise sources are correlated and their effects on spatially extended systems have†Electronic address: http://gip.dft.unipa.itL305July 29, 2005 15:39 WSPC/167-FNL 00269L306 A. Fiasconaro, D. Valenti & B. SpagnoloA. Fiasconaro, D. Valenti & B. Spagnolobeen investigated in Refs. [3] (see cited references there) and [4]. In this paper westudy the spatio-temporal evolution of an ecosystem of three interacting species: twocompeting preys and one predator, in the presence of a colored multiplicative noise.We find a nonmonotonic behavior of the average size of the patterns as a functionof the noise intensity. The effects induced by the colored noise, in comparison withthe white noise case [9], are: (i) pattern formation with a greater dimension of theaverage area, (ii) a shift of the maximum of the area of the patterns towards highervalues of the multiplicative noise intensity.2. The ModelTo describe the dynamics of our spatially distributed system, we use a coupled maplattice (CML) [9, 10] with a multiplicative noisexn+1i,j = µxni,j(1− xni,j − βnyni,j − αzni,j) + xni,jXni,j +D∑p(xnp − xni,j),yn+1i,j = µyni,j(1 − yni,j − βnxni,j − αzni,j) + yni,jYni,j +D∑p(ynp − yni,j),zn+1i,j = µzzni,j[−1 + γ(xni,j + yni,j)] + zni,jZni,j +D∑p(znp − zni,j),(1)where xni,j , yni,j and zni,j are respectively the densities of preys x, y and the predatorz in the site (i, j) at the time step n. Here α and γ are the interaction parame-ters between preys and predator, D is the diffusion coefficient, µ and µz are scalefactors.∑p indicates the sum over the four nearest neighbors in the map lattice.X(t), Y (t), Z(t) are Ornstein–Uhlenbeck processes with the statistical properties〈χ(t)〉 = 0, 〈〈χ(t)χ(t + τ)〉 =q2τce−τ/τc , (2)and〈Xni,jYmi,j 〉 = 〈Xni,jZmi,j〉 = 〈Yni,jZmi,j〉 = 0 ∀ n,m, i, j (3)where τc is the correlation time of the process, q is the noise intensity, and χ(t)represents the three continuous stochastic variables (X(t), Y (t), Z(t)), taken at timestep n. The boundary conditions are such that no interaction is present out oflattice. Because of the environment temperature, the interaction parameter β(t)between the two preys can be modelled as a periodical function of timeβ(t) = 1 + + ηcos(ωt). (4)Here η = 0.2, ω = pi10−3 and  = −0.1. The interaction parameter β(t) oscillatesaround the critical value βc = 1 in such a way that the dynamical regime of Lotka–Volterra model for two competing species changes from coexistence of the two preys(β < 1) to exclusion of one of them (β > 1). The parameters used in our simulationsare the same of [9], in order to compare the results with the white noise case.Specifically they are: µ = 2; α = 0.03; µz = 0.02, γ = 205 and D = 0.1. Thenoise intensity q varies between 10−11 and 10−2. With this choice of parametersthe intraspecies competition among the two prey populations is stronger comparedJuly 29, 2005 15:39 WSPC/167-FNL 00269Pattern Formation in Three Species Lotka–Volterra System with Colored Noise L307Pattern Formation in Three Species L tka–Volterra System with Col red Noiseto the interspecies interaction preys–predator (β  α), and both prey populationscan therefore stably coexist in the presence of the predator [11]. To evaluate thespecies correlation over the grid we consider the correlation coefficient rn betweena couple of them at the step n asrn =∑Ni,j(wni,j − w¯n)(kni,j − k¯n)[∑Ni,j(wni,j − w¯n)2∑Ni,j(kni,j − k¯n)2]1/2 , (5)where N is the number of sites in the grid (100x100), the symbols wn, kn representone of the three species concentration x, y, z, and w¯n, k¯n are the mean values ofthe same quantities in all the lattice at the time step n. From the definition (5) itfollows that −1 ≤ rn ≤ 1.3. Colored Noise EffectsWe quantify our analysis by considering the maximum patterns, defined as theensemble of adjoining sites in the lattice for which the density of the species belongsto the interval [3/4 max,max], where max is the absolute maximum of densityin the specific grid. The various quantities, such as pattern area and correlationparameter, have been averaged over 50 realizations, obtaining the mean valuesbelow reported. We evaluated for each spatial distribution, in a temporal step andfor a given noise intensity value, the following quantities referring to the maximumpattern (MP): mean area of the various MPs found in the lattice and correlation rbetween two preys, and between preys and predator.From the deterministic analysis we observe: (i) for  < 0 (β < 1) a coexistenceregime of the two preys, characterized in the lattice by a strong correlation betweenthem and the predator lightly anti-correlated with the two preys; (ii) for  > 0(β > 1) wide exclusion zones in the lattice, characterized by a strong anti-correlationbetween preys. Because of the periodic variation of the interaction parameter β(t),an interesting activation phenomenon for  < 0 takes place: the two preys, after aninitial transient, remain strongly correlated for all the time, in spite of the fact thatthe parameter β(t) takes values greater than 1 during the periodical evolution. Wefocus on this dynamical regime to analyze the effect of the noise. We found that thenoise acts as a trigger of the oscillating behavior of the species correlation r givingrise to periodical alternation of coexistence and exclusion regime. Even a very smallamount of noise is able to destroy the coexistence regime periodically in time. Thisgives rise to a periodical time behavior of the correlation parameter r, with the sameperiodicity of the interaction parameter β(t) (see Eq. (4)), which turns out almostindependent of the noise intensity and of the correlation time τc (see Fig. 1(a)). Thisperiodicity reflects the periodical time behavior of the mean area of the patterns. Anonmonotonic behavior of the pattern area as a function of time is observed for allvalues of noise intensity investigated. This behavior becomes periodically in timefor lower values of noise intensity, when higher values of correlation time τc areconsidered. In Figs. 1(b–d) we show the time evolution of the mean area of themaximum patterns, for q = 10−4 and for three values of correlation time, namelyτc = 1, 10, 100. The periodicity of the nonmonotonic behavior of the area of MPs isclearly observed.July 29, 2005 15:39 WSPC/167-FNL 00269L308 A. Fiasconaro, D. Valenti & B. SpagnoloA. Fiasconaro, D. Valenti & B. SpagnoloFig. 1. (a) The correlation coefficient between preys and predator as a function of time; (b–d)Mean area of the maximum patterns of the species as a function of time, for three values ofcorrelation time τc = 1, 10, 100 and for q = 10−4. The correlation plot (a) is quite the same forall the τc investigated.To analyze the noise induced pattern formation we focus on the correlationregime between preys r12 = 1, where pattern formation appears. In fact when thepreys are highly anticorrelated with species correlation parameter r12 = −1, a bigclusterization of preys is observed, with large patches of preys enlarging to all theavailable space of the lattice. This scenario, observed also in the white noise case [9],is confirmed by the analysis of the time series of the species. These large patchesappear, in the anticorrelation regime corresponding to the exclusion regime of thetwo preys, with smooth contours and low intensity of species density for lower noiseintensities and higher correlation time values.The study of the area of the pattern formation as a function of noise intensitywith colored noise shows two main effects: (1) the increase of the pattern dimensionand (2) a shift of the maximum toward higher values of the noise intensity. Asexpected, for low values of the correlation time we observe the same results than inthe white noise case. These effects are well visible in Fig. 2 where the three curvesshow the nonmonotonic behavior of the area of the maximum pattern as a functionof noise intensity. The interaction step here considered is 1400, which correspond tothe biggest pattern area found in our calculations. The first curve (τc = 1) is quitethe same found in the white noise case. The value of maximum in the third curve(τc = 100) is not so different from the previous one (τc = 10), because its value isapproaching the maximum possible value of 10,000 into the used grid.The pattern formation is visible in Fig. 3, where we report three patternsof the two preys and the predator for the following values of noise intensity:July 20, 2005 19:18 WSPC/167-FNL 00269Pattern Formation in Three Species Lotka–Volterra System with Colored Noise L309Fig. 2. Semi-Log plot of the mean area of the maximum patterns for all species as a function ofnoise intensity, at iteration step 1400 for the three correlation time here reported. See the text forthe values of the other parameters.Fig. 3. Pattern formation for preys and predator with homogeneous initial distribution, at timeiteration 1400 for τc = 1 and noise intensity: q = 10−11, 10−9, 10−7. r12, r13, r23, r123 representrespectively prey-prey, prey1-predator, prey2-predator and total preys-predator correlation. Seethe text for the values of the other parameters.July 29, 2005 15:39 WSPC/167-FNL 00269L310 A. Fiasconaro, D. Valenti & B. SpagnoloA. Fiasconaro, D. Valenti & B. Spagnoloq = 10−11, 10−9, 10−7 and τc = 1. The initial spatial distribution is homogeneousand equal for all species, that is xinitij = yinitij = zinitij = 0.25 for all sites (i, j). We seethat a spatial structure emerges with increasing noise intensity. At very low noiseintensity (q = 10−11), the spatial distribution appears almost homogeneous with-out strong pattern formation (see Fig. 3(a)). We considered here only structuredpattern, avoiding big clusterization of density visible in the case of anticorrelatedpreys. At intermediate noise intensity (q = 10−9) spatial patterns appear. As wecan see the structure disappears by increasing the noise intensity (see Fig. 3(c)).Consistently with Fig. 2, we find that for higher correlation time τc the qualitativeshape of the patterns shown in Fig. 3 are repeated, but with a shift of the maximumarea (darkest patterns) toward higher values of the noise intensity.4. ConclusionsThe noise-induced pattern formation in a coupled map lattice of three interactingspecies, described by generalized Lotka–Volterra equations in the presence of mul-tiplicative colored noise, has been investigated. We find nonmonotonic behavior ofthe mean area of the maximum patterns as a function of noise intensity for all thecorrelation time investigated. For increasing values of the correlation time τc weobserve an increase of the area of the pattern and a shift of the maximum valuetowards higher values of the noise intensity. The nonmonotonic behavior is alsofound for the area of the patterns as a function of the evolution time.AcknowledgmentsThis work was supported by INTAS Grant 01–0450, by INFM and MIUR.References[1] D. Valenti, A. Fiasconaro and B. Spagnolo, Stochastic resonance and noise delayedextinction in a model of two competing species, Physica A 331 (2004) 477–486.[2] A. A. Zaikin and L. Schimansky-Geier, Spatial patterns induced by additive noise,Phys. Rev. E 58 (1998) 4355–4360.[3] J. Garc´ıa-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Springer-Verlag, New York (1999).[4] J. Garc´ıa-Ojalvo and J. M. Sancho, Colored noise in spatially extended systems, Phys.Rev. E 49 (1994) 2769–2778.[5] M. Ibanes, J. M. Sancho, J. Buceta and K. Lindenberg, Noise-driven mechanism forpattern formation, Phys. Rev. E 67 (2003) 021113–1–8.[6] S. Ciuchi, F. de Pasquale and B. Spagnolo, Self regulation mechanism of an ecosystemin a non Gaussian fluctuation regime, Phys. Rev. E 54 (1996) 706–716.[7] B. Spagnolo, M. Cirone, A. La Barbera and F. de Pasquale, Noise induced effects inpopulation dynamics, J. Phys.: Cond. Matter 14 (2002) 2247–2255.[8] B. Spagnolo, A. Fiasconaro and D. Valenti, Noise induced phenomena in Lotka–Volterra systems, Fluctuation and Noise Letters 3(2) (2003) L177–L185.[9] A. Fiasconaro, B. Spagnolo and D. Valenti, Nonmonotonic behavior of spatiotemporalpattern formation in a noisy Lotka–Volterra system, Acta Physica Polonica B 35(4)(2004) 1491–1500.[10] Special issue CML models, ed. K. Kaneko [Chaos 2 (1992) 279–460].July 29, 2005 15:39 WSPC/167-FNL 00269Pattern Formation in Three Species Lotka–Volterra System with Colored Noise L311Pattern Formation in Thre Species L tka–Volterra Sys em with Col red Noise[11] A. D. Bazykin, Nonlinear dynamics of interacting populations, World Scientific,Singapore (1998).[12] J. Garc´ıa Lafuente, A. Garc´ıa, S. Mazzola, L. Quintanilla, J. Delgado, A. Cuttittaand B. Patti, Hydrographic phenomena influencing early life stages of the SicilianChannel anchovy, Fishery Oceanography 11(1) (2002) 31–44.

Nonmonotonic Pattern Formation in Three Species Lotka-Volterra System with Colored Noise

Abstract

Similar works

Full text

Available Versions

Archivio istituzionale della ricerca - Università di Palermo