The asymptotic regime of a complex ecosystem with \emph{N}random interacting
species and in the presence of an external multiplicative noise is analyzed. We
find the role of the external noise on the long time probability distribution
of the i-th density species, the extinction of species and the local field
acting on the i-th population. We analyze in detail the transient dynamics of
this field and the cavity field, which is the field acting on the $i^{th}$
species when this is absent. We find that the presence or the absence of some
population give different asymptotic distributions of these fields.Comment: 11 pages, 6 figures. To be published in Eur. Phys. J. 

Fiasconaro, A.

Spagnolo, B.

Valenti, D.

English

arXiv

A. Fiasconaro

D. Valenti

B. Spagnolo

Crossref

Asymptotic regime in N random interacting species

The asymptotic regime of a complex ecosystem with N random interacting species and in the presence of an external multiplicative noise is analyzed. We find the role of the external noise on the long time probability distribution of the ith density species, the extinction of species and the local field acting on the ith population. We analyze in detail the transient dynamics of this field and the cavity field, which is the field acting on the ith species when this is absent. We find that the presence or the absence of some population give different asymptotic distributions of these fields

Repositorio Universidad de Zaragoza

Asymptotic regime in N random interacting speciesA.Fiasconaro∗, D.Valenti and B.SpagnoloDipartimento di Fisica e Tecnologie Relative, Universita` di Palermoand INFM-CNR, Group of Interdisciplinary Physics†,Viale delle Scienze, I-90128 Palermo, Italy(Dated: September 30, 2005)AbstractThe asymptotic regime of a complex ecosystem with N random interacting species and in thepresence of an external multiplicative noise is analyzed. We find the role of the external noise onthe long time probability distribution of the ith density species, the extinction of species and thelocal field acting on the ith population. We analyze in detail the transient dynamics of this fieldand the cavity field, which is the field acting on the ith species when this is absent. We find that thepresence or the absence of some population give different asymptotic distributions of these fields.PACS numbers: 05.40.-a, 05.45.-a, 87.23.Cc, 89.75.-kKeywords: Statistical Mechanics, Population Dynamics, Noise-induced effects, Complex Systems∗ e-mail: afiasconaro@gip.dft.unipa.it† Electronic address: http://gip.dft.unipa.it1I. INTRODUCTIONIn recent years great attention has been devoted to population dynamics modelled bygeneralized Lotka-Volterra systems [1]. Ecosystems are a classic example of complex sys-tems, which became object of study as well by biologists as by physicists [2, 3]. Toolsdeveloped in the context of nonequilibrium statistical physics to analyze nonequilibriumnonlinear physical systems provide new insights and at the same time new approaches tothe comprehension of the properties of biological and many body systems. A key aspectto understand the complex behavior of ecosystems is the role of the external noise on thedynamics of such systems. Noise-induced effects in population dynamics, such as patternformation [4, 5], stochastic resonance, noise delayed extinction, quasi periodic oscillationsetc... have been investigated with increasing interest [6–10]. The dynamical behavior ofecological systems of interacting species evolves towards the equilibrium states through theslow process of nonlinear relaxation, which is strongly dependent on the random interactionbetween the species, the initial conditions and the random interaction with environment.The mathematical model here used to analyze the dynamics of N biological species withspatially homogeneous densities is the generalized Lotka-Volterra system with a Malthus-Verhulst modelling of the self regulation mechanism and with the addition of an externalmultiplicative noise source [11, 12]. We obtain the asymptotic behaviors of the probabilitydistribution of the populations for different values of external noise intensity. To analyzethe role of the external noise on the transient dynamics of the species we focus on the longtime distribution of the local field, that is the interaction term in the dynamical equationof the ith species that takes the influence of all other species into account. We find thatthe presence or the absence of some population give different asymptotic distributions ofthe local field and of the cavity field (field acting on the ith species when this is absent)in the absence of external noise. When the noise is switched on the asymptotic local andcavity fields tend to overlap and approximately superimpose each other for very high noiseintensity. Finally the long time evolution of the average number of the extinct species isreported for different values of the multiplicative noise intensity.2II. THE MODELThe dynamical evolution of our ecosystem composed by N interacting species in a noisyenvironment (climate, disease, etc...) is described by the following generalized Lotka-Volterraequations with a multiplicative noise, in the framework of Ito stochastic calculusdni(t) =[(gi(ni(t)) +∑j 6=iJijnj(t))dt+√ǫdwi]ni(t), i = 1, ..., N (1)where ni(t) ≥ 0 is the population density of the ith species at time t and the function gi(ni(t))gi(ni(t)) =(α+ǫ2)− ni(t), (2)describes the development of the ith species without interacting with other species. InEq. (1), α is the growth parameter, the interaction matrix Jij models the interaction betweendifferent species (i 6= j), and wi is the Wiener process whose increment dwi satisfy the usualstatistical properties 〈dwi(t)〉 = 0, and 〈dwi(t)dwj(t′)〉 = δijδ(t − t′)dt. We consider anasymmetric interaction matrix Jij, whose elements are randomly distributed according toa Gaussian distribution with 〈Jij〉 = 0, 〈JijJji〉 = 0, and σ2j = J2/N . Therefore ourecosystem contains 50% of prey-predator interactions (Jij < 0 and Jji > 0), 25% competitiveinteractions (Jij > 0 and Jji > 0), and 25% symbiotic interactions (Jij < 0 and Jji < 0).We consider all species equivalent so that the characteristic parameters of the ecosystem areindependent of the species. The formal solution of Eq. (1) isni(t) =ni(0)zi(t)1 + ni(0)∫ t0dt′zi(t′), (3)wherezi(t) = exp[αt+√ǫwi(t) +∫ t0dt′hi,loc(t′)]. (4)The term hi,loc(t) =∑j 6=i Jijnj(t) is the local field acting on the ith population and representsthe influence of other species on the differential growth rate. We note that the dynamicalbehavior of the ith population depends on the time integral of the process zi(t) and the timeintegral of the local field.In the absence of external noise (ǫ = 0), for a large number of interacting species we canassume that the local field hi(t) is Gaussian with zero mean and variance3σ2hi = Σj,k < JijJik >< njnk >= J2〈n2i 〉 , with < JijJik >= δjkJ2N. (5)As a consequence, in the absence of external noise, from the fixed-point equation ni(α −ni + hi) = 0, the stationary probability distribution of the populations is the sum of atruncated Gaussian distribution at ni = 0 (ni > 0 always) and a delta function for extinctspecies. The initial values of the populations ni(0) have also Gaussian distribution withmean value 〈ni(0)〉 = 1, and variance σ2n(0) = 0.03. The interaction strength J determinestwo different dynamical behaviors of the ecosystem. Above a critical value Jc = 1.1, thesystem is unstable and at least one of the populations diverges. Below Jc the system is stableand asymptotically reaches an equilibrium state. The equilibrium values of the populationsdepend both on their initial values and on the interaction matrix. If we consider a quenchedrandom interaction matrix, the ecosystem has a great number of equilibrium configurations,each one with its attraction basin. For an interaction strength J = 1 and an intrinsic growthparameter α = 1 we obtain: 〈ni〉 = 1.4387, 〈n2i 〉 = 4.514, and σ2ni = 2.44. These values agreewith that obtained from numerical simulation of Eq. (1).In the presence of external noise (ǫ 6= 0) we calculate long time probability distribution fordifferent values of the noise intensity. These are shown in the following Fig. 1. For increasingexternal noise intensity we obtain a larger probability distribution with a lower maximum(see the different scales in Figs. 1 for different noise intensity values). The distributionbecomes asymmetric for ǫ = 0.1 and tends to become a truncated delta function around thezero value (P (ni) = δ(ni) for ni > 0, and P (ni) = 0 for ni ≤ 0), for further increasing noiseintensity. The role of the multiplicative noise is to act as an absorbing barrier at ni = 0 [11].To confirm this picture we calculate the time evolution of the average number of extinctspecies for different noise intensities. This time behavior is shown in Fig. 2. We see thatthis number increases with noise intensity, and after the value ǫ = 0.1 reaches quickly thenormalized maximum value at ǫ = 10.To analyze in more detail the influence of each species on the long time dynamics ofthe ecosystem we calculate in this regime the local field hi,loc(t) and the cavity field hi,cav(t),which is the field acting on the ith population when this population is absent. The probabilitydistributions for both local and cavity fields have been obtained by simulations for differentspecies in the presence and in absence of external noise. The results are shown in the nextsection (see Fig. 3). We found that the probability distributions of the cavity fields differ4FIG. 1: Probability distribution for the species densities. The values of the external noise intensityare ǫ = 0, 0.001, 0.01, 0.1, 1, 10.FIG. 2: Time evolution of the average number of extinct species for different noise intensities.substantially from those of local fields for the same species, while in the presence of noisethe two fields overlap. To quantify this overlap between the probabilities distributions of thetwo fields we define an overlap coefficient λ(t), which is the distance between the averagevalues of the two distributions, normalized to their widths5λ(t) =h¯i,loc − h¯i,cav√σ2i,loc + σ2i,cav=dh(t)σd(t), (6)wheredh(t) = h¯i,loc − h¯i,cav , σ2d(t) = σ2i,loc(t) + σ2i,cav(t) . (7)With this definition the distributions start to overlap significantly for |λ| . 1, and becomestrongly overlapping for |λ| ≪ 1.III. RESULTS AND COMMENTSIn the calculation the following parameters have been used: α = 1.2, J = 1, σ2J =0.005, N = 200; the number of averaging experiment used is Nexp = 1000. Concerningthe initial condition the parameters are: 〈ni〉 = 1, σ2no = 0.03. The dynamics of variousspecies are different even if they are equivalent according to the parameters in the dynamicalEq. (1). However we note that to change the species index by fixing the random matrix or tochange the random matrix by fixing the species index is equivalent as regards the asymptoticdynamical regime. Fig. 3 shows for various noise intensities the local and cavity fieldsFIG. 3: Probability distribution of both the local (black circles)vand the cavity (white circles)fields for various values of noise intensity ǫ = 0, 0.001, 0.01, 0.1, 1. The graph are taken at the timet = 50 unit steps.6probability distributions at time t = 50 (a. u.). For noiseless dynamics the distributions ofthe fields for the species i = 3 appear very narrow around their mean values and very spacedeach other. By increasing the noise intensity, we observe an equal enlargement of the twodistributions, maintaining however the same mean values. At ǫ = 0.1 the two distributionsstart to overlap until, for stronger noise intensity (ǫ = 1), they superimpose each other. Theoverlap coefficient λ(t) is equal to zero. The noise makes equivalent all the species in theasymptotic regime and the absence of some species doesn’t contribute to any changes in thedynamics of all other species. The last plot in Fig. 3 (ǫ = 10) gives a delta distributionaround zero. This means that, at the time considered in our simulation (t = 50) and forthis noise intensity, all the species are extinct (see also Fig. 2).FIG. 4: Time evolution of the overlap coefficient λ(t) between the distributions of local and cavityfields for different noise intensities. The increasing behavior of the overlap coefficient as a functionof time, in the noiseless case, is due to the narrowing of the two fields distributions (σloc, σcav → 0)towards a δ-function. For a noisy environment the distributions tend to enlarge, decreasing thevalue of the overlap coefficient λ(t).The detailed time evolution of the overlap of the two distributions can be seen fromFig. 4, where it is plotted the coefficient defined in Eq. (6). For ǫ = 0 the λ(t) coefficientincreases with time. This is due to the different time behavior of the distance between themean values of the field distributions and of their standard deviation σ2i,loc and σ2i,cav. Thedistance dh(t) = h¯i,loc − h¯i,cav is almost constant in time, except a rapid initial transient(see Fig. 5, ǫ = 0), but the corresponding evolution of the distribution widths decreases7rapidly in time. This effect is due to the quenched random matrix. This behavior remainsunchanged until the noise intensity reaches the value of ǫ = 0.01. From this value of externalnoise intensity some differences start to be visible (see Figs. 4, 5 and 6), and at ǫ = 1, aftersome fluctuations, both the distance dh(t) and the overlap coefficient λ(t) reach a value closeto zero. The two field distributions are totally overlapped.FIG. 5: Time evolution of the distance dh(t) between the mean values of the fields distributionsfor different noise intensities, namely ǫ = 0, 0.001, 0.01, 0.1, 1.FIG. 6: Time evolution of the standard deviation σ2i,loc (grey circles)and σ2i,cav (black circles) ofthe field distributions for the same different noise intensities of Fig. 5.8It is worthwhile to note that the behaviors shown in Figs. 3, 4, 5 and 6 have been foundfor some species, and changing the species different evolutions of the distribution dynamicsappear and in particular of the λ(t) coefficient. This is due to complexity of our ecosystemand to the extinction process during the transient dynamics. Moreover this strange behavior,found for some populations and in the asymptotic regime, is reminiscent of the phase-transition phenomenon, and it is related to the following peculiarities of our dynamicalsystem: (i) all the populations are positive; (ii) different initial conditions drive the ecosysteminto different attraction basins; and (iii) the complex structure of the attraction basins.While in the presence of noise all the populations seem to be equivalent in the long timeregime, some populations, in the absence of external noise, have an asymptotical dynamicalbehavior such that they significantly influence the dynamics of other species. A more detailedanalysis on these points will be done in a forthcoming paper.IV. CONCLUSIONSWe analyzed the asymptotic regime of an ecosystem composed by N interacting speciesin the presence of multiplicative noise. We find the role of the noise on the asymptoticprobability distribution of populations and on the extinction process. Concerning the localand the cavity fields, a phase transition like phenomenon is observed. Their probabilitydistributions tend to overlap each other in the presence of external noise, reaching strongoverlap for high noise intensity (|λ(t)| ≈ 0), while they are separated (|λ(t)| > 1) in theabsence of noise. This phenomenon can be ascribed to the complexity of our ecosystem.AcknowledgementsThis work was supported by MIUR and INFM-CNR.[1] G. Baxter, A. J. McKane, and M. B. Tarlie, Phys Rev. E 71, 011106 (2005); A. J. McKane andT. J. Newman, Phys. Rev. Lett. 94, 218102 (2005); A. Shabunin, A. Efimov, G.A. Tsekouras,et al., Physica A 347, 117-136 (2005); K. Tokita, Phys. Rev. Lett. 93, 178102 (2004); G. J.Ackland and I. D. Gallagher, Phys. Rev. Lett. 93, 158701 (2004); R. Mankin, A. Sauga, A.9Ainsaar, et al., Phys. Rev. E 69, 061106 (2004); Y. De Decker, G.A. Tsekouras, A. Provata,et al., Phys. Rev. E 69, 036203 (2004); G.A. Tsekouras, A. Provata, C. Tsallis, Phys. Rev.E 69, 016120 (2004); A. Provata, G.A. Tsekouras, F. Diakonos, et al., Fluct. Noise Lett. 3,L241-L250 (2003).[2] J. D. Murray, Mathematical Biology, 2nd ed. (Springer, New York, 1993) 54 (1996).[3] C. Escudero, J. Buceta, F. J. de la Rubia, and Katja Lindenberg, Phys. Rev. E 69, 021908(2004); T. J. Kawecki and R. D. Holt, Am. Nat. 160, 333 (2002); Michel Droz and AndrzejPe¸kalski, Phys. Rev. E 69, 051912 (2004); I. Giardina, J. P. Bouchaud, M. Mezard, J. Phys.A: Math. Gen. 34, L245 (2001); H Rieger, J. Phys. A: Math. Gen. 22, 3447 (1989).[4] J. E. S. Socolor, S. Richards, and W. G. Wilson, Phys. Rev. E 63, 041908 (2001).[5] A. Fiasconaro, D. Valenti and B. Spagnolo, Acta Phys. Pol. B 35, 1491 (2004); D. Valenti,A. Fiasconaro and B. Spagnolo, Acta Phys. Pol. B 35, 1481 (2004); A. La Barbera and B.Spagnolo, Physica A 314, 120 (2001).[6] B. Spagnolo A. Fiasconaro and D. Valenti, Fluct. Noise Lett. 3, L177 (2003); B. Spagnoloand A. La Barbera, Physica A 315, 201 (2002); A. F. Rozenfeld Rozenfeld, C.J. Tessone, E.Albano, H.S. Wio, Phys. Lett. A 280, 45 (2001); J. M. G. Vilar and R. V. Sole´, Phys. Rev.Lett. 80, 4099 (1998).[7] See the special section on ”Complex Systems”, Science 284, 79-107 (1999); the special sectionon ”Ecology through Time”, Science 293, 623-657 (2001).[8] D. Valenti, A. Fiasconaro and B. Spagnolo, Physica A 331, 477 (2004).[9] B. Spagnolo D. Valenti, A. Fiasconaro, Math. Biosciences and Eng. 1, 185 (2004).[10] D. F. Russel, L. AQ. Wilkens and F. Moss, Nature 402, 291 (2000).[11] S. Ciuchi, F. de Pasquale and B. Spagnolo, Phys. Rev. E 54, 706 (1996); ibid. 47, 3915 (1993);P. Barrera, S. Ciuchi and B. Spagnolo, J. Phys. A: Math. Gen. 26, L559-L565 (1993).[12] B. Spagnolo, M. A. Cirone, A. La Barbera and F. de Pasquale, J. Phys.: Condens. Matter14, 2247 (2002); M. A. Cirone, F. de Pasquale and B. Spagnolo, Fractals 11, 217 (2003); B.Spagnolo, D. 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The asymptotic regime of a complex ecosystem with N random interacting species and in the presence of an external multiplicative noise is analyzed. We find the role of the external noise on the long time probability distribution of the ith density species, the extinction of species and the local field acting on the ith population. We analyze in detail the transient dynamics of this field and the cavity field, which is the field acting on the ith species when this is absent. We find that the presence or the absence of some population give different asymptotic distributions of these fields. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 200605.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion, 05.45.-a Nonlinear dynamics and nonlinear dynamical systems, 87.23.Cc Population dynamics and ecological pattern formation, 89.75.-k Complex systems,

Research Papers in Economics

FIASCONARO, Alessandro

VALENTI, Davide

SPAGNOLO, Bernardo

Archivio istituzionale della ricerca - Università di Palermo

Eur. Phys. J. B 50, 189–194 (2006)DOI: 10.1140/epjb/e2006-00058-4 THE EUROPEANPHYSICAL JOURNAL BAsymptotic regime in N random interacting speciesA. Fiasconaroa, D. Valenti, and B. SpagnoloDipartimento di Fisica e Tecnologie Relative, Universita` di Palermo and INFM-CNR, Group of Interdisciplinary Physics Vialedelle Scienze, 90128 Palermo, ItalyReceived 11 October 2005 / Received in ﬁnal form 29 November 2005Published online 16 February 2006 – c© EDP Sciences, Societa` Italiana di Fisica, Springer-Verlag 2006Abstract. The asymptotic regime of a complex ecosystem with N random interacting species and in thepresence of an external multiplicative noise is analyzed. We ﬁnd the role of the external noise on the longtime probability distribution of the ith density species, the extinction of species and the local ﬁeld actingon the ith population. We analyze in detail the transient dynamics of this ﬁeld and the cavity ﬁeld, whichis the ﬁeld acting on the ith species when this is absent. We ﬁnd that the presence or the absence of somepopulation give diﬀerent asymptotic distributions of these ﬁelds.PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.45.-aNonlinear dynamics and nonlinear dynamical systems – 87.23.Cc Population dynamics and ecologicalpattern formation – 89.75.-k Complex systems1 IntroductionIn recent years great attention has been devoted to pop-ulation dynamics modelled by generalized Lotka-Volterrasystems [1]. Ecosystems are a classic example of complexsystems, which became object of study as well by biolo-gists as by physicists [2,3]. Tools developed in the contextof nonequilibrium statistical physics to analyze nonequi-librium nonlinear physical systems provide new insightsand at the same time new approaches to the comprehen-sion of the properties of biological and many body sys-tems. A key aspect to understand the complex behaviorof ecosystems is the role of the external noise on the dy-namics of such systems. Noise-induced eﬀects in popula-tion dynamics, such as pattern formation [4,5], stochasticresonance, noise delayed extinction, quasi periodic oscil-lations etc. have been investigated with increasing inter-est [6–9]. The dynamical behavior of ecological systems ofinteracting species evolves towards the equilibrium statesthrough the slow process of nonlinear relaxation, which isstrongly dependent on the random interaction between thespecies, the initial conditions and the random interactionwith environment. Moreover biological evolution presentsthe same fundamental ingredient that characterizes phys-ical systems far from equilibrium in their route to equilib-rium, namely the disorder-order transition. Diﬀerent mod-els of evolution are reported in literature, which are usefulto describe a lot of dynamical population problems [10,11].Among them it is worthwhile to cite two bit-string modelsa e-mail: afiasconaro@gip.dft.unipa.itof population dynamics, namely the Eigen model [12] andthe Penna model [13].The mathematical model here used to analyze the dy-namics of N biological species with spatially homogeneousdensities is the generalized Lotka-Volterra system with aMalthus-Verhulst modelling of the self regulation mecha-nism and with the addition of an external multiplicativenoise source [14,15]. We obtain the asymptotic behaviorsof the probability distribution of the populations for dif-ferent values of external noise intensity. To analyze therole of the external noise on the transient dynamics of thespecies we focus on the long time distribution of the localﬁeld, that is the interaction term in the dynamical equa-tion of the ith species that takes the inﬂuence of all otherspecies into account. We ﬁnd that the presence or the ab-sence of some population gives rise to diﬀerent asymptoticdistributions of the local ﬁeld and of the cavity ﬁeld (ﬁeldacting on the ith species when this is absent) in the ab-sence of external noise. When the noise is switched on theasymptotic local and cavity ﬁelds tend to overlap and ap-proximately superimpose each other for very high noiseintensity. Finally the long time evolution of the averagenumber of the extinct species is reported for diﬀerent val-ues of the multiplicative noise intensity.2 The modelThe dynamical evolution of our ecosystem composed byN interacting species in a noisy environment (climate,disease, etc.) is described by the following generalized190 The European Physical Journal BLotka-Volterra equations with a multiplicative noise, inthe framework of Ito stochastic calculusdni(t) =⎡⎣⎛⎝gi(ni(t)) +∑j =iJijnj(t)⎞⎠ dt +√dwi⎤⎦ni(t),i = 1, ..., N (1)where ni(t) ≥ 0 is the population density of the ith speciesat time t and the function gi(ni(t))gi(ni(t)) =(α +2)− ni(t), (2)describes the development of the ith species without inter-acting with other species. In equation (1), α is the growthparameter, the interaction matrix Jij models the inter-action between diﬀerent species (i = j), and wi is theWiener process whose increment dwi satisfy the usual sta-tistical properties 〈dwi(t)〉 = 0, and 〈dwi(t)dwj(t′)〉 =δijδ(t− t′)dt. We consider an asymmetric interaction ma-trix Jij , whose elements are randomly distributed accord-ing to a Gaussian distribution with 〈Jij〉 = 0, 〈JijJji〉 = 0,and σ2j = J2/N . Therefore our ecosystem contains 50% ofprey-predator interactions (Jij < 0 and Jji > 0), 25%competitive interactions (Jij > 0 and Jji > 0), and 25%symbiotic interactions (Jij < 0 and Jji < 0). We considerall species equivalent so that the characteristic parame-ters of the ecosystem are independent of the species. Theformal solution of equation (1) isni(t) =ni(0)zi(t)1 + ni(0)∫ t0dt′zi(t′), (3)wherezi(t) = exp[αt +√wi(t) +∫ t0dt′hi,loc(t′)]. (4)The term hi,loc(t) =∑j =i Jijnj(t) is the local ﬁeld actingon the ith population and represents the inﬂuence of otherspecies on the diﬀerential growth rate. We note that thedynamical behavior of the ith population depends on thetime integral of the process zi(t) and the time integral ofthe local ﬁeld.In the absence of external noise ( = 0), for a largenumber of interacting species we can assume that the localﬁeld hi(t) is Gaussian with zero mean and varianceσ2hi,loc = Σj,k〈JijJik〉〈njnk〉 = J2〈n2i 〉 ,with 〈JijJik〉 = δjk J2N. (5)As a consequence, in the absence of external noise, fromthe ﬁxed-point equation ni(α− ni + hi) = 0, the station-ary probability distribution of the populations is the sumof a truncated Gaussian distribution at ni = 0 (ni > 0always) and a delta function for extinct species. The ini-tial values of the populations ni(0) have also Gaussiandistribution with mean value 〈ni(0)〉 = 1, and varianceσ2n(0) = 0.03. The interaction strength J determines twodiﬀerent dynamical behaviors of the ecosystem. Above acritical value Jc = 1.1, the system is unstable and at leastone of the populations diverges. Below Jc the system isstable and asymptotically reaches an equilibrium state.The equilibrium values of the populations depend bothon their initial values and on the interaction matrix. Ifwe consider a quenched random interaction matrix, theecosystem has a great number of equilibrium conﬁgura-tions, each one with its attraction basin. For an interactionstrength J = 1 and an intrinsic growth parameter α = 1we obtain: 〈ni〉 = 1.4387, 〈n2i 〉 = 4.514, and σ2ni = 2.44.These values agree with that obtained from numerical sim-ulation of equation (1).In the presence of external noise ( = 0) we calculatelong time probability distribution for diﬀerent values ofthe noise intensity. These are shown in the following Fig-ure 1. For increasing external noise intensity we obtain alarger probability distribution with a lower maximum (seethe diﬀerent scales in Figure 1 for diﬀerent noise intensityvalues). The distribution becomes asymmetric for  = 0.1and tends to become a truncated delta function aroundthe zero value (P (ni) = δ(ni) for ni > 0, and P (ni) = 0for ni ≤ 0), for further increasing noise intensity. The roleof the multiplicative noise is to act as an absorbing bar-rier at ni = 0 [14]. To conﬁrm this picture we calculate thetime evolution of the average number of extinct species fordiﬀerent noise intensities. This time behavior is shown inFigure 2. We see that this number increases with noiseintensity, and after the value  = 0.1 reaches quickly thenormalized maximum value (see Fig. 2,  = 1,  = 10).To analyze in more detail the inﬂuence of each specieson the long time dynamics of the ecosystem we calculatein this regime the local ﬁeld hi,loc(t) and the cavity ﬁeldhi,cav(t), which is the ﬁeld acting on the ith populationwhen this population is absent. The probability distribu-tions for both local and cavity ﬁelds have been obtained bysimulations for diﬀerent species in the presence and in ab-sence of external noise. The results are shown in the nextsection (see Fig. 3). We found that the probability distri-butions of the cavity ﬁelds diﬀer substantially from thoseof local ﬁelds for the same species, while in the presenceof noise the two ﬁelds overlap. To quantify this overlapbetween the probability distributions of the two ﬁelds wedeﬁne an overlap coeﬃcient λ(t), which is the distancebetween the average values of the two distributions, nor-malized to their widthsλ(t) =h¯i,loc − h¯i,cav√σ2i,loc + σ2i,cav=dh(t)σd(t), (6)wheredh(t) = h¯i,loc − h¯i,cav , σ2d(t) = σ2i,loc(t) + σ2i,cav(t). (7)With this deﬁnition the distributions start to overlap sig-niﬁcantly for |λ|  1, and become strongly overlapping for|λ|  1.A. Fiasconaro et al.: Asymptotic regime in N random interacting species 191Fig. 1. Probability distribution for the species densities. The values of the external noise intensity are  = 0, 0.001, 0.01, 0.1, 1, 10.Fig. 2. Time evolution of the average number of extinct species for diﬀerent noise intensities.3 Results and commentsIn the calculation the following parameters have beenused: α = 1.2, J = 1, σ2J = 0.005, N = 200; the num-ber of averaging experiment used is Nexp = 1000. Con-cerning the initial condition the parameters are: 〈ni〉 = 1,σ2no = 0.03. The dynamics of various species are diﬀerenteven if they are equivalent according to the parametersin the dynamical equation (1). However we note that tochange the species index by ﬁxing the random matrix orto change the random matrix by ﬁxing the species indexis equivalent as regards the asymptotic dynamical regime.Figure 3 shows for various noise intensities the local andcavity ﬁelds probability distributions at time t = 50 (au).For noiseless dynamics the distributions of the ﬁelds forthe species i = 3 appear very narrow around their meanvalues and very spaced each other. By increasing the noiseintensity, we observe an equal enlargement of the two dis-tributions, maintaining however the same mean values.At  = 0.1 the two distributions start to overlap until, forstronger noise intensity ( = 1), they superimpose eachother. The overlap coeﬃcient λ(t) is equal to zero. Thenoise makes equivalent all the species in the asymptoticregime and the absence of some species doesn’t contributeto any changes in the dynamics of all the other species.The last plot in Figure 3 ( = 10) gives a delta distribution192 The European Physical Journal BFig. 3. Probability distribution of both the local (black circles) and the cavity (white circles) ﬁelds for various values of noiseintensity  = 0, 0.001, 0.01, 0.1, 1. The graph are taken at the time t = 50 unit steps.Fig. 4. Time evolution of the overlap coef-ﬁcient λ(t) between the distributions of localand cavity ﬁelds for diﬀerent noise intensities.The increasing behavior of the overlap coeﬃ-cient as a function of time, in the noiseless case,is due to the narrowing of the two ﬁelds distri-butions (σloc, σcav → 0) towards a δ-function.For a noisy environment the distributions tendto enlarge, decreasing the value of the overlapcoeﬃcient λ(t).around zero. This means that, at the time considered inour simulation (t = 50) and for this noise intensity, all thespecies are extinct (see also Fig. 2).The detailed time evolution of the overlap of the twodistributions can be seen from Figure 4, where the coeﬃ-cient deﬁned in equation (6) is plotted. For  = 0 the λ(t)coeﬃcient increases with time. This is due to the diﬀerenttime behavior of the distance between the mean valuesof the ﬁeld distributions and of their standard deviationσ2i,loc and σ2i,cav. The distance dh(t) = h¯i,loc − h¯i,cav isalmost constant in time, except a rapid initial transient(see Fig. 5,  = 0), but the corresponding evolution ofthe distribution widths decreases rapidly in time. This ef-fect is due to the quenched random matrix. This behaviorremains unchanged until the noise intensity reaches thevalue of  = 0.01. From this value of external noise in-tensity some diﬀerences start to be visible (see Figs. 4–6),and at  = 1, after some ﬂuctuations, both the distancedh(t) and the overlap coeﬃcient λ(t) reach a value closeto zero. The two ﬁeld distributions are totally overlapped.It is worthwhile to note that the behaviors shown inFigures 3–6 have been found for some species, and chang-ing the species diﬀerent evolutions of the distribution dy-namics appear and in particular of the λ(t) coeﬃcient.This is due to complexity of our ecosystem and to the ex-tinction process during the transient dynamics. Moreoverthis strange behavior, found for some populations and inthe asymptotic regime, is reminiscent of the phase transi-tion phenomenon [16], and it is related to the following pe-culiarities of our dynamical system: (i) all the populationsA. Fiasconaro et al.: Asymptotic regime in N random interacting species 193Fig. 5. Time evolution of the distance dh(t)between the mean values of the ﬁelds distribu-tions for diﬀerent noise intensities, namely  =0, 0.001, 0.01, 0.1, 1.Fig. 6. Time evolution of the standard devia-tion σ2i,loc (grey circles)and σ2i,cav (black cir-cles) of the ﬁeld distributions for the samenoise intensities of Figure 5.are positive; (ii) diﬀerent initial conditions drive theecosystem into diﬀerent attraction basins; and (iii) thecomplex structure of the attraction basins. While in thepresence of noise all the populations seem to be equivalentin the long time regime, some populations, in the absenceof external noise, have an asymptotical dynamical behav-ior such that they signiﬁcantly inﬂuence the dynamics ofthe other species. A more detailed analysis on these pointswill be done in a forthcoming paper.4 ConclusionsWe analyzed the asymptotic regime of an ecosystem com-posed by N interacting species in the presence of mul-tiplicative noise. We ﬁnd the role of the noise on theasymptotic probability distribution of populations and onthe extinction process. Concerning the local and the cav-ity ﬁelds, a phase transition like phenomenon is observed.Their probability distributions tend to overlap each otherin the presence of external noise, reaching strong over-lap for high noise intensity (|λ(t)| ≈ 0), while they areseparated (|λ(t)| > 1) in the absence of noise. This phe-nomenon can be ascribed to the complexity of our ecosys-tem.This work was supported by MIUR and INFM-CNR.References1. G. Baxter, A.J. McKane, M.B. Tarlie, Phys Rev. E71, 011106 (2005); A.J. McKane, T.J. Newman, Phys.Rev. Lett. 94, 218102 (2005); A. Shabunin, A. Eﬁmov,G.A. Tsekouras et al., Physica A 347, 117 (2005); K.Tokita, Phys. Rev. Lett. 93, 178102 (2004); G.J. Ackland,I.D. Gallagher, Phys. Rev. Lett. 93, 158701 (2004); R.Mankin, A. Sauga, A. Ainsaar et al., Phys. Rev. E 69,061106 (2004); Y. De Decker, G.A. Tsekouras, A. Provataet al., Phys. Rev. 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The asymptotic regime of a complex ecosystem with N random
interacting species and in the presence of an external
multiplicative noise is analyzed. We find the role of the external
noise on the long time probability distribution of the
ith density species, the extinction of species and the
local field acting on the ith population. We analyze
in detail the transient dynamics of this field and the cavity
field, which is the field acting on the ith species when this
is absent. We find that the presence or the absence of some
population give different asymptotic distributions of these
fields

Asymptotic regime in N random interacting species

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