Influence of local carrying capacity restrictions on stochastic predator-prey models

We study a stochastic lattice predator-prey system by means of Monte Carlo simulations that do not impose any restrictions on the number of particles per site, and discuss the similarities and differences of our results with those obtained for site-restricted model variants. In accord with the classic Lotka-Volterra mean-field description, both species always coexist in two dimensions. Yet competing activity fronts generate complex, correlated spatio-temporal structures. As a consequence, finite systems display transient erratic population oscillations with characteristic frequencies that are renormalized by fluctuations. For large reaction rates, when the processes are rendered more local, these oscillations are suppressed. In contrast with site-restricted predator-prey model, we observe species coexistence also in one dimension. In addition, we report results on the steady-state prey age distribution.Comment: Latex, IOP style, 17 pages, 9 figures included, related movies available at http://www.phys.vt.edu/~tauber/PredatorPrey/movies

Effect of Acute Thermal Stress Exposure on Ecophysiological Traits of the Mediterranean Sponge Chondrilla nucula: Implications for Climate Change

As a result of climate change, the Mediterranean Sea has been exposed to an increase in the frequency and intensity of marine heat waves in the last decades, some of which caused mass mortality events of benthic invertebrates, including sponges. Sponges are an important component of benthic ecosystems and can be the dominant group in some rocky shallow-water areas in the Mediterranean Sea. In this study, we exposed the common shallow-water Mediterranean sponge Chondrilla nucula (Demospongiae: Chondrillidae) to six different temperatures for 24 h, ranging from temperatures experienced in the field during the year (15, 19, 22, 26, and 28 °C) to above normal temperatures (32 °C) and metabolic traits (respiration and clearance rate) were measured. Both respiration and clearance rates were affected by temperature. Respiration rates increased at higher temperatures but were similar between the 26 and 32 °C treatments. Clearance rates decreased at temperatures >26 °C, indicating a drop in food intake that was not reflected by respiration rates. This decline in feeding, while maintaining high respiration rates, may indicate a negative energy balance that could affect this species under chronic or repeated thermal stress exposure. C. nucula will probably be a vulnerable species under climate change conditions, affecting its metabolic performance, ecological functioning and the ecosystem services it provides

Complete Solution of the Kinetics in a Far-from-equilibrium Ising Chain

The one-dimensional Ising model is easily generalized to a \textit{genuinely nonequilibrium} system by coupling alternating spins to two thermal baths at different temperatures. Here, we investigate the full time dependence of this system. In particular, we obtain the evolution of the magnetisation, starting with arbitrary initial conditions. For slightly less general initial conditions, we compute the time dependence of all correlation functions, and so, the probability distribution. Novel properties, such as oscillatory decays into the steady state, are presented. Finally, we comment on the relationship to a reaction-diffusion model with pair annihilation and creation.Comment: Submitted to J. Phys. A (Letter to the editor

Fluctuations and Correlations in Lattice Models for Predator-Prey Interaction

Including spatial structure and stochastic noise invalidates the classical Lotka-Volterra picture of stable regular population cycles emerging in models for predator-prey interactions. Growth-limiting terms for the prey induce a continuous extinction threshold for the predator population whose critical properties are in the directed percolation universality class. Here, we discuss the robustness of this scenario by considering an ecologically inspired stochastic lattice predator-prey model variant where the predation process includes next-nearest-neighbor interactions. We find that the corresponding stochastic model reproduces the above scenario in dimensions 1< d \leq 4, in contrast with mean-field theory which predicts a first-order phase transition. However, the mean-field features are recovered upon allowing for nearest-neighbor particle exchange processes, provided these are sufficiently fast.Comment: 5 pages, 4 figures, 2-column revtex4 format. Emphasis on the lattice predator-prey model with next-nearest-neighbor interaction (Rapid Communication in PRE

Exactly solvable models through the empty interval method, for more-than-two-site interactions

Single-species reaction-diffusion systems on a one-dimensional lattice are considered, in them more than two neighboring sites interact. Constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E_n(t)$'s, the probability that $n$ consecutive sites are empty at time $t$. The general method of solving the time evolution equation is discussed. As an example, a system with next-nearest-neighbor interaction is studied.Comment: 19 pages, LaTeX2

Solution of classical stochastic one dimensional many-body systems

We propose a simple method that allows, in one dimension, to solve exactly a wide class of classical stochastic many-body systems far from equilibrium. For the sake of illustration and without loss of generality, we focus on a model that describes the asymmetric diffusion of hard core particles in the presence of an external source and instantaneous annihilation. Starting from a Master equation formulation of the problem we show that the density and multi-point correlation functions obey a closed set of integro-differential equations which in turn can be solved numerically and/or analyticallyComment: 2 figure

Generalized empty-interval method applied to a class of one-dimensional stochastic models

In this work we study, on a finite and periodic lattice, a class of one-dimensional (bimolecular and single-species) reaction-diffusion models which cannot be mapped onto free-fermion models. We extend the conventional empty-interval method, also called {\it interparticle distribution function} (IPDF) method, by introducing a string function, which is simply related to relevant physical quantities. As an illustration, we specifically consider a model which cannot be solved directly by the conventional IPDF method and which can be viewed as a generalization of the {\it voter} model and/or as an {\it epidemic} model. We also consider the {\it reversible} diffusion-coagulation model with input of particles and determine other reaction-diffusion models which can be mapped onto the latter via suitable {\it similarity transformations}. Finally we study the problem of the propagation of a wave-front from an inhomogeneous initial configuration and note that the mean-field scenario predicted by Fisher's equation is not valid for the one-dimensional (microscopic) models under consideration.Comment: 19 pages, no figure. To appear in Physical Review E (November 2001

Large Fluctuations and Fixation in Evolutionary Games

We study large fluctuations in evolutionary games belonging to the coordination and anti-coordination classes. The dynamics of these games, modeling cooperation dilemmas, is characterized by a coexistence fixed point separating two absorbing states. We are particularly interested in the problem of fixation that refers to the possibility that a few mutants take over the entire population. Here, the fixation phenomenon is induced by large fluctuations and is investigated by a semi-classical WKB (Wentzel-Kramers-Brillouin) theory generalized to treat stochastic systems possessing multiple absorbing states. Importantly, this method allows us to analyze the combined influence of selection and random fluctuations on the evolutionary dynamics \textit{beyond} the weak selection limit often considered in previous works. We accurately compute, including pre-exponential factors, the probability distribution function in the long-lived coexistence state and the mean fixation time necessary for a few mutants to take over the entire population in anti-coordination games, and also the fixation probability in the coordination class. Our analytical results compare excellently with extensive numerical simulations. Furthermore, we demonstrate that our treatment is superior to the Fokker-Planck approximation when the selection intensity is finite.Comment: 17 pages, 10 figures, to appear in JSTA

Commitment versus persuasion in the three-party constrained voter model

In the framework of the three-party constrained voter model, where voters of two radical parties (A and B) interact with "centrists" (C and Cz), we study the competition between a persuasive majority and a committed minority. In this model, A's and B's are incompatible voters that can convince centrists or be swayed by them. Here, radical voters are more persuasive than centrists, whose sub-population consists of susceptible agents C and a fraction zeta of centrist zealots Cz. Whereas C's may adopt the opinions A and B with respective rates 1+delta_A and 1+delta_B (with delta_A>=delta_B>0), Cz's are committed individuals that always remain centrists. Furthermore, A and B voters can become (susceptible) centrists C with a rate 1. The resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph. At mean field level, there is a continuous transition from a coexistence phase when zeta= Delta_c. In a finite population of size N, demographic fluctuations lead to centrism consensus and the dynamics is characterized by the mean consensus time tau. Because of the competition between commitment and persuasion, here consensus is reached much slower (zeta=Delta_c) than in the absence of zealots (when tau\simN). In fact, when zeta<Delta_c and there is an initial minority of centrists, the mean consensus time asymptotically grows as tau\simN^{-1/2} e^{N gamma}, where gamma is determined. The dynamics is thus characterized by a metastable state where the most persuasive voters and centrists coexist when delta_A>delta_B, whereas all species coexist when delta_A=delta_B. When zeta>=Delta_c and the initial density of centrists is low, one finds tau\simln N (when N>>1). Our analytical findings are corroborated by stochastic simulations.Comment: 25 pages, 6 figures. Final version for the Journal of Statistical Physics (special issue on the "applications of statistical mechanics to social phenomena"

Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model

Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see e.g. B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan [Nature {\bf 418}, 171 (2002)] and B. Kirkup and M. A. Riley [Nature {\bf 428}, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely the `rock-paper-scissors' or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting in regular (yet neutrally stable) oscillations of the population densities, we demonstrate that fluctuations arising in the system with a \emph{finite number of agents} drastically alter this picture and are responsible for extinction: After long enough time, two of the three species die out. As main findings we provide analytic estimates and numerical computation of the extinction probability at a given time. We also discuss the implications of our results for a broad class of competing population systems.Comment: 12 pages, 9 figures, minor correction