Spatial Patterns Emerging from a Stochastic Process Near Criticality

There is mounting empirical evidence that many communities of living organisms display key features which closely resemble those of physical systems at criticality. We here introduce a minimal model framework for the dynamics of a community of individuals which undergoes local birth-death, immigration, and local jumps on a regular lattice. We study its properties when the system is close to its critical point. Even if this model violates detailed balance, within a physically relevant regime dominated by fluctuations, it is possible to calculate analytically the probability density function of the number of individuals living in a given volume, which captures the close-to-critical behavior of the community across spatial scales. We find that the resulting distribution satisfies an equation where spatial effects are encoded in appropriate functions of space, which we calculate explicitly. The validity of the analytical formulae is confirmed by simulations in the expected regimes. We finally discuss how this model in the critical-like regime is in agreement with several biodiversity patterns observed in tropical rain forests

Metastability and anomalous fixation in evolutionary games on scale-free networks

We study the influence of complex graphs on the metastability and fixation properties of a set of evolutionary processes. In the framework of evolutionary game theory, where the fitness and selection are frequency-dependent and vary with the population composition, we analyze the dynamics of snowdrift games (characterized by a metastable coexistence state) on scale-free networks. Using an effective diffusion theory in the weak selection limit, we demonstrate how the scale-free structure affects the system's metastable state and leads to anomalous fixation. In particular, we analytically and numerically show that the probability and mean time of fixation are characterized by stretched exponential behaviors with exponents depending on the network's degree distribution.Comment: 5 pages, 4 figures, to appear in Physical Review Letter

When does cyclic dominance lead to stable spiral waves?

Species diversity in ecosystems is often accompanied by characteristic spatio-temporal patterns. Here, we consider a generic two-dimensional population model and study the spiraling patterns arising from the combined effects of cyclic dominance of three species, mutation, pair-exchange and individual hopping. The dynamics is characterized by nonlinear mobility and a Hopf bifurcation around which the system's four-phase state diagram is inferred from a complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterized by spiraling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterize a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiraling patterns are also affected by the nonlinear diffusion

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

Kinetic Anomalies in Addition-Aggregation Processes

We investigate irreversible aggregation in which monomer-monomer, monomer-cluster, and cluster-cluster reactions occur with constant but distinct rates K_{MM}, K_{MC}, and K_{CC}, respectively. The dynamics crucially depends on the ratio gamma=K_{CC}/K_{MC} and secondarily on epsilon=K_{MM}/K_{MC}. For epsilon=0 and gamma<2, there is conventional scaling in the long-time limit, with a single mass scale that grows linearly in time. For gamma >= 2, there is unusual behavior in which the concentration of clusters of mass k, c_k decays as a stretched exponential in time within a boundary layer k<k* propto t^{1-2/gamma} (k* propto ln t for gamma=2), while c_k propto t^{-2} in the bulk region k>k*. When epsilon>0, analogous behaviors emerge for gamma<2 and gamma >= 2.Comment: 6 pages, 2 column revtex4 format, for submission to J. Phys.

Exactly solvable models through the generalized empty interval method: multi-species and more-than-two-site interactions

Multi-species reaction-diffusion systems, with more-than-two-site interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E^{\mathbf a}_n(t)$'s, the expectation value of the product of certain linear combination of the number operators on $n$ consecutive sites at time $t$.Comment: 10 pages, LaTe

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.Comment: 9 pages, 2 figure

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"

Cluster approximation solution of a two species annihilation model

A two species reaction-diffusion model, in which particles diffuse on a one-dimensional lattice and annihilate when meeting each other, has been investigated. Mean field equations for general choice of reaction rates have been solved exactly. Cluster mean field approximation of the model is also studied. It is shown that, the general form of large time behavior of one- and two-point functions of the number operators, are determined by the diffusion rates of the two type of species, and is independent of annihilation rates.Comment: 9 pages, 7 figure

Voting and Catalytic Processes with Inhomogeneities

We consider the dynamics of the voter model and of the monomer-monomer catalytic process in the presence of many ``competing'' inhomogeneities and show, through exact calculations and numerical simulations, that their presence results in a nontrivial fluctuating steady state whose properties are studied and turn out to specifically depend on the dimensionality of the system, the strength of the inhomogeneities and their separating distances. In fact, in arbitrary dimensions, we obtain an exact (yet formal) expression of the order parameters (magnetization and concentration of adsorbed particles) in the presence of an arbitrary number $n$ of inhomogeneities (``zealots'' in the voter language) and formal similarities with {\it suitable electrostatic systems} are pointed out. In the nontrivial cases $n=1, 2$, we explicitly compute the static and long-time properties of the order parameters and therefore capture the generic features of the systems. When $n>2$, the problems are studied through numerical simulations. In one spatial dimension, we also compute the expressions of the stationary order parameters in the completely disordered case, where $n$ is arbitrary large. Particular attention is paid to the spatial dependence of the stationary order parameters and formal connections with electrostatics.Comment: 17 pages, 6 figures, revtex4 2-column format. Original title ("Are Voting and Catalytic Processes Electrostatic Problems ?") changed upon editorial request. Minor typos corrected. Published in Physical Review