The threshold of coexistence and critical behaviour of a predator–prey cellular automaton

Abstract

We study a probabilistic cellular automaton to describe two population biology problems: the threshold of species coexistence in a predator-prey system and the spreading of an epidemic in a population. By carrying out time-dependent simulations we obtain the dynamic critical exponents and the phase boundaries (thresholds) related to the transition between an active state, where prey and predators present a stable coexistence, and a prey absorbing state. The estimates for the critical exponents show that the transition belongs to the directed percolation universality class. In the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class. Patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed.

Stochastic lattice gas model describing the dynamics of the SIRS epidemic process

“…Using a stochastic lattice gas model, we study here the dynamics of propagation of an epidemic in a population in which the individuals are separated into three classes determined by their relative states to a given disease: susceptible (S), infected (I) and recovered (R) individuals. One of the most well known models in this context is the socalled susceptible-infected-recovered (SIR) model [1,2,3,4,5,6,7,8,9,10,11,12,13], a model for an epidemic which occurs during a time interval that is much smaller then the lifetime of the host. It describes the spreading of an epidemic process occurring in a population initially composed by susceptible individuals that become infected by contact with infected individuals.…”

Stable oscillations of a predator–prey probabilistic cellular automaton: a mean-field approach

“…In fact, the role of space in the description of population biology problems has been recognized by several authors in the last years [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. In a very clear manner, Durrett and Levin [11] have pointed out that the modelling of population dynamics systems which are spatially distributed by interacting particle systems [11,27,28,29] is the appropriate theoretical approach that is able to give the more complete description of the problem.…”

Anisotropic probabilistic cellular automaton for a predator-prey system

“…The anisotropic cellular automaton is a variation of the automaton introduced in [10,14,17,18]. Here each site of a regular square lattice interacts with its first neighbors only at two preferential directions.…”

“…In the last years a particular great effort has been done in order to understand the role of space given by a spatial structure and local interactions in the characterization of the dynamics of competing biological species systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In this context it has been studied irreversible stochastic lattice models [19][20][21] with the purpose of mimic predator-prey systems with Markovian local rules based in the Lotka-Volterra model [22,23].…”

“…Here we study the coexistence of a two-species system by considering a probabilistic cellular automaton (PCA) which is a modified version of the automaton devised in [10,14,17,18]. The model, to be called anisotropic predator-prey PCA possess local rules that are similar to the ones of the cellular automaton proposed in [24] and was introduced by us in order to explore the effect of spatial anisotropy in the temporal oscillations.…”