Self-Organized Patterns of Coexistence Out of a Predator-Prey Cellular Automaton

Abstract

We present a stochastic approach to modeling the dynamics of coexistence of prey and predator populations. It is assumed that the space of coexistence is explicitly subdivided in a grid of cells. Each cell can be occupied by only one individual of each species or can be empty. The system evolves in time according to a probabilistic cellular automaton composed by a set of local rules which describe interactions between species individuals and mimic the process of birth, death and predation. By performing computational simulations, we found that, depending on the values of the parameters of the model, the following states can be reached: a prey absorbing state and active states of two types. In one of them both species coexist in a stationary regime with population densities constant in time. The other kind of active state is characterized by local coupled time oscillations of prey and predator populations. We focus on the self-organized structures arising from spatio-temporal dynamics of the coexistence. We identify distinct spatial patterns of prey and predators and verify that they are intimally connected to the time coexistence behavior of the species. The occurrence of a prey percolating cluster on the spatial patterns of the active states is also examined.

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

“…The fundamental level is the one based on stochastic lattice gas models also called interacting particle systems [24]. These spatial structured models are relevant when the diseases spreads over a region with few individuals or when a group of individual of a given class becomes isolated, loosing contact with the other class of individuals during a certain period of time [29,30].…”

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.…”