Probabilistic Cellular Automata Describing a Biological Two-Species System

Abstract

We consider two probabilistic cellular automata to analyze the stochastic dynamics of a biological two-species system. We focus our attention on the characterization of the dynamic patterns exhibited by both models. Performing Monte Carlo simulations, we observe a time oscillating behavior that occurs at a local level.

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.…”

“…It was set up by considering the type of local dynamics with unsymmetrical rules and from the results of Monte Carlo simulations for the present PCA [10,29]. To further clarify this point we show in Fig.…”

“…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.…”