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

Abstract

Abstract.We analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator-prey system. The individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired on the processes of the Lotka-Volterra model. Two levels of mean-field approximations are set up. The simple approximation is equivalent to an extended patch model, a simple metapopulation model with patches colonized by prey, patches colonized by predators and empty patches. This approximation is capable of describing the limited available space for species occupancy. The pair approximation is moreover able to describe two types of coexistence of prey and predators: one where population densities are constant in time and another displaying self-sustained time-oscillations of the population densities. The oscillations are associated with limit cycles and arise through a Hopf bifurcation. They are stable against changes in the initial conditions and, in this sense, they differ from the Lotka-Volterra cycles which depend on initial conditions. In this respect, the present model is biologically more realistic than the Lotka-Volterra model.

Population dynamics on random networks: simulations and analytical models

“…In this case also, sustained oscillations have been identified in a small region of the phase diagram of the PA equations of a two parameter predator-prey model on the square lattice [15,16]. Not much is known about predator-prey interaction parameters [30], and this small oscillatory phase may correspond to biologically realistic parameter values for some predatorprey pairs.…”

“…The performance of the PA was compared with the perfect mixing or mean field approximation (MFA) for several different population dynamics models on lattices and other graphs [14,15,16,17,18]. In these studies, the PA was found to perform much better than the MFA in the prediction of the steady state of the system, and to give a good quantitative agreement in a large region of parameter space.…”

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.…”