Chaos and pattern formation in a spatial tritrophic food chain

The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics. The model assumes that the populations are homogeneously mixed, so that the probability that any two individuals interact is uniform and space can be ignored. In this paper we propose a spatial version of the Hastings and Powell model in which predators seek their preys only in a finite neighborhood of their home location, breaking the mixing hypothesis. Treating both space and time as discrete variables we derive a set of coupled equations that describe the evolution of the populations at each site of the spatial domain. We show that the introduction of local predator–prey interactions result in qualitatively distinct dynamics of predator and prey populations. The evolution equations for the predators involve averages over the local density of preys, whereas the equations for the preys involve double averages, where the local density of both preys and predators appear. Our numerical simulations show that local predation also leads to spontaneous pattern formation and to qualitative changes in the global dynamics of the system. In particular, depending on the size of the predation neighborhoods, the chaotic strange attractor present in the original model of Hastings and Powell can be replaced by a stable fixed point or by an attractor of simpler topology.