The role of indirect prey-taxis and interference among predators in pattern formation

It is important to find biological factors that may lead to formation of patches in the distribution of species. We build a simple model describing a consumer/predator which, besides random dispersion, searches for food by moving toward the gradient of some chemical released by prey. This mechanism is referred as indirect prey-taxis. The predator's rate of consumption is assumed to drop due to interference among predators when too many of them encounter on some aggregate of prey. The latter is assumed to have a negligible motility with its density governed by an ODE. The interference among consumers is modeled by a modification of the Beddington's de Angelis functional response. Detailed bifurcation analysis and numerical simulations of an auxiliary system indicate that nonconstant steady states imitating patches of species occur in the model provided the predator density exceeds certain threshold and taxis strength is big enough.