Multiple patterns formation for an aggregation/diffusion predator-prey system

We investigate existence of stationary solutions to an aggrega-tion/diffusion system of PDEs, modelling a two species predator-prey interac-tion. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant −α, with α > 0. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are com-posed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant α, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.