Global weak solution for a singular two-component reaction-diffusion system

Abstract

We study a singular reaction–diffusion system motivated by a dedicated diffusive predator–prey model system devised in the spatially homogeneous case by Courchamp and Sugihara [‘Modelling the biological control of an alien predator to protect island species from extinction’, Ecol. Appl. 9 (1999) 112–123]. The reactive part features a functional response to predation and a singular numerical functional response to predation specifically designed for modelling the introduction of greedy predators into a fragile or insular environment. Under some circumstances, this may lead to finite time quenching of the solution, that is, finite time extinction for both species. The aim of this work is to derive a suitable notion of global (in time) weak solution and to prove that such global weak solutions do exist. The existence part is achieved by approximating the reactive part by a more classical and nonsingular one and then passing to the limit in the resulting reaction–diffusion system. Our first result shows that this limiting process supplies global weak solutions. In the case of equidiffusivities such global weak solutions satisfy a suitable free boundary value problem.