Critical domain sizes of a discrete-map hybrid and reaction-diffusion model on hostile exterior domains

Abstract

We study a hybrid impulsive reaction-diffusion equation composed with a discrete-time map in bounded domain $\varOmega $ in space dimension $n\in \mathbb N$ . We assume that the exterior of domain is not lethal (not completely hostile) but hostile. We consider Robin boundary conditions which are used for mixed or reactive or semipermeable boundaries. Given geometry of the domain $\varOmega $ , we establish critical domain sizes for the persistence and extinction of a species. Specifically, for habitats with the shape of $n$ -hypercube and ball of fixed radius, we formulate the critical domain sizes depending on parameters of the model, including $h$ , i.e. a measure of the hostility of the external (to $\varOmega $ ) environment. For a general habitat, called Lipschitz domains, we apply isoperimetric inequalities and variational methods to find the associated critical domain sizes. We also provide applications of the main results in marine reserve, terrestrial reserve and insect pest outbreaks.