Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation

Abstract

In this paper we study the equation \[ u t = Δ ( ϕ ( u ) − λ f ( u ) + λ u t ) + f ( u ) u_t=\Delta (\phi (u) - \lambda f(u) + \lambda u_t) + f(u) \] in a bounded domain of R d \mathbb {R}^d , d ≥ 1 d\ge 1 , with homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by ϕ \phi , and total birth and mortality rates characterized by f f . We will show that the aggregating mechanism induced by ϕ ( u ) \phi (u) allows the survival of a species in danger of extinction. Numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution. This is shown to be the case for a particular version of the function ϕ ( u ) \phi (u) .