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

In this paper we study the equation u(t) = Delta(phi(u) -lambdaf(u) + lambdau(t)) + f(u) in a bounded domain of R-d, d greater than or equal to 1, with homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by phi, and total birth and mortality rates characterized by f. We will show that the aggregating mechanism induced by 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 phi(u).