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

Padrón, Víctor

doi:10.1090/S0002-9947-03-03340-3

Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation
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Trans. Amer. Math. Soc. 356 (2004), 2739-2756 Request permission

Abstract:

In this paper we study the equation \[ u_t=\Delta (\phi (u) - \lambda f(u) + \lambda u_t) + f(u) \] in a bounded domain of $\mathbb {R}^d$, $d\ge 1$, 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 $\phi (u)$.

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