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

Author: Víctor Padrón
Journal: Trans. Amer. Math. Soc. 356 (2004), 2739-2756
MSC (2000): Primary 35K70; Secondary 35R25, 92D25
Published electronically: October 21, 2003
MathSciNet review: 2052595
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Abstract: In this paper we study the equation

\begin{displaymath}u_t=\Delta(\phi(u) - \lambda f(u) + \lambda u_t) + f(u) \end{displaymath}

in a bounded domain of $\mathbb{R} ^d$, $d\ge1$, 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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Additional Information

Víctor Padrón
Affiliation: Facultad de Ciencias, Departamento de Matemáticas, Universidad de Los Andes, Mérida 5101, Venezuela

Keywords: Pseudoparabolic equation, aggregating populations, recovery
Received by editor(s): May 6, 2002
Received by editor(s) in revised form: January 22, 2003
Published electronically: October 21, 2003
Additional Notes: This research was supported in part by Consejo de Desarrollo Científico, Humanístico y Técnico (CDCHT) of the Universidad de Los Andes
Article copyright: © Copyright 2003 American Mathematical Society

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