Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\phi(u)$.

References [Enhancements On Off] (What's this?)

  • 1. W.C. Allee, Animal aggregations, University of Chicago Press, Chicago, 1931.
  • 2. D.G. Aronson, The role of diffusion in mathematical population biology: Skellam revisited, in ``Mathematics in Biology and Medicine'', Lecture Notes in Biomathematics 57, S. Levin, Springer-Verlag, Berlin,1985. MR 86g:92002
  • 3. D.G. Aronson, Density-dependent interaction-difusion systems, in ``Dynamics and Modeling of Reactive Systems'', W.E. Stewart et al., Acad. Press, 1980. MR 82a:35056
  • 4. J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London, 1960. MR 22:11074
  • 5. P. Grindrod, Models of individual aggregation in single and multispecies communities, J. Math. Biol. 26 (1988), 651-660. MR 90a:92041
  • 6. D. Grunbaum and A. Okubo, Modeling social animal aggregations, in ``Frontiers in Mathematical Biology'', Lecture Notes in Biomathematics 100, S.A. Levin, Springer-Verlag, New York, 1994. MR 96d:92001
  • 7. M.E. Gurtin and R.C. MacCamy, On the diffusion of biological populations, Mathematical biosciences 33 (1977), 35-49. MR 58:33147
  • 8. M.A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology 43 (1993), 141-158.
  • 9. M. Lizana and V. Padrón, A spatially discrete model for aggregating populations, J. Math. Biol. 38 (1999), 79-102. MR 99m:92034
  • 10. J.D. Murray, Mathematical Biology, Biomathematics 19, Springer-Verlag, 1993. MR 94j:92002
  • 11. A. Novick-Cohen and R.L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc. 324 (1991), 331-351. MR 91f:35152
  • 12. A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics 10, Springer-Verlag, New York, 1980. MR 81i:92025
  • 13. V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Commun. in Partial Differential Equations 23 (1998), 457-486. MR 99k:35185
  • 14. M. Protter and H. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. MR 86f:35034
  • 15. J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951), 196-218. MR 13:263b
  • 16. J.G. Skellam, The formulation and interpretation of mathematical models of diffusionary processes in population biology, in ``The Mathematical Theory of the Dynamics of Biological Populations'', M.S. Bartlett and R.W. Hiorns, eds., Acad. Press, New York, 1973. MR 58:20591a

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K70, 35R25, 92D25

Retrieve articles in all journals with MSC (2000): 35K70, 35R25, 92D25

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

American Mathematical Society