Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology. Abstract.Most of the fundamental elements of ecology, ranging from individual behavior to species abundance, diversity, and population dynamics, exhibit spatial variation. Partial differential equation models provide a means of melding organism movement with population processes and have been used extensively to elucidate the effects of spatial variation on populations. While there has been an explosion of theoretical advances in partial differential equation models in the past two decades, this work has been generally neglected in mathematical ecology textbooks. Our goal in this paper is to make this literature accessible to experimental ecologists.Partial differential equations are used to model a variety of ecological phenomena; here we discuss dispersal, ecological invasions, critical patch size, dispersal-mediated coexistence, and diffusion-driven spatial patterning. These models emphasize that simple organism movement can produce striking large-scale patterns in homogeneous environments, and that in heterogeneous environments, movement of multiple species can change the outcome of competition or predation. In the classical applications of PDE models to population ecology, organisms are assumed to have Brownian random motion, the rate of which is invariant in time and space. This assumption leads to the diffusion model (Okubo 1980, Edelstein-Keshet 1986, Murray 1989): au (x, y, t) (02U a2U _ _ _ _= LITERATURE CITED Allee, W. C. 1931. Animal aggregations, a study on general sociology. University of Chicago Press, Chicago, Illinois, USA. Alt, W. 1985. Models for mutual attraction and aggregation of motile individuals. Pages 33-38 in V. Capasso, E. Grosso, and S. L. Paveri-Fontana, editors. Lecture Notes in Biomathematics Number 57. Ammerman, A. J., and L. L. Cavalli-Sforza. 1984. The Neolithic transition and the genetics of populations in Europe. . 1988. Analyzing field studies of insect dispersal using two-dimensional transport equations. Environmental Entomology 17:815-820. Bertsch, M., M. E. Gurtin, D. Hilhorst, and L. A. Peletier. 1984. On interacting populations that disperse to avoid crowding: the effect of a sedentary colony. Journal of Mathematical Biology 19:1-12. Bertsch, M., M. E. Gurtin, D. Hilhorst, and L. A. Peletier. 1985. On interacting populations that disperse to avoid crowding: preservation of segregation. Journal of Mathematical Biology 23:1-13. Bradford, E., and J. P. Philip. 1970a. Stability of steady distributions of asocial populations in one dimension. Journal of Theoretical Biology 29:13-26.

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