Abstract
We analyze a discrete-time model of populations that grow and disperse in separate phases. The growth phase is a nonlinear process that allows for the effects of local crowding. The dispersion phase is a linear process that distributes the population throughout its spatial habitat. Our study quantifies the issues of survival and extinction, the existence and stability of nontrivial steady states, and the comparison of various dispersion strategies. Our results show that all of these issues are tied to the global nature of various model parameters. The extreme strategies of staying-in place and going-everywhere-uniformly are compared numerically to diffusion strategies in various contexts. We approach the mathematical analysis of our model from a functional analysis and an operator theory point of view. We use recent results from the theory of positive operators in Banach lattices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allen, L.: Persistence, extinction, and critical patch number for island population. J. Math. Biol. 24, 617–625 (1987)
Cohen, D., Murray, J. D.: A generalized diffusion model for growth and dispersal in a population. J. Math. Biol. 12, 237–249 (1981)
DeAngelis, D. L., Post, W. M., Travis, C. C.: Positive feedback in natural systems. (Biomathematics, vol. 15). Berlin Heidelberg New York: Springer 1986
Deimling, K.: Nonlinear functional analysis. Berlin Heidelberg New York: Springer 1985
Ellner, S.: Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol. 19, 169–200 (1984)
Gadgil, M.: Dispersal: population consequences and evolution. Ecology 52, 253–261 (1971)
Hamilton, W. D., May, R. M.: Dispersal in stable habitats. Nature 269, 578–581 (1977)
Hardin, D. P., Takáč, P. Webb, G. F.: A comparison of dispersal strategies for survival of spatially heterogeneous populations. SIAM J. Appl. Math. 48, 1396–1423 (1988)
Hardin, D. P., Takáč, P., Webb, G. F.: Asymptotic properties of a continuous-space discrete-time population model in a random environment. J. Math. Biol. 26, 361–374 (1988)
Hörmander, L.: The analysis of linear partial differential operators I. Berlin Heidelberg New York: Springer 1983
Horn, H. L., MacArthur, R. H.: Competition among fugitive species in a harlequin environment. Ecology 53, 749–752 (1972)
Kierstead, H., Slobodkin, L. B.: The size of water masses containing plankton bloom. J. Mar. Res. 12, 141–147 (1953)
Klinkhamer, P. G. L., de Jong, T. J., Metz, J. A. J., Val, J.: Life history tactics of annual organisms: The joint effects of dispersal and delayed germination. Theor. Popul. Biol. 32, 27–156 (1987)
Kot, M.: Diffusion-driven period-doubling bifurcation. Preprint 1987
Kot, M., Schaffer, W. M.: Discrete-time growth-dispersal models. Math. Biosci. 80, 109–136 (1986)
Krasnoselskii, M. A., Zabreiko, P. P.: Geometrical methods of nonlinear analysis. Berlin Heidelberg New York: Springer 1984
Levin, S. A.: Dispersion and population interactions. Am. Natur. 108, 207–228 (1974)
Levin, S. A., Cohen, D., Hastings, A.: Dispersal strategies in patchy environments. Theor. Popul. Biol. 26, 165–191 (1984)
Levin, S. A., Segel, L. A.: Pattern generation in space and aspect. SIAM Rev. 27, 45–67 (1985)
MacArthur, R. H., Wilson, E. O.: The theory of island biogeography. Princeton: Princeton University Press 1967
Martin, R. H.: Nonlinear operators and differential equations in Banach spaces. New York: Wiley 1976
McMurtrie, R.: Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments. Math. Biosci. 39, 11–51 (1978)
Okubo, A.: Diffusion and ecological problems: Mathematical models (Biomathematics, vol. 10). Berlin Heidelberg New York: Springer 1980
Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1974
Rudin, W.: Functional analysis. New York: McGraw-Hill 1973
Schaefer, H. H.: Topological vector spaces. Berlin Heidelberg New York: Springer 1971
Schaefer, H. H.: Banach lattices and positive operators. Berlin Heidelberg New York: Springer 1974
Skellam, J. G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)
Takáč, P.: The local stability of positive solutions to the Hammerstein equation with a nonmonotonic Nemytskii operator. Mh. Math. 106, 313–335 (1988)
Vance, R. R.: The effect of dispersal on population stability in one-species, discrete-space population growth models. Am. Natur. 123, 230–254 (1984)
Yosida, K.: Functional analysis, 3rd edn. Berlin Heidelberg New York: Springer 1971
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hardin, D.P., Takáč, P. & Webb, G.F. Dispersion population models discrete in time and continuous in space. J. Math. Biol. 28, 1–20 (1990). https://doi.org/10.1007/BF00171515
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00171515