Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T22:03:18.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Global stability in models of population dynamics with diffusion. I. Patchy environments

Published online by Cambridge University Press:  14 November 2011

H. I. Freedman  and
T. Krisztin
H. I. Freedman
Affiliation:
Applied Mathematics Institute, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
T. Krisztin
Affiliation:
Bolyai Institute, University of Szeged, Aradi Vértanúk tere 1, H-6720 Szeged, Hungary
Get access Rights & Permissions [Opens in a new window]

Synopsis

A class of models of single-species dynamics with diffusion within and between patches is considered. It is shown that under our prescribed conditions, there is a unique, positive, globally asymptotically stable steady state.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 1-2 , 1992 , pp. 69 - 84
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
2Cantrell, R. S. and Cosner, C.. Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 293–318.CrossRefGoogle Scholar
3Cantrell, R.S. and Cosner, C.. The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29 (1991), 315–338.CrossRefGoogle Scholar
4Chafee, N. and Infante, E. F.. A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 4 (1974), 17–37.CrossRefGoogle Scholar
5Freedman, H. I.. Deterministic Mathematical Models in Population Ecology (Edmonton: HIFR Consulting Ltd., 1987).Google Scholar
6Freedman, H. I., Shukla, J. B. and Takeuchi, Y.. Population diffusion in a two-patch environment. Math. Biosci. 95 (1989), 111–123.CrossRefGoogle Scholar
7Freedman, H. I. and Wu, J.. Steady state analysis in a model for population diffusion in a multi-patch environment. Nonlinear Anal, (in press).Google Scholar
8Friedman, A.. Partial Differential Equations (New York: Holt, Reinhart and Winston, 1969).Google Scholar
9Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
10Jones, D. D. and Walters, C. J.. Catastrophe theory and fisheries regulation. Fish. Res. Bd. Can. 33 (1976), 2829–2833.CrossRefGoogle Scholar
11Kuang, Y. and Freedman, H. I.. Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math. Biosci. 88 (1988), 67–84.CrossRefGoogle Scholar
12Levin, S. A.. Models of population dispersal in differential equations and applications. In Ecology, Epidemics and Population Problems, eds Busenberg, S. N. and Cooke, K. L. (New York: Academic Press, 1981).Google Scholar
13Levin, S. A.. Population models and community structure in heterogeneous environments. In Mathematical Ecology, eds Hallam, T. G. and Levin, S. A., Biomathematics 17, pp. 295–320 (Berlin: Springer, 1986).CrossRefGoogle Scholar
14Ludwig, D., Aronson, D. G. and Weinberger, H. F.. Spatial patterning of the spruce budworm. J. Math. Biology 8 (1979), 217–258.CrossRefGoogle Scholar
15Munn, R. E. and Fedorov, V.. The Environmental Assessment, IIASA Project Report, Vol. I (Laxenburg, Austria, 1986).Google Scholar
16Okubo, A.. Diffusion and Ecological Problems: Mathematical Models, Biomathematics 10 (New York: Springer, 1980).Google Scholar
17Pazy, A.. Semigroup of Linear Operators and Applications to Partial Differential Equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
18Sattinger, D. H.. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972), 979–1000.CrossRefGoogle Scholar
19Weikert, H.. Plankton and the pelagic environment. In Red Sea, eds Edwards, A. V. and Head, S. M., pp. 90–111 (New York:Pergamon, 1987).CrossRefGoogle Scholar