Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-28T23:18:25.417Z Has data issue: false hasContentIssue false
  • English
  • Français

Permanence in ecological systems with spatial heterogeneity

Published online by Cambridge University Press:  14 November 2011

Robert Stephen Cantrell ,
Chris Cosners  and
Vivian Hutson
Robert Stephen Cantrell
Affiliation:
Department of Mathematics and Computer Science, The University of Miami, Coral Gables, FL 33124, U.S.A.
Chris Cosners
Affiliation:
Department of Mathematics and Computer Science, The University of Miami, Coral Gables, FL 33124, U.S.A.
Vivian Hutson
Affiliation:
Department of Applied Mathematics, The University of Sheffield, Sheffield S102TN, U.K.
Get access Rights & Permissions [Opens in a new window]

Synopsis

A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of the problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of two reaction–diffusion equations which we interpret as a semi-dynamical system. We say that the system is permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. Our analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 3 , 1993 , pp. 533 - 559
Copyright
Copyright © Royal Society of Edinburgh 1993

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

1Alikakos, N. D.. An application of the invariance principle to reaction-diffusion equations. J. Differential Equations 33 (1979), 201–225.CrossRefGoogle Scholar
2Amann, H.. Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360 (1985), 47–83.Google Scholar
3Bhatia, N. and Szego, G.. Stability Theory of Dynamical Systems (Berlin: Springer, 1970).CrossRefGoogle Scholar
4Blat, J. and Brown, K. J., Bifurcation of steady–state solutions in predator-prey and competition systems. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21–34.CrossRefGoogle Scholar
5Brown, K. J.. Nontrivial solutions of predator-prey systems with small diffusion. Nonlinear Anal. 11 (1987), 685–689.CrossRefGoogle Scholar
6Brown, P. N.. Decay to uniform states in ecological interactions. SIAM J. Appl. Math. 38 (1980), 22–37.CrossRefGoogle Scholar
7Butler, G., Freedman, H. and Waltman, P.. Uniformly persistent dynamical systems. Proc. Amer. Math. Soc. 96 (1986), 425–430.CrossRefGoogle Scholar
8Cantrell, 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
9Cantrell, R. S. and Cosner, C.. On the steady-state problem for the Volterra-Lotka competition model with diffusion. Houston J. Math. 13 (1987), 337–352.Google Scholar
10Cantrell, R. S. and Cosner, C.. On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion. Houston J. Math. 15 (1989), 341–361.Google Scholar
11Cosner, C.. Eigenvalue problems with indefinite weights and reaction-diffusion models in population dynamics. In Reaction-Diffusion Equations, eds Brown, K. J. and Lacey, A. A. (Oxford: Oxford University Press. 1990).Google Scholar
12Cosner, C. and Lazer, A. C.. Stable coexistence states in the Volterra-Lotka competition model with diffusion. SIAM J. Appl. Math. 44(1984). 1112–1132.CrossRefGoogle Scholar
13Dancer, E. N.. On positive solutions of some pairs of differential equations. Trans. Amer. Math. Soc. 284(1984), 729–743.CrossRefGoogle Scholar
14Dancer, E. N.. On positive solutions of some pairs of differential equations II. J. Differential Equations 60 (1985), 236–258.CrossRefGoogle Scholar
15De Mottoni, P. and Rothe, F.. Convergence to homogeneous equilibrium states for generalized Volterra-Lotka systems. SIAM J. Appl. Math. 37 (1979), 648–663.CrossRefGoogle Scholar
16Dunbar, S. R., Rybakowski, K. P. and Schmitt, K.. Persistence in models of predator-prey populations with diffusion. J. Differential Equations 65 (1986), 117–138.CrossRefGoogle Scholar
17Fife, P. C.. Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28 (Berlin: Springer, 1979).CrossRefGoogle Scholar
18Friedman, A.. Partial Differential Equations (New York: Holt, Rinehart and Winston, 1969).Google Scholar
19Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, N.J.: Prentice Hall, 1964).Google Scholar
20Hale, J. K. and Waltman, P.. Persistence in infinite dimensional systems. SIAM J. Appl. Math. 20 (1989), 388–395.CrossRefGoogle Scholar
21Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981)CrossRefGoogle Scholar
22Hess, P. and Kato, T.. On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. Partial Differential Equations 5 (1980), 999–1030.CrossRefGoogle Scholar
23Hofbauer, J. and Sigmund, K.. Dynamical Systems and the Theory of Evolution (Cambridge: Cambridge University Press, 1988).Google Scholar
24Hutson, V.. A theorem on average Liapunov functions. Monatsh. Math. 98 (1984), 267–275.CrossRefGoogle Scholar
25Hutson, V.. The existence of an equilibrium for permanent systems. Rocky Mountain J. Math. 20 (1990), 1033–1040.CrossRefGoogle Scholar
26Hutson, V. and Moran, W.. Repellers in reaction-diffusion systems. Rockv Mountain J. Math. 17 (1987), 301–314.Google Scholar
27Hutson, V. and Schmitt, K.. Permanence in dynamical systems. Math. Biosci. 8 (1992), 1–71.CrossRefGoogle Scholar
28Levin, S.. Population models and community structure in heterogeneous environments. In Mathematical Ecology, eds Hallam, T. G. and Levin, S., Biomathematics 17 (Berlin: Springer, 1986).Google Scholar
29Li, L.. Coexistence theorems of steady-states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), 143–166.CrossRefGoogle Scholar
30Li, L.. Global positive coexistence of a nonlinear elliptic biological interacting model. Math. Biosci. 97 (1989), 1–15.CrossRefGoogle ScholarPubMed
31Li, L. and Logan, R.. Positive solutions to general elliptic competition models. J. Differential and Integral Equations 4 (1991) 817–834.Google Scholar
32Manes, A. and Micheletti, A. M.. Un'estensione della teoria variazionale classica degli antovalori per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 7 (1973), 285–301.Google Scholar
33Mora, X.. Semilinear parabolic problems define semiflows in Ck spaces. Trans. Amer. Math. Soc. 278(1983), 21–55.Google Scholar
34Okubo, A.. Diffusion and Ecological Problems: Mathematical Models, Biomathematics 10 (Berlin: Springer, 1980).Google Scholar
35Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations (Englewood Cliffs, N.J.: Prentice Hall, 1967).Google Scholar
36Redheffer, R. and Walter, W.. Invariant sets for systems of partial differential equations. Arch. Rational Mech. Anal. 67 (1977), 41–52.CrossRefGoogle Scholar
37Smoller, J.. Shock Waves and Reaction-Diffusion Equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
38Zeidler, E.. Nonlinear Functional Analysis and its Applications, vol. II, (Berlin: Springer, 1986).CrossRefGoogle Scholar