×

Predator-prey dynamics in models of prey dispersal in two-patch environments. (English) Zbl 0793.92014

Summary: Models are presented for a single species that disperses between two patches of a heterogeneous environment with barriers between patches and a predator for which the dispersal between patches does not involve a barrier. Conditions are established for the existence, uniform persistence, and local and global stability of positive steady states. In particular, an example that demonstrates both the stabilizing and destabilizing effects of dispersion is presented. This example indicates that a stable migrating predator-prey system can be made unstable by changing the amount of migration in both directions.

MSC:

92D40 Ecology
34D05 Asymptotic properties of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Allen, L. J.S., Persistence and extinction in Lotka-Volterra reaction diffusion equations, Math. Biosci., 65, 1-12 (1983) · Zbl 0522.92021
[2] Allen, L. J.S., Persistence, extinction, and critical patch number for island populations, J. Math. Biol., 24, 617-625 (1987) · Zbl 0603.92019
[3] Beretta, E.; Takeuchi, Y., Global stability of single-species diffusion Volterra models with continuous time delays, Bull. Math. Biol., 49, 431-448 (1987) · Zbl 0627.92021
[4] Beretta, E.; Takeuchi, Y., Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delays, SIAM J. Appl. Math., 48, 627-651 (1988) · Zbl 0661.92018
[5] Beretta, E.; Solimano, F.; Takeuchi, Y., Global stability and periodic orbits for two patch predator-prey diffusion-delay models, Math. Biosci., 85, 153-183 (1987) · Zbl 0634.92017
[7] Edelstein-Keshet, L., Mathematical Models in Biology (1988), Random House: Random House New York · Zbl 0674.92001
[8] Freedman, H. I., Single species migration in two habitats: persistence and extinction, Math. Model., 8, 778-780 (1987)
[9] Freedman, H. I.; Takeuchi, Y., Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear Anal., TMA 13, 993-1002 (1989) · Zbl 0685.92018
[10] Freedman, H. I.; Takeuchi, Y., Predator survival versus extinction as a function of dispersal in a predator-prey model with patchy environment, Appl. Anal., 31, 247-266 (1989) · Zbl 0641.92016
[11] Freedman, H. I.; Waltman, P., Mathematical models of population interaction with dispersal. I. Stability of two habitats with and without a predator, SIAM J. Appl. Math., 32, 631-648 (1977) · Zbl 0362.92006
[12] Freedman, H. I.; Waltman, P., Persistence in models of three interacting predator-prey populations, Math. Biosci., 68, 213-231 (1984) · Zbl 0534.92026
[13] Freedman, H. I.; Rai, B.; Waltman, P., Mathematical models of population interactions with dispersal. II. Differential survival in a change of habitat, J. Math. Anal. Appl., 115, 140-154 (1986) · Zbl 0588.92020
[14] Hastings, A., Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates, J. Math. Biol., 28, 181-208 (1985)
[15] Holt, R. D., Population dynamics in two patch environments: some anomalous consequences of optional habitat selection, Theor. Pop. Biol., 28, 181-208 (1985) · Zbl 0584.92022
[16] Huffaker, C. B.; Kennett, C. E., Experimental studies on predation: predation and cyclamen-mite populations on strawberries on California, Hilgardia, 26, 191-222 (1956)
[17] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Math. Biosci., 111, 1-71 (1992) · Zbl 0783.92002
[18] Kuang, Y.; Smith, H. L., Global stability in infinite delay, Lotka-Volterra type systems. Lotka-Volterra type systems, J. Differ. Equations, 102 (1993) · Zbl 0786.34077
[19] Kuang, Y.; Martin, R. H.; Smith, H. L., Global stability for infinite delay, dispersive Lotka-Volterra systems: weakly interacting populations in nearly identical patches, J. Dynam. Differ. Equations, 3, 339-360 (1991) · Zbl 0731.92029
[20] LeBlond, N. R., Porcupine Caribou Herd (1979), Canadian Arctic Resources Comm: Canadian Arctic Resources Comm Ottawa
[21] Levin, S. A., Dispersion and population interactions, Am. Nat., 1008, 207-228 (1974)
[22] Levin, S. A., Spatial patterning and the structure of ecological communities, (Ludwig, D., Some Mathematical Questions in Biology, Vol. VII (1976), Am. Math. Soc: Am. Math. Soc Providence, R.I), 1-35
[23] Levin, S. A.; Segal, L. A., Hypothesis for origin of planktonic patchiness, Nature, 259, 659 (1976)
[24] Segal, L. A.; Levin, S. A., Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions, (Lakin, W. D., Topics in Statistical Mechanics and Biophysics: A Memorial to Julius L. Jackson. Topics in Statistical Mechanics and Biophysics: A Memorial to Julius L. Jackson, AIP Conf. Proc. (1976), Amer. Inst. Phys: Amer. Inst. Phys New York), 123-152, No. 27
[25] Smith, H. L., On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46, 368-375 (1986) · Zbl 0607.92023
[26] Takeuchi, Y., Global stability in generalized Lotka-Volterra diffusion systems, J. Math. Anal. Appl., 116, 209-221 (1986) · Zbl 0595.92013
[27] Takeuchi, Y., Diffusion effect on stability of Lotka-Volterra models, Bull. Math. Biol., 46, 585-601 (1986) · Zbl 0613.92025
[28] Takeuchi, Y., Cooperative systems theory and global stability of diffusion models, Acta Appl. Math., 14, 49-57 (1989) · Zbl 0665.92017
[29] Takeuchi, Y., Diffusion-mediated persistence in two-species competition Lotka-Volterra model, Math. Biosci., 95, 65-83 (1989) · Zbl 0671.92022
[30] Takeuchi, Y., Conflict between the need to forage and the need to avoid competition: persistence of two-species model, Math. Biosci., 99, 181-194 (1990) · Zbl 0703.92024
[31] Vance, R. R., The effect of dispersal on population stability in one-species, discrete space population growth models, Am. Nat., 123, 230-254 (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.