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The effect of dispersal on permanence in a predator-prey population growth model. (English) Zbl 1032.92032

Summary: We consider a periodic predator-prey system where the prey can disperse between one patch with a low level of food and without predation and one patch with a higher level of food but with predation. We assume Volterra within-patch dynamics. Under the assumption that the average of the dispersal rate from Patch 1 to Patch 2 is less than that of the intrinsic growth rate of prey in Patch 1, we provide a sufficient and necessary condition to guarantee the prey and predator species to be permanent by using the main techniques of Z. Teng, Appl. Anal. 72, 339-352 (1999; Zbl 1031.34045).

MSC:

92D40 Ecology
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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[1] Teng, Z., Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. anal., 72, 339-352, (1999) · Zbl 1031.34045
[2] Miliski, M.; Heller, R., Influence of a predator on the optimal foraging behaviour of stickbacks (gasteropodus aculeatus L.), Nature (London), 275, 642-644, (1978)
[3] Cerri, R.D.; Fraser, D.F., Predation and risk in foraging minnows: balancing conflicting demands, Amer. nat., 121, 552-561, (1983)
[4] Miliski, M., The patch choice model: no alternative to balancing, Amer. nat., 125, 317-320, (1985)
[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
[6] Song, X.; Chen, L., Persistence and periodic orbits for two species predator prey system with diffusion, Canad. appl. math. quart., 6, 3, 233-244, (1998) · Zbl 0941.92032
[7] Luo, M.; Ma, Z., The persistence of two species Lotka-Volterra model with diffusion, J. biomath., 12, 52-59, (1997), (in Chinese) · Zbl 0898.92026
[8] Allen, L.J.S., Persistence and extinction in Lotka-Volterra reaction-diffusion equations, Math. biosci., 65, 1-12, (1983) · Zbl 0522.92021
[9] 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
[10] Chewning, W.C., Migratory effect in predation prey systems, Math. biosci., 23, 253-262, (1975) · Zbl 0301.92010
[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. math., 32, 631-648, (1977) · Zbl 0362.92006
[12] 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
[13] 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
[14] Hastings, A., Spatial heterogeneity and the stability of predator prey systems, Theor. pop. biol., 12, 37-48, (1977) · Zbl 0371.92016
[15] Hassell, M.P., The dynamics of arthropod predator-prey systems, (1978), Princeton Univ. Press Princeton, NJ · Zbl 0429.92018
[16] Holt, R.D., Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution, Theor. pop. biol., 28, 181-208, (1985) · Zbl 0584.92022
[17] Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two-patch environments, Math.biosci., 120, 77-98, (1994) · Zbl 0793.92014
[18] Levin, S.A., Dispersion and population interactions, Amer. natur., 108, 207-228, (1974)
[19] Takeuchi, Y., Global stability in generalized Lotka-Volterra diffusion systems, J. math. anal. appl., 116, 209-221, (1986) · Zbl 0595.92013
[20] Takeuchi, Y., Diffusion effect on stability of Lotka-Volterra model, Bull. math. biol., 46, 585-601, (1986) · Zbl 0613.92025
[21] Song, X.; Chen, L., Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay, Computers math. applic., 35, 6, 33-40, (1998) · Zbl 0903.92029
[22] Cui, J.; Chen, L., The effect of diffusion on the time varying logistic population growth, Computers math. applic., 36, 3, 1-9, (1998) · Zbl 0934.92025
[23] Cui, J.; Chen, L., The effects of habitat fragmentation and ecological invasion on population sizes, Computers math. applic., 38, 1, 1-11, (1999) · Zbl 0939.92033
[24] Smith, H.L., Cooperative systems of differential equation with concave nonlinearities, Nonlinear analysis, 10, 1037-1052, (1986) · Zbl 0612.34035
[25] Tineo, A., An iterative scheme for the N-competing species problem, J. diff. equ., 116, 1-15, (1995) · Zbl 0823.34048
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