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Permanence of delayed population model with dispersal loss. (English) Zbl 1093.92059

Summary: Permanence of a dispersal single-species population model where the environment is partitioned into several patches is considered. The species not only requires some time to disperse between the patches but also has some possibility to die during its dispersion. The model is described by delay differential equations. The existence of a ‘super’ food-rich patch is proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence implies permanence if each food-poor patch is chained to the super food-rich patch. Furthermore, it is proven that partial persistence is ensured if there exist food-rich patches and the dispersion of the species among the patches is small. When the dispersion is large, the partial persistence is realized in relatively small dispersion time.

MSC:

92D40 Ecology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
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[1] Allen, L.J.S., Persistence and extinction in single-species reaction-diffusion models, Bull. math. biol., 45, 209, (1983) · Zbl 0543.92020
[2] Allen, L.J.S., Persistence, extinction and critical patch number for island populations, J. math. biol., 24, 617, (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, (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, (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, (1987) · Zbl 0634.92017
[6] Cui, J.; Chen, L., The effect of diffusion on the time varying logistic population growth, Computers math. appl., 36, 1, (1998) · Zbl 0934.92025
[7] Cui, J.; Chen, L., Permanent and extinction in logistic and Lotka-Volterra systems with diffusion, J. math. anal. appl., 258, 512, (2001) · Zbl 0985.34061
[8] Cui, J.; Takeuchi, Y.; Lin, Z., Permanent and extinction for dispersal population systems, J. math. anal. appl., 298, 73, (2004) · Zbl 1073.34052
[9] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[10] 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, (1977) · Zbl 0362.92006
[11] Freedman, H.I., Single species migration in two habitats: persistence and extinction, Math. model., 8, 778, (1987)
[12] 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, (1989) · Zbl 0685.92018
[13] Hastings, A., Spatial heterogeneity and the stability of predator prey systems, Theoret. populat. biol., 12, 37, (1977) · Zbl 0371.92016
[14] Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. biosci., 120, 77, (1994) · Zbl 0793.92014
[15] Levin, S., Dispersion and population interactions, Am. nat., 108, 207, (1974)
[16] Lu, Z.; Takeuchi, Y., Global asymptotic behavior in single-species discrete diffusion systems, J. math. biol., 32, 67, (1993) · Zbl 0799.92014
[17] Takeuchi, Y., Global dynamical properties of Lotka-Volterra systems, (1996), World Scientific Singapore · Zbl 0844.34006
[18] Takeuchi, Y., Global stability in generalized Lotka-Volterra diffusion systems, J. math. anal. appl., 116, 209, (1986) · Zbl 0595.92013
[19] Takeuchi, Y., Diffusion effect on stability of Lotka-Volterra model, Bull. math. biol., 46, 585, (1986) · Zbl 0613.92025
[20] Takeuchi, Y., Cooperative system theory and global stability of diffusion models, Acta appl. math., 14, 49, (1989) · Zbl 0665.92017
[21] Takeuchi, Y., Diffusion-mediated persistence in two-species competition Lotka-Volterra model, Math. biosci., 95, 65, (1989) · Zbl 0671.92022
[22] Takeuchi, Y., Conflict between the need to forage and the need to avoid competition: persistence of two-species model, Math. biosci., 99, 181, (1990) · Zbl 0703.92024
[23] Y. Takeuchi, J. Cui, R. Miyazaki, Y. Saito, Permanence and periodic solutions of dispersal population model with time delays, preprint. · Zbl 1109.34059
[24] Teng, Z.; Lu, Z., The effect of dispersal on single-species nonautonomous dispersal models with delays, J. math. biol., 42, 439, (2001) · Zbl 0986.92024
[25] Vance, R.R., The effect of dispersal on population stability in one-species, discrete space population growth models, Am. nat., 123, 230, (1984)
[26] Wang, W.; Chen, L., Global stability of a population dispersal in a two-patch environment, Dyn. syst. appl., 6, 207, (1997) · Zbl 0892.92026
[27] Smith, H.L., Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems, (1995), AMS Providence, RI · Zbl 0821.34003
[28] Tineo, A., An iterative scheme for the N-competing species problem, J. diff. equat., 116, 1, (1995) · Zbl 0823.34048
[29] Zhao, X.Q., The qualitative analysis of N-species Lotka-Volterra periodic competition systems, Math. comp. model., 15, 3, (1991) · Zbl 0756.34048
[30] Teng, Z.; Chen, L., The positive periodic solutions of periodic Kolmogorov type systems with delays, Acta math. appl. sinica, 22, 446, (1999) · Zbl 0976.34063
[31] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press San Diego, CA · Zbl 0777.34002
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