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Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments. (English) Zbl 1066.92059

Summary: A delayed periodic Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments is investigated. By using R. E. Gaines and J. L. Mawhin’s [Coincidence degree and nonlinear differential equations. (1977; Zbl 0339.47031)] continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness and global stability of positive periodic solutions of the system. Numerical simulations are given to illustrate the feasibility of our main results.

MSC:

92D40 Ecology
34K13 Periodic solutions to functional-differential equations
37N25 Dynamical systems in biology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations

Citations:

Zbl 0339.47031
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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.; 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
[4] 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
[5] 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
[6] Beretta, E.; Kuang, Y., Convergence results in a well-known delayed predator-prey system, J. math. anal. appl., 204, 840-853, (1996) · Zbl 0876.92021
[7] Cushing, J.M., Integro-differential equations and delay models in population dynamics, (1977), Springer Heidelberg · Zbl 0363.92014
[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 appl. math., 32, 631-648, (1977) · Zbl 0362.92006
[12] Freedman, H.I.; Rai, B.; Waltman, P., Mathematical models of population interaction with dispersal: differential survival in a change of habitat, J. math. anal. appl., 115, 140-154, (1986) · Zbl 0588.92020
[13] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer Berlin · Zbl 0339.47031
[14] Gopalsamy, K., Harmless delay in model systems, Bull. math. biol., 45, 295-309, (1983) · Zbl 0514.34060
[15] Gopalsamy, K., Delayed responses and stability in two-species systems, J. austral math. soc. ser. B, 25, 473-500, (1984) · Zbl 0552.92016
[16] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht/Norwell, MA · Zbl 0752.34039
[17] Hale, J., Theory of functional differential equations, (1977), Springer Heidelberg
[18] Hastings, A., Delays in recruitment at different trophic levels: effects on stability, J. math. biol., 21, 35-44, (1984) · Zbl 0547.92014
[19] 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)
[20] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[21] 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
[22] Levin, S.A., Dispersion and population interactions, Amer. nat., 108, 207-228, (1974)
[23] Levin, S.A.; Segel, L.A., Hypothesis to explain the origin of planktonic patchness, Nature, 259, 659, (1976)
[24] Lu, Z.; Takeuchi, Y., Global asymptotic behavior in single-species discrete diffusion systems, J. math. biol., 32, 67-77, (1993) · Zbl 0799.92014
[25] MacDonald, N., Time lags in biological models, (1978), Springer Heidelberg · Zbl 0403.92020
[26] May, R.M., Time delay versus stability in population models with two and three trophic levels, Ecology, 4, 315-325, (1973)
[27] Ruan, S., Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. appl. math., 59, 159-173, (2001) · Zbl 1035.34084
[28] Skellam, J.D., Random dispersal in theoretical population, Miometrika, 38, 196-216, (1951) · Zbl 0043.14401
[29] Takeuchi, Y., Global stability in generalized lotka – volterra diffusive systems, J. math. anal. appl., 116, 209-221, (1986) · Zbl 0595.92013
[30] Takeuchi, Y., Diffusion effect on stability of lotka – volterra models, Bull. math. biol., 46, 586-601, (1986) · Zbl 0613.92025
[31] Takeuchi, Y., Diffusion-mediated persistence in two-species competition lotka – volterra model, Math. biosci., 95, 65-83, (1989) · Zbl 0671.92022
[32] 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
[33] Vance, R.R., The effect of dispersal on population stability in one-species, discrete space population growth models, Amer. nat., 123, 230-254, (1984)
[34] Wang, W.; Chen, L., Global stability of a population dispersal in a two-patch environment, Dynamic systems appl., 6, 207-216, (1997) · Zbl 0892.92026
[35] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
[36] Wang, W.; Ma, Z., Asymptotic behavior of a predator-prey system with diffusion and delays, J. math. anal. appl., 206, 191-204, (1997) · Zbl 0872.92019
[37] Xu, R.; Chen, L., Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment, Comput. math. appl., 40, 577-588, (2000) · Zbl 0949.92028
[38] Zhang, J.; Chen, L.; Chen, X., Persistence and global stability for two-species nonautonomous competition lotka – volterra patch-system with time delay, Nonlinear anal. TMA, 37, 1019-1028, (1999) · Zbl 0949.34060
[39] Zhang, Z.; Wang, Z., Periodic solution for a two-species nonautonomous competition lotka – volterra patch system with time delay, J. math. anal. appl., 265, 38-48, (2002) · Zbl 1003.34060
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