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Permanence and extinction for dispersal population systems. (English) Zbl 1073.34052
The system of differential equations $\dot{x}=f(t,x)$, $x\in\Bbb{R}^n$, is said to be {\it permanent} if there exists a compact set $K\subset \text{int}\,\Bbb{R}_+^n$ such that all solutions starting in $\text{int}\,\Bbb{R}_+^n$ ultimately enter and remain in $K$. The authors study the following predator-pray model in a patchy environment $$ \align \dot{x}_1&=x_1[b_1(t)-a_1(t)x_1-y\phi(t,x_1)]+ \sum_{j=1}^n(D_{1j}(t)x_j-D_{j1}(t)x_1),\\ \dot{x}_i&=x_i[b_i(t)-a_i(t)x_i]+ \sum_{j=1}^n(D_{ij}(t)x_j-D_{ji}(t)x_i),\ i=2,\ldots,n,\\ \dot{y}&=y[-d(t)+e(t)x_1\phi(t,x_1)-f(t)y], \endalign $$ where $x_i$ denotes the species $x$ in patch $i$; $d(t)>0$, $e(t)>0$, $b_i(t)>0$, $a_i(t)>0$, $D_{ij}(t)\ge 0$ are continuous $\omega$-periodic functions; $D_{ij}(t)$ is the dispersal coefficient of the species from patch $j$ to patch $i$, $D_{ii}(t)\equiv 0$; the predator functional response $x_1\phi(t,x_1)$ is bounded as $x_1\to\infty$, $\phi(t,x_1)\ge 0$, $\partial\phi(t,x_1)/\partial{x_1}\le0$ and $\partial(x_1\phi(t,x_1))/\partial{x_1}\ge0$. Necessary and sufficient conditions for the permanence of the above system are presented. Sufficient conditions for the permanence of the single-species system in the absence of a predator ($y=0$) is also obtained. The approach is based on the well-known properties of the periodic logistic model $\dot{z}=z(b(t)-a(t)z)$. The biological interpretion of the main results is given.

MSC:
34D05Asymptotic stability of ODE
34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)
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References:
[1] Allen, L. J. S.: Persistence and extinction in single-species reaction -- diffusion models. Bull. math. Biol. 45, 209-227 (1983) · Zbl 0543.92020
[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] Cerri, R. D.; Fraser, D. F.: Predation and risk in foraging minnows: balancing conflicting demands. Amer. nat. 121, 552-561 (1983)
[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.; 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] Cui, J.; Chen, L.: The effect of diffusion on the time varying logistic population growth. Comput. math. Appl. 36, 1-9 (1998) · Zbl 0934.92025
[8] Cui, J.; Chen, L.: Permanence and extinction in logistic and Lotka -- Volterra systems with diffusion. J. math. Anal. appl. 258, 512-535 (2001) · Zbl 0985.34061
[9] Cushing, J. M.: Integro-differential equations and delay models in population dynamics. Lecture notes in biomath. 20 (1977) · Zbl 0363.92014
[10] Freedman, H. I.: Deterministic mathematical models in population ecology. (1980) · Zbl 0448.92023
[11] Freedman, H. I.; Moson, P.: Persistence definitions and their connections. Proc. amer. Math. soc. 109, 1025-1032 (1990) · Zbl 0695.34049
[12] 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
[13] Freedman, H. I.: Single species migration in two habitats: persistence and extinction. Math. model. 8, 778-780 (1987)
[14] 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
[15] Hastings, A.: Spatial heterogeneity and the stability of predator prey systems. Theor. population biology 12, 37-48 (1977) · Zbl 0371.92016
[16] Johnson, M. L.; Gaines, M. S.: Evolution of dispersal: theoretical models and empirical tests using birds and mammals. Annu. rev. Ecol. syst. 21, 449-480 (1990)
[17] Johst, K.; Brandl, B.: Evolution of dispersal: the importance of the temporal order of reproduction and dispersal. Proc. roy. Soc. London ser. B 264, 23-30 (1997)
[18] 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
[19] Levin, S.: Dispersion and population interactions. Amer. nat. 108, 207-228 (1974)
[20] Levin, S.; Segel, L. A.: Hypothesis to explain the origin of planktonic patchiness. Nature 259, 659 (1976)
[21] Lu, Z.; Takeuchi, Y.: Global asymptotic behavior in single-species discrete diffusion systems. J. math. Biol. 32, 67-77 (1993) · Zbl 0799.92014
[22] Milinski, M.: The patch choice model: no alternative to balancing. Amer. nat. 125, 317-320 (1985)
[23] Milinski, M.; Heller, R.: Influence of a predator on the optimal foraging behaviour of stickbacks (Gasteropodus aculeatus L.). Nature (London) 275, 642-644 (1978)
[24] Murray, J. D.: Mathematical biology. (1993) · Zbl 0779.92001
[25] Skellam, J. G.: Random dispersal in theoretical populations. Biometrika 38, 196-218 (1951) · Zbl 0043.14401
[26] Smith, H. L.: Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems. Math. surveys monogr. (1995) · Zbl 0821.34003
[27] Takeuchi, Y.: Global dynamical properties of Lotka -- Volterra systems. (1996) · Zbl 0844.34006
[28] Takeuchi, Y.: Global stability in generalized Lotka -- Volterra diffusion systems. J. math. Anal. appl. 116, 209-221 (1986) · Zbl 0595.92013
[29] Takeuchi, Y.: Diffusion effect on stability of Lotka -- Volterra model. Bull. math. Biol. 46, 585-601 (1986) · Zbl 0613.92025
[30] Takeuchi, Y.: Cooperative system theory and global stability of diffusion models. Acta appl. Math. 14, 49-57 (1989) · Zbl 0665.92017
[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] Teng, Z.; Lu, Z.: The effect of dispersal on single-species nonautonomous dispersal models with delays. J. math. Biol. 42, 439-454 (2001) · Zbl 0986.92024
[34] Thieme, H. R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. biosci. 166, 173-201 (2000) · Zbl 0970.37061
[35] Vance, R. R.: The effect of dispersal on population stability in one-species, discrete space population growth models. Amer. nat. 123, 230-254 (1984)
[36] Wang, W.; Chen, L.: Global stability of a population dispersal in a two-patch environment. Dynam. systems appl. 6, 207-216 (1997) · Zbl 0892.92026
[37] Wang, W.; Fergola, P.; Tenneriello, C.: Global attractivity of periodic solutions of population models. J. math. Anal. appl. 211, 498-511 (1997) · Zbl 0879.92027
[38] Zhang, X.; Chen, L.; Neumann, A. U.: The stage-structured predator -- prey model and optimal harvesting policy. Math. biosci. 101, 139-153 (2000) · Zbl 0961.92037
[39] Zhao, X. -Q.: The qualitative analysis of N-species Lotka -- Volterra periodic competition systems. Math. comput. Modelling 15, 3-8 (1991) · Zbl 0756.34048