## Permanence and extinction for dispersal population systems.(English)Zbl 1073.34052

The system of differential equations $$\dot{x}=f(t,x)$$, $$x\in\mathbb{R}^n$$, is said to be permanent if there exists a compact set $$K\subset \text{int}\,\mathbb{R}_+^n$$ such that all solutions starting in $$\text{int}\,\mathbb{R}_+^n$$ ultimately enter and remain in $$K$$.
The authors study the following predator-pray model in a patchy environment \begin{aligned} \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], \end{aligned} where $$x_i$$ denotes the species $$x$$ in patch $$i$$; $$d(t)>0$$, $$e(t)>0$$, $$b_i(t)>0$$, $$a_i(t)>0$$, $$D_{ij}(t)\geq 0$$ 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 $$x_1\to\infty$$, $$\phi(t,x_1)\geq 0$$, $$\partial\phi(t,x_1)/\partial{x_1}\leq0$$ and $$\partial(x_1\phi(t,x_1))/\partial{x_1}\geq0$$. 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:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D25 Population dynamics (general)

### Keywords:

predator-pray system; permanence; periodic solution
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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., vol. 20, (1977), Springer-Verlag Berlin · Zbl 0363.92014 [10] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Dekker New York · 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), Springer-Verlag Heidelberg · 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), Amer. Math. Soc. · Zbl 0821.34003 [27] Takeuchi, Y., Global dynamical properties of lotka – volterra systems, (1996), World Scientific Singapore · 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) [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
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