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Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge. (English) Zbl 1186.34062
From the introduction: The aim of this paper is to consider instability and global stability properties of the equilibria and the existence and uniqueness of limit cycles of the following model with Holling type II functional response incorporating a constant prey refuge $$m$$
\begin{aligned} & \dot x=\alpha x\left(1-\frac xk\right)-\frac{\beta(x-m)y}{1+a(x-m)}\,,\\ & \dot y=-dy + \frac{c\beta(x-m)y}{1+a(x-m)}\,,\end{aligned}\tag{1}
where $$x, y$$ denote prey and predator population respectively at any time $$t,d,k,\alpha,\beta,a,c,m$$ are positive constants.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 92D25 Population dynamics (general)
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##### References:
 [1] Holling, C.S., The functional response of predators to prey density and its role in mimicry and population regulations, Memoirs of the entomological society of Canada, 45, 3-60, (1965) [2] González-Olivares, E.; Ramos-Jiliberto, R., Dynamic consequences of prey refuges in a simple model system: more prey,fewer predators and enhanced stability, Ecological modelling, 166, 135-146, (2003) [3] Hassel, M.P., The dynamics of arthropod predator – prey systems, (1978), Princeton University Press Princeton · Zbl 0429.92018 [4] Kar, T.K., Stability analysis of a prey – predator model incorporating a prey refuge, Communications in nonlinear science and numerical simulation, 10, 681-691, (2005) · Zbl 1064.92045 [5] Huang, Y.J.; Chen, F.D.; Li, Z., Stability analysis of a prey – predator model with Holling type III response function incorporating a prey refuge, Applied mathematics and computation, 182, 672-683, (2006) · Zbl 1102.92056 [6] McNair, J.N., The effects of refuges on predator – prey interactions: A reconsideration, Theoretical population biology, 29, 38-63, (1986) · Zbl 0594.92017 [7] McNair, J.N., Stability effects of prey refuges with entry – exit dynamics, Journal of theoretical biology, 125, 449-464, (1987) [8] Ko, W.; Ryu, K., Qualitative analysis of a predator – prey model with Holling type II functional response incorporating a prey refuge, Journal of differential equations, 231, 534-550, (2006) · Zbl 1387.35588 [9] Kar, T.K., Modelling and analysis of a harvested prey – predator system incorporating a prey refuge, Journal of computational and applied mathematics, 185, 19-33, (2006) · Zbl 1071.92041 [10] Collings, J.B., Bifurcation and stability analysis of a temperature-dependent mite predator – prey interaction model incorporating a prey refuge, Bulletin of mathematical biology, 57, 63-76, (1995) · Zbl 0810.92024 [11] Sih, A., Prey refuges and predator – prey stability, Theoretical population biology, 31, 1-12, (1987) [12] Krivan, V., Effects of optimal antipredator behavior of prey on predator – prey dynamics: the role of refuges, Theoretical population biology, 53, 131-142, (1998) · Zbl 0945.92021 [13] Zhang, Z.F.; Ding, T.R.; Huang, W.Z.; Dong, Z.X., Qualitative theory of differential equations, (1985), Science Publishing Company Beijing
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