Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges. (English) Zbl 1160.92043

Summary: The effects of refuges used by prey on a predator-prey interaction with a class of functional responses are studied by using an analytical approach. The refuges are considered as two types: a constant proportion of prey and a fixed number of prey using refuges. We evaluate the effects with regard to the local stability of the interior equilibrium point, the values of the equilibrium density and the long-term dynamics of the interacting populations. The results show that the effects of refuges used by prey increase the equilibrium density of the prey population while decrease that of predators. It is also proved that the effects of refuges can stabilize the interior equilibrium point of the considered model, and destabilize it under a very restricted set of conditions which is in disagreement with previous results in this field.


92D40 Ecology
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
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[1] Holling, C. S., Some characteristics of simple types of predation and parasitism, Can. Entomol., 91, 385 (1959)
[2] Hassel, M. P.; May, R. M., Stability in insect host-parasite models, J. Anim. Ecol., 42, 693 (1973)
[3] Maynard Smith, J., Models in Ecology (1974), Cambridge University: Cambridge University Cambridge · Zbl 0312.92001
[4] Hassell, M. P., The Dynamics of Arthropod Predator-Prey Systems (1978), Princeton University: Princeton University Princeton, NJ · Zbl 0429.92018
[5] Hoy, M. A., Almonds (California), (Helle, W.; Sabelis, M. W., Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, vol. 1B (1985), Elsevier: Elsevier Amsterdam)
[6] McNair, J. M., The effects of refuges on predator-prey interactions: a reconsideration, Theor. Popul. Biol., 29, 38 (1986) · Zbl 0594.92017
[7] Sih, A., Prey refuges and predator-prey stability, Theor. Popul. Biol., 31, 1 (1987)
[8] Ives, A. R.; Dobson, A. P., Antipredator behavior and the population dynamics of simple predator-prey systems, Am. Nat., 130, 431 (1987)
[9] Ruxton, G. D., Short term refuge use and stability of predator-prey models, Theor. Popul. Biol., 47, 1 (1995) · Zbl 0812.92023
[10] Hochberg, M. E.; Holt, R. D., Refuge evolution and the population dynamics of coupled host-parasitoid associations, Evol. Ecol., 9, 633 (1995)
[11] Taylor, R. J., Predation (1984), Chapman and Hall: Chapman and Hall New York
[12] Kr˘ivan, V., Effects of optimal antipredator behavior of prey on predator-prey dynamics: the role of refuges, Theor. Popul. Biol., 53, 131 (1998) · Zbl 0945.92021
[13] González-Olivars, E.; Ramos-Jiliberto, R., Dynamics consequences of prey refuges in a simple model system: more prey, few predators and enhanced stability, Ecol. Model., 166, 135 (2003)
[14] Kar, T. K., Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10, 681 (2005) · Zbl 1064.92045
[15] Huang, Y.; Chen, F.; Zhong, L., Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput., 182, 672 (2006) · Zbl 1102.92056
[16] May, R. M., Stability and Complexity in Model Ecosystems (1974), Princeton University: Princeton University Princeton, NJ
[17] Collings, J. B., Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 57, 63 (1995) · Zbl 0810.92024
[18] Freedman, H. I., Deterministic Mathematical Model in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
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