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Stability analysis of a prey-predator model incorporating a prey refuge. (English) Zbl 1064.92045

The author analyzes a Lotka-Volterra type predator-prey model with Michaelis-Menten type functional responce. In this model, the population density of the prey is resource limited and each predator’s functional responce to the prey approaches a constant as the prey population increases and a spatial refuge protects a constant proportion of prey from predation. A refuge can be important for the biological control of a pest, but increasing the amount of refuge can increase prey densities and lead to population outbreaks.

Conditions for existence and stability of the equilibria and persistent criteria for the system are proposed. It is proved that exactly one stable limit cycle occurs in this system when the positive equilibrium is unstable; and local asymptotic stability of the positive equilibrium implies its global asymptotic stability.

34D05Asymptotic stability of ODE
92D25Population dynamics (general)
34C60Qualitative investigation and simulation of models (ODE)