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Global dynamics of a predator-prey model with Hassell-Varley type functional response. (English) Zbl 1160.34046

The authors present a systematic global qualitative analysis to a general predator-prey model with Hassell-Varley type functional response. The authors show that the predator free equilibrium is a global attractor only when the predator death rate is greater than its growth ability. The positive equilibrium exists if the above relation reverses. In cases of practical interest, authors show that the local stability of the positive steady state implies its global stability with respect to positive solutions. For terrestrial predators that form a fixed number of tight groups, authors show that the existence of an unstable positive equilibrium in the predator-prey model implies the existence of an unique nontrivial positive limit cycle.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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