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Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting. (English) Zbl 1277.34059

Summary: The bifurcation analysis of a predator-prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov-Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
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