Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. (English) Zbl 1094.34024

Authors’ abstract: The ratio-dependent predator-prey model exhibits rich interesting dynamics due to the singularity of the origin. The objective of this paper is to study the dynamical properties of the ratio-dependent predator-prey model with nonzero constant rate harvesting. For this model, the origin is not an equilibrium. It is shown that numerous kinds of bifurcation occur for the model, such as the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov-Takens bifurcation, the homoclinic bifurcation, and the heteroclinic bifurcation, as the values of parameters of the model vary. Hence, there are different parameter values for which the model has a limit cycle, or a homoclinic loop, or a heteroclinic orbit, or a separatrix connecting a saddle and a saddle-node. These results reveal far richer dynamics compared to the model with no harvesting.


34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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