Volume 19, Number 4, Winter 2011
BIFURCATIONS IN A RATIO-DEPENDENT
Dedicated to Prof. H. I. Freedman on the occasion of his 70th birthday
PREDATOR-PREY MODEL WITH PREY
LILI CHEN, YILONG LI AND DONGMEI XIAO
Abstract. The dynamics and bifurcations of a class of ratio-dependent predator-prey models with prey harvesting are investigated. The existence of all ecological feasible equilibria for this model is determined and the topological classifications of these equilibria are derived. It is proved that the model can undergo Hopf bifurcation and Bogdanov-Takens bifurcation near the corresponding positive equilibrium as some parameters of the model vary, and unstable oscillation can be observed. Some numerical simulations are provided to support our theoretical results. These theoretical conclusions reveal the effect of constant harvesting rate on the coexistence of the two species, which not only provides predication whether the two species will suffer from extinction one after the other but also gets insight into the optimal management of exploitation resources.