Dynamics in a ratio-dependent predator–prey model with predator harvesting

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Abstract

The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator–prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov–Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737–753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as “pathological behavior” for a ratio-dependent predator prey model in [P. Yodzis, Predator–prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51–58].

Keywords

Predator–prey system
Predator harvesting
Coexistence
Bifurcations
Limit cycle

Cited by (0)

1

Research supported by the NNSF of China (No. 10231020) and Program for New Century Excellent Talents in University.

2

Research supported by the NNSF of China (No. 10371043).

3

Research supported by the NNSF of China (No. 10371072).