# American Institute of Mathematical Sciences

November  2013, 18(9): 2283-2313. doi: 10.3934/dcdsb.2013.18.2283

## Dynamics of a ratio-dependent predator-prey system with a strong Allee effect

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China 2 Department of Mathematics, University of Louisville, Louisville, KY 40292

Received  September 2012 Revised  May 2013 Published  September 2013

A ratio-dependent predator-prey model with a strong Allee effect in prey is studied. We show that the model has a Bogdanov-Takens bifurcation that is associated with a catastrophic crash of the predator population. Our analysis indicates that an unstable limit cycle bifurcates from a Hopf bifurcation, and it disappears due to a homoclinic bifurcation which can lead to different patterns of global population dynamics in the model. We study the heteroclinic orbits and determine all possible phase portraits when the Bogdanov-Takens bifurcation occurs. We also provide the conditions for nonexistence of limit cycle under which the global dynamics of the model can be determined.
Citation: Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283
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