Nonlinear diffusion effect on bifurcation structures for a predator-prey model

Abstract

We study a nonlinear diffusive predator-prey model with modified Leslie-Gower and Holling-type II functional responses. Making use of global bifurcation theory, we obtain two sufficient conditions for the existence of positive solutions and describe the coexistence region $R$. Moreover, we find that the coexistence region $R$ spreads as $\beta$ increases and narrows for large $\alpha$. At last, we derive the nonlinear effect of large $\beta$ on bifurcation structures in the special case of $\alpha=0$. Some a priori estimates for positive solutions will play an important role in the proof.