Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model

This paper deals with the effect of the protection zone Ω0 on coexistence of the species for a ratio-dependent predator-prey model. We obtain a critical value λ1N(bδ(x),Ω) which is less than the well-known critical value λ1D(Ω0) obtained in the previous literatures. Furthermore, we show that if λ>λ1N(bδ(x),Ω), then the prey persists regardless of the growth rates of the predator; while if λ≤λ1N(bδ(x),Ω), then there exists a real number μ⁎, such that the prey is ultimately extinct when μ>μ⁎. As for λ<λ1N(bδ(x),Ω) and μ<μ⁎, the curve of the positive steady state solutions of the model emanating from (λ,0;−c) ends at a singular point (0,0;μ2). Meantime, by using the Lyapunov-Schmidt reduction method, we obtain a fine profile of its bifurcation diagrams, and the uniqueness or multiplicity of its positive steady state solutions. In addition, as generally expected, the chances of survival of the prey will increase with the size of the protection zone.