Effect of the protection zone in a diffusive ratio-dependent predator–prey model with fear and Allee effect

Abstract

Abstract In this paper, we study the influence of a protection zone for the prey on a diffusive predator–prey model with fear factor and Allee effect. The prior estimate, global existence, nonexistence of nonconstant positive solutions and bifurcation from semitrivial solutions are well discussed. We show the existence of a critical patch value λ 1 D ( Ω 0 ) $\lambda ^{D}_{1}(\Omega _{0})$ of the protection zone, described by the principal eigenvalue of the Laplacian operator over Ω 0 $\Omega _{0}$ with Neumann boundary conditions. When the mortality rate of the predator μ ≥ d 2 λ 1 D ( Ω 0 ) $\mu \geq d_{2}\lambda ^{D}_{1}(\Omega _{0})$ , we show that the semitrivial solutions ( 1 , 0 ) $(1,0)$ and ( θ , 0 ) $(\theta,0)$ are unstable and there is no bifurcation occurring along respective semitrivial branches.

Keywords

• Protection zone
• Fear factor
• Allee effect
• Bifurcation
• Ratio-dependent predator–prey model