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 lambda ^{D}_{1}(Omega _{0}) of the protection zone, described by the principal eigenvalue of the Laplacian operator over Omega _{0} with Neumann boundary conditions. When the mortality rate of the predator mu geq d_{2}lambda ^{D}_{1}(Omega _{0}), we show that the semitrivial solutions (1,0) and (theta,0) are unstable and there is no bifurcation occurring along respective semitrivial branches.
Highlights
The predator–prey model is one of the most basic models to study the interspecific relationship, which is still being investigated widely [1, 2]
We mainly show some dynamical behavior of the model (1.2)
The results further illustrate that the asymptotic property of solutions only depends on the strong Allee effect
Summary
The predator–prey model is one of the most basic models to study the interspecific relationship, which is still being investigated widely [1, 2]. In 2014, Cui and Shi [8] studied a diffusive predator–prey system with a strong Allee effect They analyzed the dynamics and steady state solutions of the system. Wang et al [20] first proposed a predator–prey model incorporating the cost of the fear effect into prey reproduction in 2016 Their mathematical results show that high levels of fear can stabilize the predator–prey model by excluding the existence of periodic solutions. When the reserve is established, the predator spends more time looking for prey, predator growth rate is a function of the ratio of prey to predator abundance This case can be modeled as a ratio-dependent function, which is more reasonable in the predator–prey model due to the absence of biological control paradox [18, 24]. A diffusive predator–prey model considering fear, Allee effect, and prey protection zone is as follows:.
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