Protection zone in a diffusive predator–prey model with Beddington–DeAngelis functional response

Abstract

In any reaction–diffusion system of predator–prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To save the endangered prey species, some human interference is useful, such as creating a protection zone where the prey could cross the boundary freely but the predator is prohibited from entering. This paper studies the existence of positive steady states to a predator–prey model with reaction–diffusion terms, Beddington–DeAngelis type functional response and non-flux boundary conditions. It is shown that there is a threshold value $\theta _0$ which characterizes the refuge ability of prey such that the positivity of prey population can be ensured if either the prey’s birth rate satisfies $\theta \ge \theta _0$ (no matter how large the predator’s growth rate is) or the predator’s growth rate satisfies $\mu \le 0$, while a protection zone $\Omega _0$ is necessary for such positive solutions if $\theta <\theta _0$ with $\mu >0$ properly large. The more interesting finding is that there is another threshold value $\theta ^*=\theta ^*(\mu ,\Omega _0)<\theta _0$, such that the positive solutions do exist for all $\theta \in (\theta ^*,\theta _0)$. Letting $\mu \rightarrow \infty$, we get the third threshold value $\theta _1=\theta _1(\Omega _0)$ such that if $\theta >\theta _1(\Omega _0)$, prey species could survive no matter how large the predator’s growth rate is. In addition, we get the fourth threshold value $\theta _*$ for negative $\mu$ such that the system admits positive steady states if and only if $\theta >\theta _*$. All these results match well with the mechanistic derivation for the B-D type functional response recently given by Geritz and Gyllenberg (J Theoret Biol 314:106–108, 2012). Finally, we obtain the uniqueness of positive steady states for $\mu$ properly large, as well as the asymptotic behavior of the unique positive steady state as $\mu \rightarrow \infty$.