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 [Formula: see text] 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 [Formula: see text] (no matter how large the predator's growth rate is) or the predator's growth rate satisfies [Formula: see text], while a protection zone [Formula: see text] is necessary for such positive solutions if [Formula: see text] with [Formula: see text] properly large. The more interesting finding is that there is another threshold value [Formula: see text], such that the positive solutions do exist for all [Formula: see text]. Letting [Formula: see text], we get the third threshold value [Formula: see text] such that if [Formula: see text], prey species could survive no matter how large the predator's growth rate is. In addition, we get the fourth threshold value [Formula: see text] for negative [Formula: see text] such that the system admits positive steady states if and only if [Formula: see text]. 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 [Formula: see text] properly large, as well as the asymptotic behavior of the unique positive steady state as [Formula: see text].

The role of protection zone on species spreading governed by a reaction–diffusion model with strong Allee effect

“…It is recognized that protection zones provide many economic, social, environmental, and cultural values. The role of protection zone in preventing population from extinction has been investigated in [11,12,14,17,20,21,22,23,32,33,34,40,50,53] and the references therein for reaction-diffusion models; we note that bounded habitats are assumed in those works.…”

Effects of dispersal for a predator-prey model in a heterogeneous environment

“…Hence, the only possibility is that U ∞ > 0 in Ω and v ∞ > 0 in Ω 1 . This shows that for any fixed λ > 0, (7) with µ = 0 has a positive solution.…”

“…However the dynamical behavior is changed once the protection zone is beyond the critical size: the two species can coexist even if µ is large, moreover, for sufficiently large µ, the two species stabilize at a unique positive stationary solution. Besides, the effect of a protection zone has been studied for predator-prey model with strong Allee effect [2], Leslie predator-prey model [6], ratio-dependent predator-prey model [23], Beddington-DeAngelis predator-prey model [7,21], and Lotka-Volterra competition model [5]. Finally, we point out that a related but different situation for the Holling type II predator-prey model, called a degeneracy, is studied in [4,[12][13][14] and references therein.…”

“…For any λ, µ > 0, (7) has two semi-trivial solutions: (λ, 0) and (0, µ). Thus, (7) exists two curves of these solutions in the space of (λ, U, v):…”

“…Proposition 2. For each given µ > 0, (λ * , 0, µ) is a bifurcation point of (7) where positive solutions bifurcate from Γ v , moreover, positive solutions of (7) near (λ * , 0, µ) ∈ R × X 1 are expressed by…”