Abstract
In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.
Highlights
Where Ω is the outward unit normal vector of the boundary zΩ in RN(N > 1). u and v are the densities of prey and predator, respectively. a and d are the intrinsic growth rate of prey u and predator v, respectively. c and e are capturing rate to predator and conversion rate of prey captured by a predator, respectively. m denotes the saturation coefficient, and kv is the inhibition of functional response function on behalf of predators. e parameters a, c, m, k are assumed to be only positive constants, while d can be negative
Two local bifurcation solution branches depending on the bifurcation parameter d > 0 are obtained by local bifurcation theory as follows
Ua is a positive solution of Equation (2), which demonstrates that − Δ − a + 2b(x)ua is a positive operator, and φ1 (− Δ − a + 2b(x)ua)− 1(− euaφ1/mua + 1) > 0. e range can be represented as R(F(w,v)
Summary
We prove that (d, w, v) (d, 0, 0) is a local bifurcation point of system (1), where d λΩ1 (− eua/mua + 1). Ua is a positive solution of Equation (2), which demonstrates that − Δ − a + 2b(x)ua is a positive operator, and φ1 (− Δ − a + 2b(x)ua)− 1(− euaφ1/mua + 1) > 0. Is where d is ad(etde,r0m, θinde)dloucnailqbuiefulyrcbaytioan pλoΩ1in(tcθodf/ksyθsdte+m1)(.1), Set χ v − θd, define the operator. Let A1 mu + k(χ + θd)+ 1, B1 u(mu + 1), C1 (χ + directly calculating the Frechet derivative of the operator, we θd)[k(χ + θd) + 1], D1 k(χ + θd)(2mu + 1) + mu + 1.
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