With Cross-diffusion Predator Model The Existence And Stability Of Bifurcation Solutions,

This paper is concerned with a prey-predator system with cross-diffusion of fractional type:WhereΩis a bounded domain in Rⁿ with smooth boundary (?)Ω; b, c,λ,μ,βare positive constants; u and v, respectively,represent the population densities of prey and predator species; In the reaction terms,λandμare birth rates of prey and predator, respectively. Furthermore, b and c denote the prey-predator interactions. We rewrite the second function of (1): v_t =(?)?(?), the negative coefficient of (?)_u describes an tendency such that the spatial diffusion due to the population pressure of the predator is weakened in a high density area of the prey.First, we study the existence of positive nontrivial steady-states for the following system:then (2) is reduced to the semilinear problem: We know: ifλ>λ₁, (3) has a semitrivial solution (θ_λ,0). By applying the abstract local bifurcation theorem of Crandall and Rabinowitz [4], we can obtain the existence of positive solution of (3) which bifurcates from a semitrivial solution(θ_λ,0).In order to analyse the structure of the positive solution, we make use of the formula (I.6.3) in reference [14] to calculate(?)(0) > 0. Thus, we obtain the relationship between positive solution bifurcating from semitrivial solution and cross-diffusion coefficientβ, which is that ifβ>β₁, positive solution of (3) bifurcates from the semitrivial solution curve {(θ_λ,0,β)}.The main results about the existence of positive steady-state:Theorem 1: For each fixedλ>λ₁, ifλsatisfies:then there existsβ₁∈(0, +∞), ifβ₁ <β<β₁ +ε₀, positive solutions of (3) bifurcate from a semitrivial solution (θ_λ,0,β₁). Precisely speaking, the positive solutions of (3) near (θ_λ,0,β₁) can be expressed aswhere w₂ > 0,x∈Ω;(?)(0) > 0;ε₀,δare small enough positive constants. Here,' denotes (?).Second, we study the stability of positive nontrivial steady-state for the system (1) with cross-diffusion of fractional type.We renter to§1.7.The Principle of Exchange of Stability in [14]. And the most important, by making use of the formula (I.7.40) in [14], we verify the condition of the principle of the linearized stability. Thus, we can obtain the locally asymp-totical stability of (u(s), V(s)) which bifurcates from semitrivial solution (θ_λ, 0).The main results about the stability of positive steady-state:Theorem 2: Ifβ(s) >β₁(0 < s <δ), the positive steady-state of (1) bifurcating from semitrivial solution (θ_λ,0) is locally asymptotically stable.