Abstract
In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establisha prioriestimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.
Highlights
IntroductionWe investigate the existence and non-existence of the non-constant positive stationary solutions for the following general Gause-type predator-prey system with constant self- and cross-diffusions
In this paper, we investigate the existence and non-existence of the non-constant positive stationary solutions for the following general Gause-type predator-prey system with constant self- and cross-diffusions ut − d1∆[(1 + d3v)u] = ug(u) − p(u)v, vt − d2∆[(1 + d4u)v] = v(−d + cp(u)), u(x, 0) = u0(x) ≥ 0, v(x, 0) = v0(x) ≥ 0, ∂ν u = ∂ν v = 0,(x, t) ∈ Ω × (0, ∞), (x, t) ∈ Ω × (0, ∞), x ∈ Ω, (x, t) ∈ ∂Ω × (0, ∞), (1.1)where Ω ⊂ RN (N ≥ 1 be an integer) is a bounded domain with smooth boundary ∂Ω, ν is the outward unit normal vector on the boundary ∂Ω with ∂ν = ∂ ∂ν, and the homogeneous
This paper investigates the existence and non-existence of non-constant positive solutions for a generalized Gause-type predator-prey system with self- and cross-diffusions under homogeneous Neumann boundary condition, in which the cross-diffusions are included in such away that the prey runs away from the predator and the predator moves away from a large group of preys
Summary
We investigate the existence and non-existence of the non-constant positive stationary solutions for the following general Gause-type predator-prey system with constant self- and cross-diffusions. In [9], without taking into account the cross-diffusions, the authors studied the qualitative behavior of non-constant positive solutions on a general Gause-type predator-prey system subject to the homogeneous Neumann boundary condition, in which with diffusion rates are constants. The results show that the non-existence and existence of the non-constant positive steady-state solutions are affected by the self- and cross-diffusion rates. As far as we know, there are few works related system (1.1) with the cross-diffusions for both predator and prey Inspired by these works, we investigate the existence of the non-constant positive solutions of system (1.1) by using the well-known Leray-Schauder degree theory. We discuss the Turing instability of system (1.1) mainly by considering the influence of the diffusion terms
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