Abstract
We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.
Highlights
The dynamics of a predator-prey model in a homogeneous environment can be described by the following reactiondiffusion equations: ∂u (x, t) ∂t = uF (u) − G (u) V + D1Δu, (1) ∂V (x, t) ∂tVQ (u, V) + D2ΔV, where u(x, t) and V(x, t) are the densities of prey and predator at time t and position x ∈ Ω ⊂ Rm, respectively
In this paper, we always assume that m < b; that is, we only focus on the case of weak Allee effect
We propose and analyze the dynamics of a reaction-diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey
Summary
The dynamics of a predator-prey model in a homogeneous environment can be described by the following reactiondiffusion equations:. It is easy to verify that model (12) has the following nonnegative constant steady-state solutions:. We have the following lemma regarding the existence of the positive constant steady-state solution of model (12). 2k1 + 1 − c − b < 0, 0 < m∗ < m < b, model (12) has a unique positive constant steady-state solution E3 = (u∗, V∗). I=1 where Xi = ⨁dj=im E(μi)Xij. Let E = (u, V) be any arbitrary constant steady-state solution of model (12). The local stability of the constant steady-state solution can be analyzed as follows. That + k2 the ) of positive constant steady-state solution E3 = model (12) is uniformly asymptotically stable
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