Abstract
The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory.
Highlights
Three-species reaction-diffusion models with Holling-type II functional response have been a familiar subject for the analysis
Aziz-Alaoui and Okiye [4] focused on a twodimensional continuous time dynamical system modeling a predator-prey food chain and gave the main result of the boundedness of solutions, the existence of an attracting set, and the global stability of the coexisting interior equilibrium, which was based on a modified version of the Leslie-Gower scheme and Holling-type II scheme
Ko and Ryu [19] showed that the predator-prey model with Leslie-Gower functional response had no nonconstant positive solution in homogeneous environment, but the system with a general functional response might have at least one nonconstant positive steady state under some conditions
Summary
Three-species reaction-diffusion models with Holling-type II functional response have been a familiar subject for the analysis. Ko and Ryu [19] showed that the predator-prey model with Leslie-Gower functional response had no nonconstant positive solution in homogeneous environment, but the system with a general functional response might have at least one nonconstant positive steady state under some conditions. Zhang and Zhao [20] analyzed a diffusive predator-prey model with toxins under the homogeneous Neumann boundary condition, including the existence and nonexistence of nonconstant positive steady states of this model by considering the effect of large diffusivity. Hu and Li [23] were concerned about a strongly coupled diffusive predator-prey system with a modified Leslie-Gower scheme and established the existence of nonconstant positive steady states. Motivated by the mentioned above, we consider a threespecies reaction-diffusion model with a modified LeslieGower and Holling-type II scheme under the homogeneous Neumann boundary condition as follows:. In the last two sections, we have a discussion about the nonexistence and existence of nonconstant positive steady states under different conditions
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