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.

中文翻译：

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修正Leslie-Gower格式的三种反应扩散模型的定性分析

获得了在诺伊曼边界条件下具有改进的莱斯利-高尔（Leslie-Gower）方案的三种种群反应扩散模型的定性分析。分别讨论了ODE系统和PDE系统常数解的存在性和稳定性。然后，通过最大原理和Harnack不等式给出正稳态的先验估计。此外，通过使用庞加莱不等式导出了非恒定正稳态的不存在。最后，基于Leray-Schauder度理论建立了非恒定正稳态的存在。