具有Holling-Tanner项反应扩散模型正平衡解的存在性

Coexistence of Positive Steady-state Solutions for a Class of Reaction-diffusion Model with Holling-Tanner Term

本文首先运用极值原理以及Hopf边界引理讨论了一类具有Holling-Tanner项的反应扩散模型的平衡态系统在第三边值条件下的正解的先验估计。然后运用拓扑不动点理论,将Banach空间的Leray-Schauder度推广到正锥上,分析了系统正解的存在性。随后利用锥上紧算子的不动点指数计算方法,结合线性算子的谱性质,极值原理和上下解方法,得到了平衡态系统的正解存在的充要条件。

This paper is concerned with the coexistence of steady states for a kind of reaction-diffusion model with Holling-Tanner term under the third boundary conditions. By means of maximum principles and Hopf boundary lemma, the prior estimates of the strict positive solutions are given at first. Then by using the topological fixed point theorems and extending Leray-Schauder degree on Banch spaces to positive cones, the existence of positive solutions of the system are considered. At last, by calculating the indices of fixed points of compact operators in cones and combining with spectrum analysis of operators, maximum principles and lower-upper solutions methods, the necessary and sufficient conditions for the positive solutions of the steady system are obtained.