具有Allee效应的捕食-食饵扩散模型定性分析

Qualitative Analysis of a Diffusive Predator-prey Model with Allee Effect

本文讨论一类具有B-D反应函数和Allee效应的捕食-食饵扩散模型正解的存在性、唯一性和多重性.首先运用不动点指数理论得到了正解存在的充分条件.接着利用特征值的变分原理给出了正解的唯一性条件.最后通过分析极限系统的正解,运用不动点指数理论、分歧理论和扰动理论确定了正解的确切重数和稳定性.讨论结果表明:只要Allee效应常量满足适当关系且捕食者的增长率较大,当参数满足适当条件时系统存在唯一正解,当食饵的增长率在一定范围内时系统恰好存在两个正解.

The existence,uniqueness and multiplicity of positive solutions to a diffusive predator-prey model with B-D functional response and Allee effect are discussed.By the fixed point index theory,the sufficient conditions for the existence of positive solutions are obtained.Secondly,the conditions for the uniqueness of positive solutions are given by the variational characterization of the lowest eigenvalue.Finally,based on the analysis of positive solutions to two limiting systems,the exact multiplicity and stability of positive solutions are determined by means of the combination of the fixed point index theory,bifurcation theory and perturbation theory of eigenvalues.When the Allee effect constants meet appropriate relationship and the growth rate of the predator is large,the results show that the system has only a unique positive solution when the parameters satisfy certain conditions,and has exactly two positive solutions when the growth rate of the prey lies in a certain range.