具有Crowley-Martin反应函数和Michaelis-Menten型收获项的捕食-食饵扩散模型正解的存在性和唯一性

Existence and uniqueness of positive solutions for the predator-prey diffusion model with Crowley-Martin response function and Michaelis-Menten harvesting

目的在齐次Dirichlet边界条件下研究一类具有Crowley-Martin反应函数和Michaelis-Menten型收获项的捕食-食饵扩散模型的动力学行为。方法运用不动点指数理论得到了正解的存在条件,并通过线性算子的扰动理论、度理论和稳定性理论考察捕食者对食饵的捕获率充分小时正解的唯一性和稳定性。结果给出了模型正解存在且唯一稳定的充分条件。结论当食饵的增长率较大、人类对捕食者的收获率较小且捕食者对食饵的捕获率充分小时,两物种不但共存而且系统存在唯一稳定的正解。

Purposes—To study the dynamic behaviors of a predator-prey model with Crowley-Martin response function and Michaelis-Menten harvesting under homogeneous Dirichlet boundary conditions.Methods—The conditions for the existence of positive solutions are obtained by the fixed point index theory.Then,by virtue of the combination of the linear operator perturbation theory,degree theory and stability theory,the uniqueness and stability of positive solution are investigated when the predator's capture rate is sufficiently small.Result—The sufficient conditions for the existence,uniqueness and stability of positive solutions are given.Conclusion—When the growth rate of the prey is large,the harvesting rate of human for predator is small and the predator's capture rate is sufficiently small,the predator and prey can coexist,and the model has a unique stable positive solution.