具食饵趋向性和加法Allee效应的捕食系统解的性质

Properties of Solutions for Predator-prey Model with Prey-taxis and Additive Allee Effect

为了研究加法Allee效应对捕食-食饵模型的影响,且考虑到捕食者总会朝着食饵聚集的地方迁移,本文建立了一类在齐次Neumann边界条件下具有加法Allee效应和食饵趋向性的捕食模型。首先,应用最大值原理,得到了模型正解的先验估计。其次,给出平衡点的存在性,并利用线性化算子的特征值理论得到了平衡点的稳定性。最后,以趋化系数为分歧参数,应用Crandall-Rabinowitz局部分歧理论研究了发自正常数解的局部分歧解,并进一步证明在该平衡点处发生跨临界分歧,得到了系统非常数正解的存在性。研究结果表明:在弱Allee效应下,趋化系数对共存解的稳定性有重要的影响。

In order to study the effect of additive Allee effect on predator-prey model,and considering that the predator would always migrate to the place where the prey gathered,a predator-prey model with additive Allee effect and prey-taxis under homogeneous Neumann boundary conditions was established.Firstly,the comparison principle of elliptic equations was applied to obtain a priori estimates for the positive solution of the model.Secondly,the existence of the equilibria was given,and the stability of the equilibria was obtained by using the eigenvalue theory of linearization operators.Finally,regarding the chemotaxis coefficient as the bifurcation parameter,the local bifurcation solutions emanating from the normal constant solution were studied by using the Crandall-Rabinowitz local bifurcation theory.It was further proved that a transcritical bifurcation occured at this equilibrium point,thereby established the existence of non-constant positive solutions for the system.The results showed that the chemotaxis coefficient had an important influence on the stability of coexistence solutions under the weak Allee effect.