具时滞的Crowley-Martin型捕食者-食饵扩散系统的动力学性质分析

The Crowley-Martin functional response in the predator-prey system characterizes a phenomenon of interference among predators to search and process preys, and predicts that the interference plays an important influence in the feeding rate. Diffusive systems with time delay describe the law of development of the objective things with temporal variation and spatial distribution and have significant applications in many fields. However, to the best of our knowledge, there exist few results to consider the dynamical properties of the diffusive predator-prey system with the Crowley-Martin functional response and time delay. In this project, we systematically explore the dynamical properties of the diffusive predator-prey system with the Crowley-Martin functional response and time delay. We establish the existence and stability of the steady states and the existence and bifurcation properties of Hopf-Turing bifurcation with Neumann boundary condition. In particular, we analyze the dynamical properties of the system with the two-dimensional bounded region, and would reveal the effect of the changes of the space dimension on the dynamical behavior of the system. Therefore, the research project can provide a theoretical basis for the steady development of the ecosystem.

捕食者-食饵系统中的Crowley-Martin型功能性反应刻画了捕食者之间竞争猎取食饵时产生相互干扰的现象，并且在食饵丰富的情况下这种干扰会对捕食率产生重要的影响。具有时滞的扩散系统描述了客观事物随着时间变化和空间分布的发展规律，广泛应用于实际问题的研究之中。然而目前研究具时滞的Crowley-Martin型捕食者-食饵扩散系统的动力学性质的结果并不多见，有许多问题尚待解决。本项目拟研究具时滞的Crowley-Martin型捕食者-食饵扩散系统的动力学性质。建立在Neumann边界条件下系统稳态解的存在性与稳定性、Hopf分支与Turing分支的存在性及分支性质等。特别着重分析在2维有界区域上系统的动力学性质，从而有助于揭示空间维数的变化对系统动力学行为的影响。因而本项目的研究能够为种群生态系统的持续稳定发展提供理论依据。

本项目针对一类带有Crowley-Martin型功能性反应函数和时滞的捕食者-食饵扩散系统，研究其动力学性质和分支理论，建立了在Neumann边界条件下系统内部平衡点的存在性与稳定性，获得了系统在此类平衡点附近的Hopf分支与Turing分支的存在性条件及分支性质等，揭示系统的多解性条件，并分析出空间维数的变化对系统的动力学行为的影响，丰富了微分方程的分支理论。