具扩散和时滞的Holling-Tanner 捕食食饵系统动力学分析

Law of food diversity population growth can be described by the diffusive Holling-Tanner predator-prey system, which is better than the traditional prey-predator system. To study the dynamical behaviors of the diffusive Holling-Tanner system is the one of the key topic in differential equations, which has great value in theory and practical significance. However, many related results come from the method of numerical simulations and numerical analysis and the deeper theoretical study is relatively rare. The thesis is mainly focus on the diffusive Holling-Tanner predator-prey system with delay or the ratio-dependent functional response. By employing the bifurcation theory of infinite dimensional dynamical systems, the upper-lower solutions method and the comparison principle and maximum principle of partial differential equations and numerical simulations, we will study the following research contents: 1) Turing instability induced by diffusion and Hopf bifurcations and steady state bifurcation by delays and the existence of the global steady state bifurcation; 2) dissipativeness and uniform persistence of the system and global attractivity of the positive coexistence. The aim of the study is to investigate how diffusion and delay can affect the qualitative and topological structure and to provide the reasonable mathematical explanation for the certain biological phenomena.

具扩散项的Holling-Tanner捕食食饵系统在刻画食物多样性种群增长规律时，优于传统的捕食被捕食系统，对其动力学行为的研究是微分方程热门问题之一，具极强的实际背景和理论价值。然而大多数工作仍建立在数值模拟和数值分析基础上，深入的理论研究还较少。本课题主要针对具扩散的时滞Holling-Tanner捕食食饵系统和基于比率依赖的Holling-Tanner模型，利用无穷维动力系统分支理论、偏微分方程上下解方法、极值原理、比较原理、数值计算等工具，研究以下内容：1）扩散导致的Turing不稳定性，时滞导致的Hopf分支及稳态分支以及稳态分支的大范围存在性；2）系统的耗散性、一致持久性及正共存态的全局吸引性。本项目旨在揭示“扩散”和“时滞”是如何影响系统的定性结构和拓扑结构，并为某些生物现象提供合理的数学解释。

具扩散项的Holling-Tanner 捕食食饵系统在刻画食物多样性的种群增长规律时，优于传统的捕食食饵系统。本课题主要针对具扩散项的比率依赖的Holling-Tanner 捕食食饵系统和具非局部时滞的反应扩散方程，利用稳定性理论、Hopf分支和稳态分支定理，研究平衡点的稳定性，空间齐次、非齐次周期解的存在性。具体分为三部分：1. 运用特征值分析、上下解方法得到比率依赖的Holling-Tanner系统正常值平衡点的局部稳定性和全局稳定性；运用Hopf分支定理得到空间齐次和非齐次周期解的存在性，并在某些临界值扩散导致Turing不稳定性的发生。2. 针对具分布时滞的反应扩散方程，以平均时滞为分支参数，研究扩散和时滞对系统的动力学行为的影响。结果表明，平衡点的绝对和条件稳定性区域由线性系统的系数来决定。平均时滞在强核条件下能导致稳定开关，而在弱核条件下则不能。最后把理论结果分别应用到Logistic模型、果蝇模型和具Allee效应的种群模型，数值模拟验证理论结果。3. 针对具非局部时滞的反应扩散方程，运用特征值分析和复平面理论，研究了在各类加权函数下平衡点的稳定性和Hopf分支的存在性，并通过中心流形定理、规范型理论给出确定分支周期解的方向、稳定性和周期的方法，最后把理论结果应用到food-limited 种群模型，数值模拟验证了连接不稳定的空间非齐次周期解和稳定的空间齐次周期解的轨道的存在性。