一类时滞反应扩散种群动力系统的动力学研究

In the real-life world, the prey-predator relation is an important population relation and it has important theoretical and real meanings for the protection of environment and the sustainable development of resource to study the dynamics embodied by the prey-predator relation amongst species. In recent years, all kinds of prey-predator systems of Lotka-Volterra types with different functional response functions have been given extensive attention and deep investigation by many researchers and lots of interesting results have been also obtained. Although the stability and bifurcation of the modified Leslie-Gower prey-predator systems with Michaelis-Menten type prey harvesting, which sever as an important prey-predator system, have been considered by some authors, the obtained conclusions when studying the dynamics of these systems are all relative rough since the complexity of the models themselves and hence they cannot reflect the complex dynamical behaviors of the systems under consideration. Based on these facts, according to the research experience and the research results in recent years obtained by this project group, we shall investigate the stability and bifurcation of the modified Leslie-Gower prey-predator delayed reaction-diffusion systems with Michaelis-Menten type prey harvesting in order to further grasp the complex dynamical behaviors of this kind of systems.

食饵-捕食关系是现实世界中一种重要的种群关系, 研究种群间食饵-捕食关系所体现的动力学对环境保护和资源的可持续发展都有重要的理论和现实意义。近年来, 各种具有不同功能反应函数的Lotka-Volterra型食饵-捕食系统已经被众多学者给予了广泛的关注和深入的研究, 并且得到了许多有趣的结果。具有Michaelis-Menten型食饵收获的修正Leslie-Gower食饵-捕食系统作为一种重要的食饵-捕食系统, 虽然有学者考虑了它的稳定性和分支, 但由于模型本身的复杂性使得在研究这些系统的动力学行为时所得到的结论比较粗糙, 不能真正体现模型的复杂动力学行为。鉴于此, 本项目组拟根据多年的研究经验和所取得的研究成果研究具有Michaelis-Menten型食饵收获的修正Leslie-Gower食饵-捕食时滞反应扩散系统的稳定性和分支, 以进一步探索此类系统的复杂动力学行为。

在本项目的实施过程中，项目组主要利用动力系统的定性分析理论、泛函微分方程理论与非线性泛函分析等现代数学工具，分析了时滞常微分系统、齐次Neumann边界条件或齐次Dirichlet边界条件下时滞反应扩散系统的动力学行为。. 对于具有离散时滞的化学反应Lengyel-Epstein系统、具有延迟反馈的谐振子模型和齐次Dirichlet边界条件下具有一个瞬间反馈控制项和两个时滞反馈控制项的广义Logistic种群模型，发现时滞参数的变化会引起相关平衡点的多次稳定性切换现象。对于齐次Neumann边界条件下具有集群行为的食饵-捕食者鱼群模型，发现两鱼群的自扩散仍然能够导致正常数稳态解的Turing不稳定性、Turing分支与Turing斑图。探索了齐次Neumann边界条件下空间扩散具有延迟的单种群模型正常数稳态解的动力学行为。获得了齐次Neumann边界条件下具有Michaelis-Menten型食饵收获的修正Leslie-Gower时滞反应扩散食饵-捕食者系统正常数稳态解全局渐近稳定的条件。. 对于所考虑时滞常微分系统、齐次Neumann边界条件或齐次Dirichlet边界条件下时滞反应扩散系统中所出现的Hopf分支，借助于偏泛函微分方程的规范型理论和中心流形定理计算出了相应Hopf分支的规范型，进而获得了判断Hopf分支方向和分支周期解稳定性的显式公式。借助于Matlab软件包和微分方程的数值解法对于所获得的理论结论给予了适当的数值验证。