食饵趋向性对捕食-食饵模型动力学行为的影响

Prey-taxis represents the movement of predators towards the area with higher density of prey population, such response mechanism can be described by some reaction-diffusion-advection PDE models. Since such equations are nonlinear coupled systems, they are usually very challenging to analyze, but they often yield quite interesting mathematical phenomena. This project is aim to analyze the qualitative behavior of the global solution of parabolic system and the steady-state solution of the corresponding elliptic system, we hope to reveal the effects of the prey-taxis on the dynamic behavior of such models, and to explore the spatiotemporal evolution of the interaction and the mutual restriction among biological populations. In particular, we mainly study the following issues: (1) Hopf bifurcation of non-constant equilibrium solutions and asymptotic behavior of non-constant equilibrium solutions with large parameters for Gause-type predator-prey model; (2) the global-in-time existence of global solutions and global asymptotic stability of positive constant solution for Leslie-type predator-prey model; (3) steady-state bifurcation and Hopf bifurcation of non-constant equilibrium solutions and asymptotic behavior of non-constant equilibrium solutions with large parameters for Leslie-type predator-prey model. The research results not only can simulate the dynamic characteristics of the species itself, but also can more reasonably describe the ecological phenomenon of directional motion.

食饵趋向性表示捕食者会朝着食饵密度高的区域定向运动，这样的反应机制可以用反应-扩散-对流模型的偏微分方程来描述。由于这类方程是非线性的耦合型方程组，使得理论分析十分具有挑战性，但也往往会产生一些有趣的数学现象。本项目拟通过分析抛物系统整体解和椭圆系统稳态解的定性行为，揭示食饵趋向性对这类模型动力学行为的影响，探索生物种群间相互作用和相互制约的时空演化规律。特别地，我们将主要研究并解决以下科学问题：(1)Gause型捕食-食饵模型非常数稳态解的Hopf分歧及大参数下的渐近行为；(2)Leslie型捕食-食饵模型全局时间整体解的存在性及正常数解的全局渐近稳定性；(3)Leslie型捕食-食饵模型稳态解的稳态分歧、Hopf分歧及大参数下的渐近行为。以期所得研究成果不仅能模拟物种本身的动力学特征，还能更加合理地描述定向运动这样的生态现象。

数学模型方法越来越引起微生物生态学和生物数学领域研究者的重视。反应扩散模型既有空间扩散非线性性，又有复杂的相互作用关系，这导致该类模型的理论分析会遇到极大地困难和挑战，也引起了众多国内外数学家和数学工作者的极大关注和兴趣。综合利用分歧理论、拓扑度理论、特征值理论、算子摄动理论，并结合各种极限系统性质及问题自身特点，研究了几类具有食饵趋向或交叉扩散的捕食-食饵模型，建立了模型共存解的存在性和随参数变化时共存解的渐近行为，分析了食饵趋向或交叉扩散对模型稳态解与长时间动力学行为的影响，探索生物种群间相互作用和相互制约的时空演化规律，揭示具有食饵趋向或交叉扩散捕食-食饵模型呈现的共存解存在性、多稳态等现象。