几类捕食者－食饵趋化模型的动力学性质研究

The mathematical models of chemotaxis have been extensively studied. Chemotaxis models are mathematical models arising from chemistry, medicine and biology, etc. They have profound applied background. The phenomenon of chemotaxis or similar behavior in nature have been widely observed and they are closely related to human life, for example, the growth of cancer cells, the regeneration of blood vessels, and Alzheimer's disease, etc. Hence, The studies of chemotaxis models can promote the cross integration between mathematics, chemistry and biology, and play a positive role in the development of these disciplines. In this project, we focus on the dynamical properties of several kinds of predator-prey models with taxis such as prey-taxis and predator-taxis. The global existence and boundedness of the classical solutions of three models will be proved by energy method or the structure method of auxiliary function, etc. And the stability of constant steady state solutions for these three models will be obtained by Lyapunov functional or linearized stability, etc. For the predator-prey model with predator-taxis and cross-diffusion, we will also show Turing instability, the existence of non-constant positive steady state solutions and local stability of the bifurcation branches.

趋化模型的研究受到广泛关注。趋化模型是化学、医学和生物学等领域中一些问题的数学模型，它有着丰富的应用背景。自然界中趋化现象普遍存在，且与人类生活息息相关，比如癌细胞的生长、阿尔茨海默症、血管的再生等等。因此，研究趋化模型能够促进数学与化学和生物学等学科之间的交叉融合，对这些学科的发展起着积极的作用。本项目以生物趋化现象为背景，主要研究几类捕食者－食饵趋化模型的动力学性质，将用能量估计及构造辅助函数等方法研究三类模型解的全局存在性及一致有界性；并通过构造合适的Lyapunov泛函以及线性稳定性分析等方法获得这三类捕食者－食饵趋化模型常值平衡解的稳定性；在此基础上深化对带有捕食者趋化项和交叉扩散项的捕食者－食饵系统的研究，以期得到系统在某些条件下的Turing不稳定性、系统非常值正稳态解的存在性以及系统分歧解分支的局部稳定性。

本项目研究了两类带有趋化项的捕食者－食饵模型和一类带间接信号的趋化模型，给出了这三类模型解的全局存在性及有界性。同时，对于带趋化项和营养级的捕食者－食饵模型，证明了该模型稳态解的全局渐近稳定性。主要的工作如下：.(1)对于带有趋化项和营养级的捕食者－食饵模型，用能量估计和半群理论的方法证明对任意空间维数n和任意的趋化敏感系数χ和ξ，捕食者－食饵模型的解是全局存在且一致有界的，再通过构造合适的Lyapunov函数，建立在某些条件下模型稳态解的全局渐近稳定性定理。.(2)对于带有增长项项和间接信号项的趋化模型，在空间维数n和参数满足一定条件的情况下，从基于半群理论的L^p估计法获得∇v的L^p有界出发，借助能量估计得到∫_Ω u^p ,∫_Ω |∇v|^2q ,∫_Ω w^(p+1) 的不等式，进而证明‖w‖_(L^(p+1) )的有界性, 再由半群理论获得∇v是L^∞有界的，从而通过Moser迭代法有了模型解的全局存在性和一致有界性。.(3)对于即带有间接捕食者趋化项又带有间接食饵趋化项的一般的捕食者－食饵模型，用能量估计的方法证明了在一些假设条件下，捕食者－食饵模型的解是全局存在且一致有界的 。.本项目完成了预期的研究目标，通过对本项目的实施增加了对趋化模型的进一步了解，项目负责人以第一作者身份发表3篇高水平学术论文，均以标注“国家自然科学基金资助项目”及项目编号。