Spatial and temporal dynamics of nonlinear systems with applications

该研究计划旨在研究具有空间和时间异质性的非线性演化系统的全局动力学，其灵感来自于种群生物学和流行病学。我们将研究稳定状态和周期解的存在性、唯一性、分支和全局稳定性，持久性和灭绝，全局收敛性和吸引子，行波的存在性、唯一性和稳定性，以及传播速度。* 在本计划中，我们还建议获得具有弱紧性和非自治种群模型的基本再生比的抽象 * 进化系统的传播速度和行波的一般结果。研究方法涉及动力系统、Liapunov函数、上下解、扰动和分叉、比较原理、斜积半流、线性化、非线性分析和数值模拟。这项研究的新奇在于用动力系统方法来处理具有 * 空间和时间异质性的演化方程。该研究有望建立时空动力学和生物入侵的数学理论，并发展非自治动力系统理论。这项研究的进展也有助于回答一些重要的问题，如一个相互作用的群体是否以及如何能够持续存在（避免灭绝），或者一种传染病是否在一个群体中成为地方病，估计生物入侵和流行病传播的速度，以及理解时空模式和波传播等生物现象。

This research program is aimed at the global dynamics of nonlinear evolution systems with spatial and temporal *heterogeneities, which are inspired by population biology and epidemiology. We will study the existence, *uniqueness, bifurcations, and global stability of steady states and periodic solutions, persistence and extinction, *global convergence and attractors, the existence, uniqueness and stability of traveling waves, and spreading speeds. *In this program, we also propose to obtain general results on spreading speeds and traveling waves for abstract *evolution systems with weak compactness and basic reproduction ratios for non-autonomous population models.**The investigation methods involve dynamical systems, Liapunov functions, upper-lower solutions, perturbations *and bifurcations, comparison principle, skew-product semiflows, linearization, nonlinear analysis, and numerical *simulations. The novelty of the research lies in the dynamical systems approach to evolution equations with *spatial and temporal heterogeneity.**The proposed research is expected to establish the mathematical theory of spatial-temporal dynamics and biological *invasions, and to develop the theory of non-autonomous dynamical systems. Progress in this research can also help *to answer important questions such as whether and how a community of interacting populations can persist (avoid *extinction) or whether an infectious disease becomes endemic in a population, to estimate the speeds of biological *invasion and epidemic spreading, and to understand biological phenomena such as spatial-temporal patterns and *wave propagations.