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的非线性演化系统的全局动力学 受种群生物学和流行病学的启发，我们将研究存在， 唯一性，分歧，全局稳定性的稳定状态和周期解，持久性和灭绝， 全局收敛性和吸引子，行波解的存在性、唯一性和稳定性，传播速度。 在这个程序中，我们还提出了获得一般结果的传播速度和行波的抽象 具有弱紧性和基本再生比的非自治种群模型的发展系统。 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，数值 模拟研究的新奇在于动力系统方法的发展方程， 时空异质性。 该研究有望建立时空动力学和生物学的数学理论 入侵，并发展非自治动力系统的理论。这项研究的进展也有助于 为了回答一些重要的问题，比如一个相互作用的群体是否以及如何能够持续下去（避免 灭绝）或传染病是否在人群中流行，以估计生物灭绝的速度。 入侵和流行病传播，并了解生物现象，如时空模式和 波的传播

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.