Spatial and temporal dynamics of evolution systems with applications

该研究计划旨在研究具有时空异质性的非线性演化系统的全局动力学，其灵感来自种群生物学和流行病学。我们将研究稳定状态的存在性，唯一性，分支和全局稳定性，概周期解和整体解，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。在这个项目中，我们还建议获得一般性结果的抽象单调系统的行波点火结构，传播动力学在时空异质介质和移动的环境中，周期非局部扩散系统的主要特征值，和非自治人口模型的基本再生比。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、正算子、非线性分析和数值模拟。研究的新奇在于动力系统方法的发展方程的空间和时间的异质性。该研究有望建立时空动力学和生物入侵的数学理论，并发展主特征值和基本繁殖率理论。该项目的进展还有助于回答一些重要问题，例如相互作用的种群是否以及如何持续存在（避免灭绝），或者传染病是否在种群中流行，估计生物入侵和流行病传播的速度，并了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, almost periodic solutions and entire solutions, persistence and extinction, global convergence and attractors, the existence, uniqueness and stability of traveling waves, and spreading speeds. In this project we also propose to obtain general results on traveling waves for abstract monotone systems with ignition structure, propagation dynamics in space-time heterogeneous media and shifting environments, principal eigenvalues for periodic nonlocal dispersal systems, 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, positive operators, nonlinear analysis, and numerical simulations. The novelty of the research lies in the dynamical systems approach to evolution equations with spatial and temporal heterogeneities. The proposed research is expected to establish the mathematical theory of spatial-temporal dynamics and biological invasions, and to develop the theory of principal eigenvalues and basic reproduction ratios. Progress in this project 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 propagation.