Spatial and temporal dynamics of evolution systems with applications

受种群生物学和流行病学的启发，研究具有时空异质性的非线性演化系统的全局动力学问题。我们将研究定态、概周期解和整体解、持续生存和灭绝、全局收敛和吸引子、行波的存在、唯一性和稳定性以及传播速度。在这个项目中，我们还得到了关于具有点火结构的抽象单调系统的行波，时空非均匀介质和移动环境中的传播动力学，周期非局部扩散系统的主本征值，以及非自治种群模型的基本再生率的一般结果。*研究方法涉及动力系统，Liapunov函数，上下解，扰动和分支，比较原理，斜积半流，线性化，正算子，非线性分析和数值模拟。这项研究的创新之处在于用动力系统方法研究具有时空异质性的演化方程。*本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比理论。这一项目的进展还有助于回答一些重要问题，如相互作用的种群能否以及如何生存(避免灭绝)，或传染病是否在种群中成为地方病，估计生物入侵和流行病传播的速度，以及了解诸如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.**