Spatial and temporal dynamics of nonlinear systems with applications

受种群生物学和流行病学启发，本研究项目旨在研究具有时空异质性的非线性进化系统的全局动力学。我们将研究稳态和周期解的存在性、唯一性、分岔性和全局稳定性，持续和消光，全局收敛和吸引子，行波的存在性、唯一性和稳定性，以及传播速度。在本计划中，我们还提出了关于非自治种群模型中具有弱紧致性和基本繁殖比的抽象进化系统的传播速度和行波的一般结果。研究方法包括动力系统、李雅普诺夫函数、上下解、摄动和分岔、比较原理、斜积半流、线性化、非线性分析和数值模拟。该研究的新颖之处在于用动力系统方法求解具有时空异质性的演化方程。该研究旨在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。这项研究的进展还可以帮助回答一些重要问题，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成为地方病，估计生物入侵和流行病传播的速度，以及理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 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.