描述气候变化中入侵物种演化的自由边界问题

This project aims to study some reaction diffusion equations with the free boundary, which comes from the problem of the species invasion. This work intends to focus on the impact of the heterogeneity of climate change on the qualitative properties of the solutions of such equations, which mainly investigates the three following problems. First, the existence, uniqueness and regularity of the solution of the free boundary problem and the long time behavior of the free boundary problem are studied, especially the regularity of the free boundary in high dimensional space. Second, the influence of climate warming on the convergence of the solution of a competitive system with free boundary conditions is discussed. The convergence of the solution and the interdependence of competitive relations, free boundaries and environmental heterogeneity are discussed specially. Finally, the reaction diffusion equation with different free boundary conditions is considered in the periodic environment. The influence of different free boundary conditions on the asymptotic shape of the solution of the problem is studied as spreading happens. The model of this project includes free boundary and environmental heterogeneity, which makes the model closer to the real invasion of species, but also brings the essential research difficulties. In the qualitative analysis of the new models, we not only need to be flexible in the use of classical methods, but also to explore new ideas and methods. This promotes the development of the basic theory of differential equations to a certain extent.

本项目旨在研究具有自由边界条件的反应扩散方程，它来源于生态学中的物种入侵问题，着重探讨气候变化所造成的异质性对这类方程解的定性性质的影响，主要研究以下三个问题：首先，考虑高维空间中具有自由边界条件的反应扩散方程解的存在唯一性及正则性和长时间行为，重点探讨高维空间中自由边界的正则性。其次，分析气候变暖对具有自由边界条件的反应扩散竞争系统解收敛性的影响，重点讨论解的收敛性和竞争关系、自由边界及环境异质性的相互依赖关系。最后，在周期环境中，考虑具有不同自由边界条件反应扩散方程，当蔓延发生时，重点研究不同自由边界条件对问题解的渐近形状的影响。本项目的模型包含了自由边界和环境异质性，这使得模型更接近于物种的真实入侵但也带来了本质的研究难点。在对新模型进行定性分析时，不仅要求我们要灵活运用经典的理论方法，还要探索新的思路和方法，这会在一定程度上促进偏微分方程基础理论的发展。

随着环境的不断恶化，生物入侵和传染病传播已经成为生物数学家研究的热点问题。通过建立简单的数学模型，对生物入侵和传染病传播进行定性与定量分析，这样不仅有助于更深入地理解自然界中观察到物种传播的复杂现象，也能进一步地推动生物进化理论知识的发展。本项目主要围绕非均匀环境中的两类反应扩散模型的动力学行为展开讨论。第一类问题是具有保护区域和自由边界条件的反应扩散问题，利用不动点定理和抛物初边值问题的标准理论得到了所研究问题全局解的存在唯一性。通过构造合适的上下解和非线性泛函分析理论，重点研究了保护区域和自由边界条件对反应扩散问题解的渐近行为的影响。这个研究对濒危物种如何建立合适的自然保护区进行保护提供了理论依据。第二类问题是具有不同疾病发生率的反应扩散问题，通过定义基本再生数，利用动力学理论和李雅诺普方法给出解长时间的阈值行为。另外，当染病平衡点存在时，利用先验估计和椭圆问题经典理论重点研究了扩散率和疾病发生率对染病平衡态空间分布的影响。研究结果能够为传染病防控策略的制定和部署给出很好的指导意见。