Mathematical Studies of Spatial Bistability in Ecological Systems

不同种类物质的空间演化、扩散和相互作用可以产生复杂的时空格局。 本计画探讨生物科学中的空间扩散动力学及相关的模式形成问题。 物种间的典型相互作用是消费者-资源（捕食者-被捕食者）型的，由双稳定性和空间扩散引起的现象包括水有限生态系统的灾难性变化、草原荒漠化和缺水、海洋物种种群恢复的成功或失败，以及具有深刻生物化学意义的自催化化学反应。 在实验或自然界中观察到的时空格局往往伴随着生态系统的动态变化，如湖泊、草原、沙漠、海洋等生态系统的自组织斑块，或正反馈的生物化学反应中的点、条、带状空间格局。 在非线性椭圆、抛物型偏微分方程、分支理论和无穷维动力系统中发展了新的数学工具来理解复杂时空动力学中的定态和演化问题。重点是分析和理解无法通过线性化得到的非线性现象，建立不受特定数学模型约束的普遍自然规律，为海洋生物种群崩溃、生态环境从草原和森林向沙漠和干旱地区的灾难性转变、全球气候变化等现象提供科学解释。 该项目的一个具体组成部分是模拟和模拟本地牡蛎（或其他海洋物种）种群，这为州和联邦政府通过重建切萨皮克湾几个地点的栖息地珊瑚礁来恢复牡蛎种群的努力提供了直接指导。 本工程的理论研究为人工鱼礁的建造和仔鱼投放设计提供了具体的建议。 为威廉和玛丽学院等院校的本科生和弗吉尼亚海洋科学研究所的海洋科学研究生提供定性和数学分析及生态建模方面的培训，拓宽学员的背景和视角，增强世纪数学科学和海洋科学人才队伍。 反映数学生物学前沿知识的新课程材料已经开发，并可通过互联网和其他出版物提供给学术界。

Spatial evolution, dispersal and interaction of different species of substances could generate complex spatiotemporal patterns. This project investigates the spatial bistable dynamics and related pattern formation problems occurring in biological sciences. Typical interactions between species are of consumer-resource (predator-prey) type, and phenomena caused by bistability and spatial dispersal include catastrophic shifts in water-limited ecosystems, desertification of grassland with water shortage, success or failure of marine species population restoration, and also autocatalytic chemical reactions with deep biochemistry implications. Often accompanying the bistable dynamics are the spatiotemporal patterns observed in experiments or in nature, such as the self-organized patchiness of many ecosystems like lakes, grassland, deserts and oceans, or spot, stripe and labyrinths spatial patterns in biochemical reactions with positive feedback. New mathematical tools in nonlinear elliptic, parabolic partial differential equations, bifurcation theory and infinite dimensional dynamical systems are developed to understand the stationary and evolution problems in the complex spatiotemporal dynamics. Emphasis is on analyzing and undertanding the nonlinear phenomena which cannot be obtained through linearization, and formulating universal natural principles not bounded by specific mathematical models.Research in this project helps to give scientific explanation to phenomena such as marine species population collapse, catastrophic environment shifts from grassland and forest to desert and arid areas, and global climate changes. A specific component of the project is to model and simulate the native oyster (or other marine species) population, which provides direct guidance to state and federal government efforts to restore oyster population by reconstructing habitat reefs in several Chesapeake Bay locations. Theoretical investigation in this project provides specific suggestions to the design of reef construction and larvae release. Training on qualitative and mathematical analysis and ecological modeling is provided to undergraduate students in College of William and Mary and other institutes, and graduate students in marine sciences from Virginia Institute of Marine Science, which broadens trainees' background and perspective, and enhances the mathematical science and marine science workforce in the 21st century. New curriculum material reflecting the cutting-edge knowledge on mathematical biology is developed and available to the academic community from the internet and other publications.