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.
For many physical or biological events, a slight change of initial state or parameter can lead to dramatically different outcome in the long term. For example, change of climate makes the ecosystem switching from grassland to desert, and building oyster reef with different height leads to successful or failed restoration effort. Mathematical models have been used to describe such phenomenon of alternative stable states in natural world. Our research was aimed at developing and analyzing partial differential equation models which describe bistable dynamics and related spatialtemporal pattern formations. Our results show that in a consumer-resource type model with bistable structure for the resource, either the consumer and resource bothe become extinct, or they can reach a stable state which is in equilibrium or persistent oscillatory state. The spatial movement of the substance can generate spatially nonhomogeneous patterns. Another result from our research is to show the rich dynamics and patterns generated from the interaction of different movement patterns such as diffusion, advection, chemical-induced movement, nonlocal dispersal, birth-death dynamics, and the effect of time delay. A team of researchers from College of William and Mary and University of Wyoming developed a spatial movement model of mistletoes and mistletoe-eating birds to explain the spreading of mistletoes with birds as carrier and the patchiness of mistletoe distribution (see Fig. 1). Mathematical tools in nonlocal evolution equations are developed to achieve the analytical results. Models of interaction between live oyster, oyster shell and sediment were established as a joint effort from researchers and undergraduate students from College of William and Mary and Virginia Institute of Marine Science (see Fig. 2). The model provides scientific support for the effectiveness of high relief oyster reef, which validated the large scale oyster reef restoration effort by federal and state government in Chesapeake Bay (see Fig. 3).
