The investigator develops and studies some reaction-diffusion models of population ecology and of pattern formations of biochemical systems. The spatial distribution of populations can be affected by depressed growth rate at lower density, which is called the Allee effect. For instance, it can be caused by shortage of mates, lack of effective pollination, or harvesting. The main goal is to gain insight into the ways that Allee and harvesting effects and dispersal behavior influence the persistence and extinction of species and the success or failure of biological invasions. New mathematical tools in nonlinear elliptic and parabolic partial differential equations and nonlinear functional analysis are developed to study the complicated dynamics. Emphasis is on the nonlinear phenomena that can not be obtained through linearization and on patterns in higher spatial domains that can not be observed in simplified one-dimensional problems. The project is guided by a series of practical problems, from fishery management, animal aggregation, and autocatalytic reactions. A second line of the project is focused on a pattern formation mechanism, generalizing the pioneer work of Alan Turing. A nonlinear and higher spatial dimensional mechanism is established to refine Turing pattern theory, which has been successfully applied to phenomenological models, empirical substrate-inhibition systems, and autocatalytic reactions. One part of the project studies the impact of human interference on the natural evolution of marine species. In the second part of the project, the investigator considers the origin of the generation of complex patterns found everywhere in the nature, like seashells and spotty patterns on a zebra or a leopard. These studies aim to understand and control many important natural phenomena, such as the persistence and extinction of commercial marine species, and complex pattern formations in biochemical processes. Results of the project can be helpful for formulating policy on managing commercial fisheries and forests, and for developing new biotechnology simulations of natural life forms.
