Grant: Discovery Grants Program - Individual

Population models for species are used to study the behaviors of the populations or the interaction of two or more species. One important topic is to determine under what circumstances the species either survive or go extinct. According to human needs, the exploitation of biological resources and the harvesting of populations are commonly practiced in fishery, forestry, and wildlife management. To predict whether species will become extinct and to obtain insight into the optimal management of renewable resources, one needs to consider models which incorporate harvesting rates. The aim is to determine a harvestable quantity of the species without having the population die out. According to Clark's book (C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, second edition, John Wiley & Sons, New York, 1990), the management of renewable resources is thought to be based on the maximum sustainable yield (MSY). If the populations of the species are harvested by some process of over-exploitation (that is, harvesting rate is strictly greater than the MSY), then the species could become extinct. Mathematically, models of populations of species are often governed by one or more first order ordinary differential equations or by parabolic or elliptic partial differential equations (PDEs) including Laplacian boundary value problems. These models play an important role in modern applicable mathematics. The long-term objectives of my research program are to propose and analyze new models arising in mathematical biology and ecology, and establish new theories in differential equations and apply these and other known theories to treat these models. My short term objectives are to rigorously analyze some models such as population models of single species including diffusive Nicholson blowfly models, and of two species such as predator-prey models and chemostat models in PDE forms. Emphasis will be placed on PDE models with harvesting rates and finding the MSYs of these models since there is little study on the MSYs of the PDE models even in one-dimensional cases. The main difficulty is that maximum principles cannot be applied.