Elliptic partial differential equations with applications to population models with harvesting rates

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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.