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

种群模型用于研究种群的行为或两个或多个种群之间的相互作用。一个重要的话题是确定物种在什么情况下生存或灭绝。根据人类的需要，生物资源的开发和种群的收获在渔业、林业和野生动物管理中很常见。为了预测物种是否会灭绝，并洞察可再生资源的最佳管理，人们需要考虑纳入收割率的模型。这样做的目的是在不使种群灭绝的情况下确定物种的可收获数量。根据克拉克的书(C.W.Clark，数学生物经济学，可再生资源的最优管理，第二版，John Wiley&Sons，纽约，1990)，可再生资源的管理被认为是基于最大可持续产量(MSY)。如果该物种的种群是通过某种过度开发的过程(即收获速度严格地大于MSY)来收获的，那么该物种可能会灭绝。

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.