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。 在数学上，物种种群模型通常由一个或多个一阶常微分方程组或包括拉普拉斯边值问题的抛物型或椭圆型偏微分方程(PDE)来描述。这些模型在现代应用数学中起着重要的作用。 我的研究计划的长期目标是提出和分析数学、生物学和生态学中出现的新模型，并建立微分方程中的新理论，并应用这些理论和其他已知的理论来处理这些模型。我的短期目标是严格分析一些模型，如单种群模型，包括扩散的Nicholson苍蝇模型，以及两种群模型，如捕食者-食饵模型和偏微分方程形式的恒化器模型。由于即使在一维情形下，关于偏微分方程模型的MSY的研究也很少，因此本文将重点放在具有收获率的偏微分方程模型以及求出这些模型的MSY上。主要的困难是不能应用最大值原则。

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.