Collaborative Research: Nonlinear Nonlocal First Order Hyperbolic Problems in Population Models

研究人员考虑了一个一般系统的非线性非局部双曲方程描述动态的几个相互作用的人口。 该项目的目标是发展双曲方程组解的存在唯一性、长时间行为和数值逼近的理论。 分析和数值方法相结合，用于理解复杂的拟议模型的动力学。 为了研究该系统解的存在唯一性，主要采用了两种方法。 研究解的适定性的第一种方法是通过经典守恒律的有限差分方法。 第二种方法来研究的适定性和长时间行为的解决方案是基于单调逼近和比较结果，已成功地建立了半线性的情况下，由调查人员。 虽然类似的方法已被用于研究其他非线性偏微分方程模型，但它们的应用到特殊情况下的一般非线性和非局部模型考虑在这个项目中只进行了研究人员。 此外，本文还提出了一种新的非线性非局部方程组的反问题的二阶方法，它利用真实的数据和模型之间的联系来估计未知参数。 同时，开发了一个数值模拟软件包，用于模拟所提出的模型。 许多人口和他们的相互作用与mesmenthave使用结构化人口的方法，其中感兴趣的结构是由内部特征，如年龄或大小。 例如，具有分布率的大小结构种群模型已成功地用于描述加州稻田中食蚊鱼的动态。 结构化种群方法也被用来模拟层次结构种群的动态，其中个体之间的差异对资源的可利用性有直接影响。 另一个应用是藻类聚集背景下浮游动物和浮游植物之间的捕食者-猎物相互作用。 一般来说，为了分析、管理和控制种群的动态，必须了解种群进化与环境之间的相互作用。 在这个项目中，研究人员研究了一个一般的结构化人口模型。 由于其复杂性，分析和数值方法相结合的发展，以调查这样的人口的动态。 特别是，提供了一个数值包来模拟所提出的模型。 此外，还介绍了根据野外资料估计种群内个体生长和死亡的技术。 由于所提出的模型的通用性，结果有助于回答人口动态中的非线性现象的问题。 此外，所得到的数值方法可供种群生物学家用来研究一般种群模型的动力学行为。

The investigators consider a general system of nonlinearnonlocal hyperbolic equations describing the dynamics of severalinteracting populations. The goal of the project is to developtheories on the existence-uniqueness, long-time behavior, andnumerical approximation of solutions to the hyperbolic system. Acombination of analytical and numerical methods is used tounderstand the dynamics of the complex proposed model. To studyexistence and uniqueness of solutions to this system, two majorapproaches are adopted. The first approach to study thewell-posedness of solutions is via the finite difference methodused for classical conservation laws. The second approach toinvestigate the well-posedness and long-time behavior ofsolutions is based on the monotone approximation and comparisonresults that have been successfully established for thesemilinear case by the investigators. Although similar approacheshave been used in studying other nonlinear partial differentialequation models, their applications to special cases of thegeneral nonlinear and nonlocal model considered in this projecthave only been carried out by the investigators. In addition, anumerical methodology is developed for an inverse problemgoverned by the proposed nonlinear nonlocal system of equations.It uses a connection between real data and the model to estimateunknown parameters. Meanwhile, a numerical package is developedfor simulating the proposed model. Many populations and their interactions with the environmenthave been modeled using the structured population approach, wherethe structures of interest are induced by internalcharacteristics such as age or size. For example, size-structuredpopulation models with distributed rates have been successfullyused to describe the dynamics of mosquitofish in California ricefields. The structured population approach has also been used tomodel the dynamics of hierarchically structured populations inwhich the differences between individuals have a direct effect onthe availability of resources. Another application is thepredator-prey interaction between zooplankton and phytoplanktonwithin the context of algal aggregation. Generally speaking, inorder to analyze, manage, and control the dynamics of apopulation, it is necessary to understand the interactionsbetween the population evolution and its environment. In thisproject, the investigators study a general structured populationmodel. Due to its complexity, a combination of analytical andnumerical methods is developed to investigate the dynamics ofsuch a population. In particular, a numerical package to simulatethe proposed model is provided. Furthermore, certain techniquesare introduced to estimate the growth and mortality forindividuals within the population from field data. Because of thegenerality of the proposed model, the results help to answerquestions about nonlinear phenomena in population dynamics. Inaddition, the resulting numerical method can be used bypopulation biologists to investigate the dynamical behavior ofgeneral population models.