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

研究人员考虑了一个非线性非局部双曲方程的一般系统，该系统描述了几个相互作用种群的动力学。该项目的目标是发展双曲系统解的存在唯一性、长时间行为和数值逼近的理论。分析和数值方法的结合被用来理解复杂模型的动力学。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用经典守恒律的有限差分法。研究解的适定性和长时间行为的第二种方法是基于单调近似和比较结果，这些结果已经由研究人员成功地建立了这些半线性情况。虽然类似的方法已用于研究其他非线性偏微分方程模型，但它们在本项目中考虑的一般非线性和非局部模型的特殊情况下的应用仅由研究人员进行。此外，本文还研究了由所提出的非线性非局部方程组控制的反问题的数值方法。它利用实际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程序。许多种群及其与环境的相互作用已经使用结构化种群方法建模，其中感兴趣的结构是由年龄或大小等内部特征引起的。例如，具有分布率的大小结构种群模型已被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分层结构种群的动态，其中个体之间的差异对资源的可用性有直接影响。另一个应用是在藻类聚集的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，有必要了解种群进化与其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，我们发展了一种结合解析和数值的方法来研究这种种群的动态。特别地，提供了一个数值包来模拟所提出的模型。此外，还引入了一些技术，根据野外数据估计种群内个体的生长和死亡率。由于所提模型的通用性，所得结果有助于回答种群动力学中非线性现象的问题。此外，所得到的数值方法可以被种群生物学家用来研究一般种群模型的动态行为。

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.