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

研究人员考虑了一个描述几个相互作用的种群的动力学的非线性非局部双曲型方程的一般系统。该项目的目标是发展关于双曲型方程组解的存在唯一性、长期行为和数值逼近的理论。采用解析和数值方法相结合的方法来理解所提出的复杂模型的动力学。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来研究经典守恒律的有限差分法。研究解的适定性和长时间性态的第二种方法是基于单调逼近和比较，这些结果已经被研究者成功地建立在线性情形下。虽然类似的方法已经被用来研究其他非线性偏微分方程模型，但它们在本项目中所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完成。此外，对于由所提出的非线性非局部方程组控制的反问题，发展了数值方法，它利用实际数据和模型之间的联系来估计未知参数。同时，开发了一个用于模拟该模型的数值程序包。许多种群及其与环境的相互作用都是用结构化种群方法来模拟的，其中感兴趣的结构是由年龄或大小等内在特征引起的。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蚊虫的动态。结构化种群方法也被用来模拟分级结构种群的动态，在这种动态中，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，有必要了解种群演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，我们发展了一种解析和数值相结合的方法来研究这种种群的动力学。文中还给出了模拟该模型的数值程序包。此外，还引入了某些技术来根据现场数据估计种群内个体的生长和死亡。由于模型的通用性，所得结果有助于回答种群动力学中有关非线性现象的问题。此外，所得到的数值方法可用于绕过种群生物学家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.