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

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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.