NSF Award Abstract - #0211453

AWSFL008-DS3

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

Abstract

The investigators consider a general system of nonlinear nonlocal hyperbolic equations describing the dynamics of several interacting populations. The goal of the project is to develop theories on the existence-uniqueness, long-time behavior, and numerical approximation of solutions to the hyperbolic system. A combination of analytical and numerical methods is used to understand the dynamics of the complex proposed model. To study existence and uniqueness of solutions to this system, two major approaches are adopted. The first approach to study the well-posedness of solutions is via the finite difference method used for classical conservation laws. The second approach to investigate the well-posedness and long-time behavior of solutions is based on the monotone approximation and comparison results that have been successfully established for the semilinear case by the investigators. Although similar approaches have been used in studying other nonlinear partial differential equation models, their applications to special cases of the general nonlinear and nonlocal model considered in this project have only been carried out by the investigators. In addition, a numerical methodology is developed for an inverse problem governed by the proposed nonlinear nonlocal system of equations. It uses a connection between real data and the model to estimate unknown parameters. Meanwhile, a numerical package is developed for simulating the proposed model. Many populations and their interactions with the environment have been modeled using the structured population approach, where the structures of interest are induced by internal characteristics such as age or size. For example, size-structured population models with distributed rates have been successfully used to describe the dynamics of mosquitofish in California rice fields. The structured population approach has also been used to model the dynamics of hierarchically structured populations in which the differences between individuals have a direct effect on the availability of resources. Another application is the predator-prey interaction between zooplankton and phytoplankton within the context of algal aggregation. Generally speaking, in order to analyze, manage, and control the dynamics of a population, it is necessary to understand the interactions between the population evolution and its environment. In this project, the investigators study a general structured population model. Due to its complexity, a combination of analytical and numerical methods is developed to investigate the dynamics of such a population. In particular, a numerical package to simulate the proposed model is provided. Furthermore, certain techniques are introduced to estimate the growth and mortality for individuals within the population from field data. Because of the generality of the proposed model, the results help to answer questions about nonlinear phenomena in population dynamics. In addition, the resulting numerical method can be used by population biologists to investigate the dynamical behavior of general population models.