Doctoral Dissertations

Qingping Deng

Date of Award

5-2000

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Thomas G. Hallam

Committee Members

Michel W. Beuy,

Abstract

This dissertation has developed an individual-based, physiologically structured model for a fish population with diffusive and advective movement in a spatial environment. It incorporates spatio-temporal processes and individual processes simultaneously into the population dynamic model of a McKendrick-von Foerster type partial differential equation. Anindividualfish is physiologically structured according to age, lipid and structure (protein and carbohydrates). Fish are assumed to be immobile in their embryonic stage and the fish begin to feed and might move after the embryonic stage. Advective processes are induced by environmental heterogeneity, in which fish move toward neighboring areas with different levels of, for instance, resource density or/and chemical toxicant concentration. The population dynamic model is complicated, in that it is a mixed type partial differential equation that combines a quasi-linear hyperbolic equation in the embryonic stage and degenerate parabolic equation in the older life stage.

Some mathematical aspects of the model of primary interest have been discussed. The existence of a local weak solution has been shown. By the constructive analysis used to demonstrate the existence of a local solution, a computational scheme for the mathematical model has been developed. For the individual growth model, we simply use the implicit Runge-Kutta method. For the population dynamic model of partial differential problem, we use a characteristic finite difference method in the age-time domain and a finite element method with numerical integration and upwind modification in the spatial domain. Furthermore, the numerical scheme has been proved to yield numerical approximations with optimal error estimates and produce biologically reasonable approximate solutions as well.

The mathematical and computational models have been used to study a specific model of a population of rainbow trout, Oncorhynchus mykiss, in a spatial environment. We Have investigated numerically the dynamics of spatio-temporal population distribution variations as they are viewed through the fish population density, total fish biomass, total fish age, total fish lipid, total fish structure (protein) and total fish protected protein. Furthermore, the model has also been used to study the effects of a spatially distributed nonpolar narcotic chemical on a rainbow trout population. The combined effects of lethal and sublethal toxicant effects have been considered.

The methodologies and conclusions in this dissertation can be extended immediately into other populations and even some community settings, such as the fish-Daphnia predator-prey model if Daphnia are assumed to be immobile.

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