Structured Population Models: Numerical Methods and Application to Frogs Infected with Chytridiomycosis

Abstract

The main objective of this dissertation is to study numerical methods and applications of structured population models. In Chapter 1, a quasilinear hierarchically size-structured population model is presented. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. The solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this chapter we developed a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method was provided. We also developed a high resolution second order scheme to compute the measure-valued solution of the model and performed a comparative study between the two schemes. In Chapter 2, a model that describes the dynamics of a frog population infected with chytridiomycosis disease is presented. Chytridiomycosis is an emerging disease caused by the fungal pathogen Batrachochydrium dendrobatidis (Bd) that poses a serious threat to frog populations worldwide. Several studies have shown that inoculation of bacterial species Janthinobacterium lividum (Jl) can mitigate the impact of the disease. However, there are many questions regarding this interaction that are unknown. Therefore, a mathematical model of a frog population infected with chytridiomycosis was developed to investigate how the inoculation of Jl could reduce the impact of Bd on frogs. The model also illustrates the important role of temperature in the diseases epidemiology. The model simulation results suggest possible control strategies for Jl to limit the impact of Bd in various scenarios. Finally, in Chapter 3, concluding remarks and discussion on our future work is provided