CNCS Graduate Certificate Recipient

Aaron C.W. Ashih

Thesis Title: Spatial and Stochastic Models for Population Growth with Sexual and Asexual Reproduction

Ph.D. Final Defense Date:  10 April 2001

Ph.D. Dissertation Committee:

Dr. David G. Schaeffer, Supervisor
Dr. William G. Wilson, Co-Supervisor
Dr. Gregory F. Lawler
Dr. Michael C. Reed

Abstract:

This dissertation is a work in three parts:

Persistence in the Continuous-Deterministic Model. We build and analyze a two-sex deterministic reaction-diffusion model. The model has separate variables for males (m), unfertilized females (u), and fertilized females (f). The reaction terms are analogous to the well known logistic equation, but here they are bistable (populated n_+ and unpopulated 0). The addition of spatial diffusion yields traveling waves connecting n_+ and 0. A population is said to persist if the traveling wave invades the unpopulated space --- this corresponds to a positive traveling wave speed. We find a good approximation to predict persistence: if the stable population size, n_+, is more than twice the critical population size, n_-, then local prevalence will invade.

Extinction Time in the Discrete-Stochastic Model. We examine the influence of Allee effects (negative population growth below a critical value) on the extinction time. We proceed by applying perturbation analysis to the birth-death process. Although asymptotic expansion does not converge, Pade approximants prove remarkably accurate. We compute an approximation to expected extinction time, illustrating its dependence on parameters.

Diffusion Approximations to the Discrete-Stochastic Model. The analysis here closely parallels the work in the previous section. We improve the standard diffusion approximation by modifying the diffusion factor, replacing an arithmetic mean of birth and death rates with a logarithmic mean. The new result for expected extinction time gives a better approximation than the analogous result in the discrete analysis. Further, the result is intuitive to interpret. Extinction time strongly depends on the ratio of the stable-population density to the critical-population density.