Search

We formulate a delayed reaction-diffusion model that describes competition between two species in a stream. We divide each species into two compartments, individuals inhabiting on the benthos and individuals drifting in the stream. Time delays are incorporated to measure the time lengths from birth to maturity of the benthic populations. We assume that the growth of population takes place on the benthos and that dispersal occurs in the stream. Our system consists of two linear reaction-diffusion equations and two delayed ordinary differential equations. We study the dynamics of the non-spatial model, determine the existence and global stability of the equilibria, and provide conditions under which solutions converge to the equilibria. We show that the existence of traveling wave solutions can be established through compact integral operators. We define two real numbers and prove that they serve as the lower bounds of the speeds of traveling wave solutions in the system.

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.

This thesis introduces mathematical models to understand the role of peer influenceon the spread of alcohol abuse. Our first model involves an integro-differential equation (IDE) model and is based on the Braun et al [BWPBG] network model. IDEproblems often arise in studying non-local interactions between populations [M]. Weprovide a mathematical formula to explain the relationship between a constant population resilience level and the initial average peer influence, a detailed analysis of a regularized partial differential equation approximation to the IDE problem traveling wave solutions, and a mathematical model for treatment that would be available to severe alcoholics.We also propose a new network model that shows similar dynamics with the one in Braun et al (2006). However, this new approach does not have a certain degeneracy we observed in the Braun model. This work is the first step toward the development of continuum models for the dynamics of social networks. There are many exciting new directions to pursue including applications in obesity and gang violence as well as evolution of network connection dynamics.