This paper will treat the bifurcation and numerical simulation of rotating wave (RW) solutions of the FitzHugh-Nagumo (FHN) equations. These equations are often used as a simple mathematical model of excitable media. The dependence of the solutions on a uniformly applied current, and also on the diffusion coefficients or domain size will be studied. Ranges of applied current and/or diffusion coefficients in which RW solutions are observed will be described using bifurcation theory and continuation methods. The bifurcation of time-periodic solutions of these FHN equations without diffusion is described first. Similar methods are then used to find RW solutions on a circular ring and numerical simulations are described. These results are then extended to investigate RW solutions on annular rings of finite cross-section. Scaling arguments are used to show how the existence of solutions depends on either the diffusion coefficient or on the size of the region.

Poikilothermic ectotherms have evolved behaviours that help them maintain or regulate their body temperature (T (b)) around a preferred or 'set point' temperature (T (set)). Thermoregulatory behaviors may range from body positioning to optimize heat gain to shuttling among preferred microhabitats to find appropriate environmental temperatures. We have modelled movement patterns between an active and non-active shuttling behaviour within a habitat (as a biased random walk) to investigate the potential cost of two thermoregulatory strategies. Generally, small-bodied ectotherms actively thermoregulate while large-bodied ectotherms may passively thermoconform to their environment. We were interested in the potential energetic cost for a large-bodied ectotherm if it were forced to actively thermoregulate rather than thermoconform. We therefore modelled movements and the resulting and comparative energetic costs in precisely maintaining a T (set) for a small-bodied versus large-bodied ectotherm to study and evaluate the thermoregulatory strategy.

Prostate cancer is one of the most prevalent types of cancer among men. It is stimulated by the androgens, or male sexual hormones, which circulate in the blood and diffuse into the tissue where they stimulate the prostate tumor to grow. One of the most important treatments for advanced prostate cancer has become androgen deprivation therapy (ADT). In this paper we present three different models of ADT for prostate cancer: continuous androgen suppression (CAS), intermittent androgen suppression (IAS), and periodic androgen suppression. Currently, many patients in the U.S. receive CAS therapy of ADT, but many undergo a relapse after several years and experience adverse side effects while receiving treatment. Some clinical studies have introduced various IAS regimens in order to delay the time to relapse, and/or to reduce the economic costs and adverse side effects. We will compute and analyze parameter sensitivity analysis for CAS and IAS which may give insight to plan effective data collection in a future clinical trial. Moreover, a periodic model for IAS is used to develop an analytical formulation for relapse times which then provides information about the sensitivity of relapse to the parameters in our models.

No abstract

No abstract