2:15-2:40 Lih-Ing Roeger, Department of Mathematics and Statistics, Texas Tech University
Discrete May-Leonard Competition Models Annual Plants Competition
The famous May-Leonard (M-L) competition model is well understood. However, the discrete-time version of the M-L competition model does not behave exactly the same as its continuous counterpart. In this talk, we will present two discrete-time M-L competition models that are derived from annual plant competition. The local dynamics of these two discrete-time models applicable to three competition plant species are shown to have properties similar to the M-L differential equation model. The dynamics of the two discrete models are shown to be similar. However, they are not dynamically consistent with the continuous M-L models. Unlike the continuous M-L model, the Hopf bifurcations of the discrete M-L models are not degenerate. The continuous M-L model is the limiting case of the discrete models.
2:45-3:10 V. L. Kocic and J. Williams, Xavier University of Louisiana
Dynamics of the Discrete Model of West Nile Epidemics
The system of non-autonomous nonlinear difference equations models the spread of the West Nile Encephalitis. The disease is transmitted by mosquitoes to both birds and humans; mosquitoes can be infected only from birds; infected birds and infected humans can recover, while infected mosquitoes can not recover. The model composed of twelve difference equations representing the effects of the Wes Nile-Like Virus on populations of birds, humans, and mosquitoes. The model includes the effects of spraying to control the population of mosquitoes as a main tool for controlling the epidemics. The model was originally introduced by Thomas and Urena, and later modified and generalized by Darensburg and Kocic. In this paper we study the global behavior of the model, and in particular we focus on the effects of different spraying strategies to the dynamics of the model.
3:15-3:40 Jeffrey Edmunds, University of Mary Washington
The Influence of Stage Structure on Models of Competing Species
Mathematical models of competing species have been used historically in support of the theory of competitive exclusion, which say that two similar species competing for the same resource cannot coexist indefinitely. Recent work with a variety of stage-structured, discrete models has provided a number s counter-examples to this theory. I will discuss three particular models; a two-species version of the LPA model, which was derived to study insect populations; a two-species version of the classic Ricker model with juvenile/adult stage-structured incorporated; and finally, the Leslie/Gower competition model with juvenile/adult stage-structure incorporated.
3:45-4:10 Rebecca Culshaw, University of Texas at Tyler
Combining Immunotherapy and Antiviral Treatment of HIV: a Control-Theoretic Approach
We present an optimal control model of combined treatment of HIV-1. The state system consists of three ordinary differential equations. The treatments are modeled by two controls, one of which simulating therapy that directly enhances the population of DC4+ lymphocytes as well as other immune cells. The optimal control and optimality system are characterized and numerical simulations of the system under optimal treatment are given.