Dynamics of a Population Model Controlling the Spread of Plague in Prairie Dogs Catalin Georgescu The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD USA Catalin.Georgescu@usd.edu Dan Van Peursem The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD 57069 USA dpeursem@usd.edu Abstract: The infestation of prairie dog population with the deadly Yersinia Pestis has been recently the object of investigation of numerous biologists due to the potentially high risk of contamination of the human population. In the present paper we construct and analyze a model describing the interaction between the prairie dog population (as the host population) and the flea population (as the vector population) and derive the existence (under certain conditions impose to the main parameter) of an equilibrium that plays the role of a global attractor. Key–Words: Stability of Equilibria, Global attractor, Bifurcation. 1 Background. In recent years, prairie dog populations have been hit hard by the plague caused by Yersinia pestis. The literature suggests that total colonies can be wiped out upon infestation and the disease is rapidly spread from one colony to the next (see [1],[2],[6]). There are reasons to expect that the spread of plague from face to face contact or through respiratory methods is not the predominate mode of transferring the disease as Hoogland et al. have reported (see [5]) that upon killing fleas with the insecticide Pyraperm the spread of the disease was stopped for a time. The agreed upon mode of transmission of the vector is through in- fected fleas hosting on the prairie dogs or other mam- mals. Seery et al. and Hoogland et. al have shown that by using insecticides on an infected colony one can stop the spread of an epizootic and can be used to save the colony. We present a mathematical model that will allow one to show analytically what kind of control one needs on the fleas in order to prevent an epizootic once the plague has been manifested in a colony. The model shows that when one starts with healthy and infected prairie dogs and fleas, there are three equilibrium points that can develop. We will consider α, the ratio between the normal death rates of flea population and prairie dog population, as the working parameter of the problem. We will show that if this parameter stays smaller than a certain value α 0 , the dynamics of the system is governed by three equi- libria: there is the case where healthy and infected prairie dogs and fleas can coexist and it is an asymp- totically stable equilibrium, there is the case where all infected fleas and infected prairie dogs are wiped out, and there is the case where only the healthy prairie dogs exist, these last two equilibria being unstable. If the parameter exceeds the bifurcation point α 0 , one witness a collapse of equilibria to only healthy prairie dogs existing and the change in stability of the last two mentioned above equilibria. 2 The Origins of the Model. The model will consist of a typical host-vector model with the prairie dogs acting as the host population and the flea population being the vector population. Let P be the total prairie dog population, S being the num- ber of infected or sick prairie dogs, F be the total flea population, and D be the number of diseased or in- fected fleas. The governing equations are: dP dt = r p  1 − P k p  P − d r P, (1) dF dt = r f  1 − F mP  F − d f F, (2) dS dt = t p  P − S P  D − c 1 d r S, (3) dD dt = t f S P (F − D) − c 2 d f D. (4) RECENT ADVANCES IN APPLIED MATHEMATICS AND COMPUTATIONAL AND INFORMATION SCIENCES - Volume I ISBN: 978-960-474-071-0 ISSN: 1790-5117 78