IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 1 Ver. IV (Jan. - Feb. 2016), PP 21-40 www.iosrjournals.org DOI: 10.9790/5728-12142141 www.iosrjournals.org 21 | Page Modeling and Analysis of a Prey-Predator System with Disease in Predator M.V. Ramana Murthy, Dahlia Khaled Bahlool Departments of Mathematics, College of Science, Osmania University, Hyderabad, India Abstract: In the present paper a prey-predator model with disease that spreads among the predator species only is proposed and investigated. It is assumed that the disease is horizontally transmitted by contact between the infected predator and the susceptible predator. The local and global stability analyses are carried out. The persistence conditions of the model are established. Local bifurcation analyses are performed. Numerical simulation is used extensively to detect the occurrence of Hopf bifurcation and confirm our obtained analytical outcomes. Keywords: Prey-Predator, Stability, Local bifurcation, Hopf bifurcation, Persistence. I. Introduction A prey-predator type of interaction among species can be clearly seen in many ecological systems throughout the world, such as a deer-lion relation. In nature, prey and predator species exhibit fluctuation of abundance or population increase and decrease. The study of this fluctuation that is in apparently stable patterns has long been of interest to animal conservationists and mathematicians. Consequently the dynamics of prey-predator interactions have been studied extensively in the last three decades see for example [1-4] and the references therein. The evolution of disease in natural populations has always been an important field of both theoretical and experimental studies due to their effects on the existence populations. Although, most of the previous studies have been focused on the interactions between the pathogen and its host [5-7], it is obvious that populations are generally involved in complex trophic interactions with other populations. Therefore, this should be taken in to account when one construct mathematical model of the evolution of diseases in real ecosystems. Indeed it has been shown that predation of infected populations can both increase and decrease the infection prevalence [8-9]. Accordingly, mathematical epidemiology to study the dynamics of diseases spread has become an interesting topic of research study and received much attention from scientists after the pioneering work of Kermack-McKendrick. A number of mathematical models of disease spread have been introduced relevant to the type of diseases, for example SI, SIS, SIR, SEIR, SEIRS [10-14] and references therein. Eco-epidemiology is a rather new branch of study, merging features of interacting populations among which a transmissible disease spreads. It can be viewed as the coupling of an ecological prey- predator (or competition) model and an epidemiological SI, SIS or SIRS model. Following Anderson and May (1982) who were the first to propose an eco-epidemiological model by merging the ecological prey-predator model introduced by Lotka and Volterra, and the epidemiological SIR model introduced by Kermack and McKendrick, many works have been devoted to the study of the effects of a disease on a prey-predator system [15-18] and references therein. Keeping the above in view, most of the previous studies focused on the disease in prey- predator system with vertical transmitted of disease. However, in this paper an eco-epidemiological model consisting of prey-predator model with horizontally transmitted of disease within predator population is proposed and studied. II. The Model Formulation In this section a mathematical model, which describes the dynamical behavior of a prey- predator system with horizontally transmitted infectious disease in predator, is proposed and analyzed. Consequently, in order to formulates this model the following hypotheses are considered