Journal of Mathematics and System Science 6 (2016) 395-408 doi: 10.17265/2159-5291/2016.10.003 A Leslie-Gower Holling Type-II Predator-Prey Mathematical Model with Disease in Prey Population Incorporating a Prey Refuge P. Mandal, N. Das and S. Pal 1. Department of Mathematics, Hooghly Mohsin College, Chinsurah, Hooghly, India. 2. Department of Mathematics, Maulana Azad College, Kolkata, India. 3. Department of Mathematics, University of Kalyani, Nadia, India. Received: July 25, 2016 / Accepted: August 23, 2016 / Published: October 25, 2016. Abstract: We formulate and analyze a predator-prey model followed by Leslie–Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey population has been classified into two categories, namely susceptible prey, infected prey where as the predator population remains free from infection. To investigate the behaviour of prey population we incorporate prey refuge in this model. Since the prey refuge decreases the predation rate then it has an important effect in our predator-prey interaction model. We have discussed the existence of various equilibrium points and local stability analysis at those equilibrium points. We investigate the Hopf-bifurcation analysis about the interior equilibrium point by taking the rate of infection parameter and the prey refuge parameter as bifurcation parameters. The numerical analysis is carried out to support the analytical results and to discuss some interesting results that our model exhibits. Keywords: Predator and prey, Disease transmission, Prey refuge, Stability, Hopf-bifurcation. 1. Introduction In theoretical ecology the dynamical relationship between prey and predator is one of the important themes from the point of view of its existence. Also mathematical modelling is essential to understand the dynamical behaviour of such system. Anderson and May [1], Hadeler and Freedman [2] analyzed eco-epidemic models where predator population is infected through eating prey. The mathematical models, like [3, 4, 5, 6, 7] on prey–predator system have been studied by the researchers, out of which the interesting dynamics of Holling Tanner model [7] plays an important role in the theoretical ecology. Leslie [8,9] introduced prey–predator model in which the carrying capacity of the predators environment is proportional to the number of prey. Incorporating the Corresponding author: S. Pal, Department of Mathematics, University of Kalyani, Nadia, India. Holling type-II functional response the modified Leslie Gower prey-predator model was obtained, for example [10]. On the other hand spatial prey refuge is one of the more relevant behavioral traits that affect the dynamics of predator-prey systems. Some of the researchers [6, 10, 11, 12, 13] have investigated the influence of prey refuge and they concluded that the refuge used by the prey has a stabilizing effect on the prey-predator interaction. In epidemiology, the incidence rate in mathematical models of infectious disease [14, 15] play an important role. In [16] the authors analyzed an eco–epidemic model incorporating a prey refuge in predator-prey system in which we observe the very interesting dynamics corresponding to refuge term. Recently, the effect of disease on ecological systems is an important issue from mathematical as well as experimental point of view as the effect of infectious disease on the ecological D DAVID PUBLISHING