Random dispersal in a predator-prey-parasite model

Similar to the model studied in [11], it is assumed that the parasite under consideration is a microparasite so that the parasite population is not explicitly modeled in the interaction. There are only two interacting species prey and predator in the model. Individuals in each species are classified as either infected or uninfected. Let x1 and x2 denote the uninfected and infected prey populations respectively, and y1 and y2 be the corresponding predators. The prey population in the absence of the predator and the parasite is modeled by a simple logistic equation with per capita growth rate r and carrying capacity 1/q. It is assumed that the infected prey does not reproduce. We use a simple Holling type I functional response, and let e denote the predator conversion rate. Since infected prey may increase its likelihood of being preyed upon due to the disease, we let θ1 denote the factor that affects the predator-prey interaction. The predator preys on both infected and uninfected prey indiscriminately when θ1 = 1. If θ1 < 1, then the infected prey has a less chance of being captured. The infected prey will be more likely to be preyed upon if θ1 > 1. The natural death rates of the infected prey and predator are denoted by d1 and d2 respectively. The disease related mortality rates of the prey and predator populations are denoted by α1 and α2, respectively. These parameters are assumed to be constants. Disease can be transmitted in two ways. An uninfected prey will become infected if it makes contact with an infected predator. Similarly, an uninfected predator will become infected if it contacts with an infected prey.