Numerical solution of a spatio-temporal predator-prey model with infected prey

A spatio-temporal eco-epidemiological model is formulated by combining an available non-spatial model for predator-prey dynamics with infected prey [D. Greenhalgh and M. Haque, Math. Meth. Appl. Sci., 30 (2007), 911-929] with a spatio-temporal susceptible-infective (SI)-type epidemic model of pattern formation due to diffusion [G.-Q. Sun, Nonlinear Dynamics, 69 (2012), 1097-1104]. It is assumed that predators exclusively eat infected prey, in agreement with the hypothesis that the infection weakens the prey, making it available for predation otherwise we assume that the predator has essentially no access to healthy prey of the same species. Furthermore, the movement of predators is described by a non-local convolution of the density of infected prey as proposed in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369-400]. The resulting convection-diffusion-reaction system of three partial differential equations for the densities of susceptible and infected prey and predators is solved by an efficient method that combines weighted essentially non-oscillatory (WENO) reconstructions and an implicit-explicit Runge-Kutta (IMEX-RK) method for time stepping. Numerical examples illustrate the formation of spatial patterns involving all three species.