On a population pathogen model incorporating species dispersal with temporal variation in dispersal rate.

In the present paper, we consider a mathematical model of ecosystem population interaction where the population suffers from a susceptible-infectious-susceptible disease. Dispersal of both the susceptible and the infective is incorporated using reaction-diffusion equations. We first study the stability criteria of the basic (non-spatial) model around the disease-free and the infected steady states. We find that the loss rate of the infective species controls disease prevalence. Also without predation pressure, the disease will continue to exist among the population. Then we analyze the spatial model with species dispersal in constant as well as in time-varying form. It is observed that though constant dispersal is unable to generate diffusion-driven instability, dispersal with sinusoidal variation in dispersion rate can generate diffusive instability when the wave number of the perturbation lies within a given range. Numerical simulations are performed to illustrate analytical studies.