Spatially heterogeneous eco-epidemic model: Stabilizing role of non-local disease transmission

Abstract

Interaction between prey and predator in the presence of an infectious pathogen is the main focus of this article. A non-local transmission, that encompasses the possibility of acquiring infection from a distanced potential infected individual, is incorporated by utilizing a convolution of a spatial kernel function of compact support with the spatial distribution of the infected population. The spatial kernel function characterizes the likelihood of contracting an infection from an infected individual located within a certain range and its compact support indicates the extent of the non-local disease transmission. The model is governed by a system of semilinear parabolic integro-differential equations whose global-in-time classical solution has been established. The basic reproduction numbers of two distinct disease-free homogeneous steady states are derived for the non-local diffusive eco-epidemiological model. The associated non-spatial model undergoes a supercritical Hopf-bifurcation and possesses a stable limit cycle bifurcated from the stable endemic equilibrium point. The Turing instability conditions of the endemic homogeneous steady state are derived for both the local and non-local systems. A wide variety of spatio-temporal solutions over a one-dimensional spatial domain have been explored for both the local model and the non-local model with parabolic kernel function. It has been observed that an increase in the extent of the non-local disease transmission increases the parametric region of the stable spatially homogeneous solution. On the other hand, the parametric region for the spatially heterogeneous solutions, characterizing the patchy distributions of interacting populations, shrinks with an increase in the extent of the non-local disease transmission.