1Department of Mathematical Sciences, Nasarawa State University Keffi, Nigeria; Nigeria
2Department of Statistics, Waziri Umaru Federal Polytechnic Birnin-Kebbi, Nigeria. Nigeria
In this work, we analyse a pair of two-dimensional coupled reaction-diusion equations known as the Gray-Scott model, in which spot patterns have been observed. We focus on stationary patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. A complete bifurcation study of these solutions is presented. The main mathematical techniques employed in this analysis of the stationary patterns is the Turing instability theory. This paper addresses the question of how popula- tion diusion aects the formation of the spatial patterns in the Gray-Scott model by Turing mechanisms. In particular, we present a theoretical analysis of results of the numerical simulations in two dimensions. Moreover, there is a critical value for the system within the linear regime. Below the critical value the spatial patterns are impermanent, whereas above it stationary spot patterns can exist over time. We have observed the formation of spatial patterns during the evolution, which are sparsely isolated ordered spot patterns that emerge in the space. In this research we focuse on three areas: rst, the biology; second, the mathematics and third, the application. We use these spatial patterns to understand the nature of disease spread and that means to understand the mechanism of interaction of the populations. There remains uncertainty in the mechanisms surrounding the genesis of how epidemics spread in their spatial enveronment. The role of mathematical modelling in understanding the spread and control of epidemics can never be over emphised.