Executive Summary : | The main objective of the project is essentially to study multiscale reaction-diffusion population models with local and non local interactions. The project intends to consider both linear and nonlinear diffusion in the models to analyse their effects on the system dynamics respectively. Theoretical study of existence, stability of solutions and corresponding bifurcation analysis of the patterns will be performed. Extensive numerical simulations will be carried out to validate the theoretical results. The project is primarily focused to present a detailed study on the existence and stability of the stationary, non stationary and traveling wave solutions of the multiscale reaction- diffusion models and to draw a comparison with the results of its counterpart having one timescale. Another endeavour is to explore all possible bifurcation scenarios and the structure of the patterns for the said type of reaction diffusion systems. Although there are some works available in the literature the are many gap areas are still prevailing such as lack of studies on multiscale spatio-temporal models specially in presence of nonlocal interactions, understanding how different scaling affects the stability of the solution of a general reaction diffusion equation when the diffusion is linear or nonlinear, poor literature on the existence of traveling waves and pulses theoretically for multiscale reaction diffusion systems in case of local or nonlocal interactions, bifurcation analysis of spatio-temporal patterns for the multiscale models. In view of these challenging question this project aims at addressing some these questions. To do so we plan to formulate various multiscale two dimensional population models including diffusion and study them with help of multiple scale perturbation analysis, slow fast analysis and other relevant tools. A successful completion of this project will give a platform to address many physical, chemical and biological problems. |