Research Projects

Parabolic reaction-diffusion (PRD) equations have been widely used to study the spatial distribution of interacting species and their corresponding patterns. However, they often fail to accurately describe spatial dispersal due to species often exhibiting inertia, resulting in delayed effects in their spatial movement. This leads to the development of hyperbolic reaction transport equations, which have been used to extend the PRD system to remove undesirability. This project proposal aims to develop general criteria for a two-species HRD system to investigate instabilities due to self and cross-diffusions and species-specific inertial time. The study aims to understand how cross-diffusion and species-dependent inertia jointly influence pattern-forming instabilities in the system. The developed theory will be applied to reveal the way pattern-forming instabilities are influenced by biological and chemical systems' underlying kinetics. The results will be extended to a three-species system and then applied to an HRD network system to reveal emergent properties not present in individual systems. This investigation can be seen as a framework for studying diffusive instabilities in a particular HRD system that becomes a PRD system without inertia.