The Impact of Spatially Localized Feed Parameter Perturbations on Reaction-Diffusion Systems*

The Gray-Scott Model is a prominent reaction-diffusion system that comprises two non-real chemical species, u and v, which interact through complex reactions and transformations. By manipulating parameter values within the model, spatiotemporal patterns emerge that mirror essential biological, chemical, and physical processes, including DNA oligomer formation, skin pigmentation development, and oscillatory chemical reactions. While extensive studies have typically focused on fixed parameter values, this research innovatively introduces specially defined functional forms that spatially vary key parameters throughout the lattice. Specifically, by employing sinusoidal and other oscillatory functions to modulate parameter values, we can seamlessly integrate the dynamics and behaviors characteristic of diverse parameter settings within a single reaction framework. This novel approach not only reveals a spectrum of unique and intricate patterns but also enhances the model's applicability, extending its relevance to a broader range of natural phenomena. Through this exploration, we aim to deepen our understanding of reaction-diffusion systems and their potential implications in both theoretical research and practical applications.