Optimization of Spatiotemporal Dynamics in the Gierer-Meinhardt Model using a PD Control Strategy

While the spatiotemporal dynamics of reaction-diffusion systems have been extensively studied, the control of spatiotemporal evolution dynamics in reaction-diffusion systems remains an unresolved issue. The Gierer-Meinhardt reaction-diffusion system is a typical mathematical model to describe chemical and biological phenomena. By controlling the Turing patterns within the system, it is possible to make relatively accurate predictions and regulate the development trends of the reactants. However, there has been relatively limited research on the control of Turing patterns in Gierer-Meinhardt systems. In this paper, we propose a proportional differential control strategy for the Gierer-Meinhardt system with diffusion terms, aiming to regulate the spatiotemporal dynamics of the model through the controller.

I. Introduction

In 1952, Turing first utilized reaction-diffusion equations to elucidate the process of pattern formation on biological surfaces [1]. Reaction-diffusion systems can be employed to describe differential and spatial patterns in the fields of biology and chemistry. Turing patterns emerge from the interplay of reactions and diffusion, having the potential to disrupt the uniform stability of reaction-diffusion models and give rise to the formation of Turing patterns. This phenomenon is referred to as diffusion-induced Turing instability. Inspired by Turing’s work, the Gierer-Meinhardt (GM) reaction-diffusion model was introduced to investigate relatively simple molecular mechanisms involved in autocatalysis and cross-catalysis in 1972 [2]. In 1974, Gierer and Meinhardt derived the sufficient conditions to ensure the formation of spatial patterns [3]. Since then, the GM model has found extensive application in modeling various biological and chemical reaction processes.