# Pattern formation in reaction-diffusion systems in the presence of short-term memory

## TL;DR

This paper investigates how short-term memory influences pattern formation in reaction-diffusion systems, extending traditional models beyond Markovian assumptions to better understand and potentially control spatio-temporal patterns.

## Contribution

It introduces a non-Markovian reaction-diffusion model, derives a complex Ginzburg-Landau equation incorporating memory effects, and demonstrates how memory alters pattern properties and stability.

## Key findings

- Memory significantly affects pattern properties.
- Derived a memory-dependent complex Ginzburg-Landau equation.
- Numerical simulations confirm the impact of memory on patterns.

## Abstract

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatio-temporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrate that memory changes the properties of the spatio-temporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatio-temporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory.