Parameter Identification in Reaction-Diffusion-Chemotaxis Systems from Single Turing Pattern Amplitudes

Abstract

Turing patterns encode key information about biological mechanisms, yet traditional inverse problems rely on non-biological data like boundary measurements, neglecting the patterns themselves. We introduce a new direction that directly uses the amplitudes of Turing patterns for parameter identification. Motivated by stripe-like biological patterns, we study two 1D models: one with density-dependent chemotaxis ($\chi (n,c)={\chi }_{0}n$) and one with ratio-dependent chemotaxis ($\chi (n,c)={\chi }_{0}n/c$). We present a framework that uses the spatial amplitude profile of a stationary pattern to recover system parameters, including wavelength, diffusion constants, and chemotactic and kinetic coefficients---offering a biologically grounded paradigm for reverse-engineering pattern formation. The core contribution is a proof-of-concept showing that, under a Fourier truncation at mode $M=3$, the amplitude data can reconstruct the parameter combinations $\{d_n{k}^2,d_c{k}^2,{\chi }_0{k}^2,r\}$ for both models. These uniquely determine the ratios $d_c/d_n$ and ${\chi }_0/d_n$, while individual parameters are recovered up to an overall scale. This is a purely theoretical work; numerical validation and stability analysis are left for future research. The framework is extensible to other models with suitable modifications.

Keywords: Inverse reaction-diffusion equations, Turing patterns, Turing instability, periodic solutions, sinusoidal form