Unveiling Biological Models Through Turing Patterns

Turing patterns play a fundamental role in morphogenesis and population dynamics, encoding key information about the underlying biological mechanisms. Yet, traditional inverse problems have largely relied on non-biological data such as boundary measurements, neglecting the rich information embedded in the patterns themselves. Here we introduce a new research direction that directly leverages physical observables from nature--the amplitude of Turing patterns--to achieve complete parameter identification. We present a framework that uses the spatial amplitude profile of a single pattern to simultaneously recover all system parameters, including wavelength, diffusion constants, and the full nonlinear forms of chemotactic and kinetic coefficient functions. Demonstrated on models of chemotactic bacteria, this amplitude-based approach establishes a biologically grounded, mathematically rigorous paradigm for reverse-engineering pattern formation mechanisms across diverse biological systems.