Abstract
Parameter identification in pattern formation models from a single
experimental snapshot is challenging, as traditional methods often require
knowledge of initial conditions or transient dynamics -- data that are
frequently unavailable in experimental settings. In this study, we extend the
recently developed statistical approach, Correlation Integral Likelihood (CIL)
method to enable robust parameter identification from a single snapshot of an
experimental pattern. Using the chlorite-iodite-malonic acid (CIMA) reaction --
a well-studied system that produces Turing patterns -- as a test case, we
address key experimental challenges such as measurement noise, model-data
discrepancies, and the presence of mixed-mode patterns, where different spatial
structures (e.g., coexisting stripes and dots) emerge under the same
conditions. Numerical experiments demonstrate that our method accurately
estimates model parameters, even with incomplete or noisy data. This approach
lays the groundwork for future applications in developmental biology, chemical
reaction modelling, and other systems with heterogeneous output.