Applied Analysis Seminar Variational structure, bifurcations, and pattern selection in a new mechanochemical model of Hydra

A central question in mathematical biology is how complex biological patterns emerge from simple physical and biochemical interactions. Understanding the mechanisms underlying symmetry breaking and organiser formation remains a major challenge, requiring a close interplay between experiments and mathematical modelling.

In this talk, we will address these questions using the freshwater polyp Hydra, a classical model organism for regeneration and pattern formation. We will briefly review different concepts proposed to explain symmetry breaking in Hydra and discuss the mathematical challenges they raise. The main focus will be a new mechanochemical PDE model inspired by experiments on regenerating epithelia. The model couples morphogen dynamics and tissue mechanics through a feedback loop between tissue stretching and morphogen production. For an exponential elasticity–morphogen coupling, the system admits a variational structure, providing a rigorous analytical framework for studying pattern formation. Bifurcation and stability analysis further reveal robust symmetry breaking, the emergence of stable single-peaked organiser-like patterns, and bistable regimes generated by fold bifurcations. Unlike classical Turing mechanisms, this patterning principle does not require long-range diffusive inhibition or a second diffusible inhibitor.

Theoretical predictions are in close agreement with experimental observations and suggest that mechanical feedback may represent a fundamental principle of epithelial pattern formation. The Hydra example illustrates how mathematical modelling can help uncover general principles of biological self-organisation while opening new questions in the analysis of nonlinear PDEs and pattern selection.