Turing patterns on growing surfaces
We investigate models for biological pattern formation via reaction-diﬀusion systems posed on continuously evolving surfaces. The nonlinear reaction kinetics inherent in the models and the evolution of the spatial domain mean that analytical solutions are generally unavailable and numerical simulations are necessary. In the ﬁrst part of the talk, we examine the feasibility of reaction-diﬀusion systems to model the process of parr mark pattern formation on the skin surface of the Amago trout. By simulating a reaction-diﬀusion system on growing surfaces of diﬀering mean curvature, we show that the geometry of the surface, speciﬁcally the surface curvature, plays a central role in the patterns generated by a reaction-diﬀusion mechanism. We conclude that the curvilinear geometry that characterises ﬁsh skin should be taken into account in future modelling endeavours. In the second part of the talk, we investigate a model for cell motility and chemotaxis. Our model consists of a surface reaction-diﬀusion system that models cell polarisation coupled to a geometric evolution equation for the position of the cell membrane. We derive a numerical method based on surface ﬁnite elements for the approximation of the model and we present numerical results for the migration of two and three dimensional cells.