Numerical simulations of reaction-diffusion systems with Neumann boundary conditions (NBC) on growing domains by Maini et al. exhibit square and roll patterns that are usually associated with bifurcations from a trivial equilibrium on a square lattice. However, these patterns change as the domain grows. In this talk, I will discuss several of these transitions between different types of squares and between squares and rolls, using tools from bifurcation theory with symmetry and dynamical systems. To understand these transitions, we will need to discuss two issues: the speed at which the domain size changes and the relations between NBC and periodic boundary conditions (PBC) on a square.
We have found that a generic continuous transition can occur between two types of squares. Also, the transition between squares and rolls can generically occur either via steady-states and time-periodic states (standing waves), or via a jump. Moreover, interestingly, the symmetry constraints induced by NBC are important in understanding which solutions exist and which solutions are stable. Therefore, I will also point out interesting differences between the NBC and PBC problems.
Our future goal is to use this research work to construct reaction-diffusion systems on growing square domains that show prescribed transitions between different patterns.