Figure 9: Image of a marine angelfishFor comparison with numerical simulation we show a grey-scale image of a marine angel fish (excluding tail).
Figure 10: Simulated patterns of a marine angelfishNumerical solution of the steady state activator (u) profile for the Gierer-Meinhardt system (bounding box is [0,2]-by-[0,2]). Associated unstructured mesh has 36803 nodes and 72856 triangles. The standard Galerkin FEM using continuous piecewise linear basis functions was employed with zero flux boundary conditions. The numerical scheme is 2nd order and semi-implicit (in time). The initial conditions were taken to be random variations about the steady states of the corresponding spatially homogeneous system.
Figure 11: Simulated patterns on a schematic mammal coatNumerical solution of the steady state activator (u) profile for the Gierer-Meinhardt system (bounding box is [0,3]-by-[0,2]). Associated unstructured mesh has 7687 nodes and 14768 triangles. The standard Galerkin FEM using continuous piecewise linear basis functions was employed with zero flux boundary conditions. The numerical scheme is 2nd order and semi-implicit (in time). The initial conditions were taken to be random variations about the steady states of the corresponding spatially homogeneous system. |
Reaction-diffusion equations for Turing patternsIn addition to spatiotemporal pattern formation phenomena such as rotating spiral waves and target patterns, there is the possibility of stationary pattern formation ('Turing patterns'). These arise from an initial unpatterned state and relie on significant differences between diffusion coefficients and thus this mechanism is also called 'diffusion driven instability'. Morphogenesis, or embryonic development, is a specific area in biology where numerous reaction-diffusion systems have been used to model the 'pre-pattern' that determines animal skin patterns, for example, in shells, mammals, fish, and butterflies. For concreteness I have been studying the mathematical and numerical analysis of a specific two-component activator-inhibitor system introduced by Gierer and Meinhardt (Kybernetik, Vol.12, pp.30-39, 1972). There are significant challenges in both these tasks. For example, the equation are difficult to analyses mathematically and may even blow-up for certain initial data. And from a numerical point of view standard first order time-stepping schemes are insufficient for accurately capturing the highly oscillatory (in space) solutions.The model reaction-diffusion system has the nondimensional form 
where 'u(x,t)' and 'v(x,t)' denote the activator and inhibitor concentrations at position 'x' and time 't'. The parameters 'r', 'mu', 'alpha', 'Du' and 'Dv' are positive parameters. We assume that the activator diffusion coefficient 'Du' must be smaller than the inhibitor diffusion coefficient 'Dv', which is necessary for Turing patterns to form. In figure 9 we show a grey-scale image of a marine angel fish which is similar to the numerical simulation (Figure 10) of the Gierer-Meinhardt system on the same domain. Another numerical simulation is shown in Figure 11, which shows a series of regularly spaced spots on a schematic mammal coat. |
Last updated: May 18, 2006 by Marcus R Garvie