Figure 9: Image of a marine angelfish
For comparison with numerical simulation we show a greyscale image
of a marine angel fish (excluding tail).
Figure 10: Simulated patterns of a marine angelfish
Numerical solution of the steady state activator (u) profile for the
GiererMeinhardt 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 semiimplicit (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 coat
Numerical solution of the steady state activator (u) profile for the
GiererMeinhardt 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 semiimplicit (in time). The initial conditions were taken
to be random variations about the steady states of the corresponding
spatially homogeneous system.

Reactiondiffusion equations for Turing patterns
In 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 reactiondiffusion
systems have been used to model the 'prepattern' 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
twocomponent activatorinhibitor system introduced by Gierer and Meinhardt
(Kybernetik, Vol.12, pp.3039, 1972). There are significant challenges in both
these tasks. For example, the equation are difficult to analyses mathematically and
may even blowup for certain initial data. And from a numerical point of view
standard first order timestepping schemes are insufficient for accurately capturing
the highly oscillatory (in space) solutions.
The model reactiondiffusion 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 greyscale image of a marine angel fish which is similar to
the numerical simulation (Figure 10) of the GiererMeinhardt 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.
