  We present a fully adaptive multiresolution scheme for spatially
two-dimensional, possibly degenerate reaction-diffusion systems, focusing on
combustion models and models of pattern formation and chemotaxis in
mathematical biology. Solutions of these equations in these applications
exhibit steep gradients, and in the degenerate case, sharp fronts and
discontinuities. The multiresolution scheme is based on finite volume
discretizations with explicit time stepping. The multiresolution representation
of the solution is stored in a graded tree. By a thresholding procedure, namely
the elimination of leaves that are smaller than a threshold value, substantial
data compression and CPU time reduction is attained. The threshold value is
chosen optimally, in the sense that the total error of the adaptive scheme is
of the same slope as that of the reference finite volume scheme. Since chemical
reactions involve a large range of temporal scales, but are spatially well
localized (especially in the combustion model), a locally varying adaptive time
stepping strategy is applied. It turns out that local time stepping accelerates
the adaptive multiresolution method by a factor of two, while the error remains
controlled.
