ABSTRACT
Suppose that parameters k1, . . . , k4, V1, V2 have been chosen to satisfy the inequality requirements given in this paper for pattern formation and fixation. Temporarily setting V1 and V2 to zero to look at the corresponding eigenvalues in the absence of diffusion, both eigenvalues will have negative real part, since parameters were chosen inside the pattern region. Thus,
form and become fixed due to linear effects it is necessary and sufficient that the real part of at least one eigenvalue become positive. Since increasing diffusion levels from 0 to V1 and V2, respectively, decreases the quantity outside the radical on the righthand side of (28), it must be the case that the radical is real-valued in the presence of diffusion at levels V1 and V2, with the radicand increased enough by the presence of this diffusion to push the real part of one eigenvalue positive. Thus, with the diffusion at levels V1 and V2, the eigenvalues are real-valued, with 1 > 0 and 2 < 0. It is the positive
1 which is responsible for the growth of patterns, as solutions corresponding to 1 will grow, rather than decay, with time.
