By considering the two cases (did4 — d2d3) < 0 and (did4 — d2d3) > 0 show that the minimum of H does not lie in the first quadrant of the kx2 — ky2 plane and that diffusion-driven instability will first occur, for increasing ratios d3/d1, d4/d2 on one of the axial boundaries of the positive quadrant.

By setting k^, k2 equal to 0 in turn in the expression for H show that the conditions for diffusion-driven instability are d4 fu + d2gv > 0, d3 fu + dx g V > 0 (d4 fu - d2gv)2 + 4d2d4 fvgu > o, (d3 fu - d1 gv)2 + 4d1d3 fvgu > 0

so for it to occur (d3/d1) > dc and/or (d4/d2) > dc where 1

Now consider the rectangular domain 0 < x < a, 0 < y < b with zero flux boundary conditions with a, b constants with a sufficiently greater than b so that the domain is a relatively thin rectangular domain. Show that it is possible to have the first unstable mode 2 bifurcation result in a striped pattern along the rectangle if the diffusion coefficient ratio in one direction exceeds the critical ratio. (Such a result is what we would expect intuitively since if only one ratio, d4/d2 > dc, then the diffusion ratio in the x-direction is less than the critical ratio and we would expect spatial variation only in the y-direction, hence a striped pattern along the rectangle. A nonlinear analysis of this problem shows that such a pattern is stable. It further shows that if both ratios exceed the critical ratio a stable modulated (wavy) stripe pattern solution can be obtained along the rectangle.)

11. Suppose that a two-species reaction diffusion mechanism in u and v generates steady state spatial patterns U (x ), V (x ) in a one-dimensional domain of size L with zero flux boundary conditions ux _ vx _ 0 at both boundaries x _ 0 and x _ L. Consider the heterogeneity functions defined by

Hg(w) _ — wx dx, Hs(w) _ — [Wx - Hg(w)]2 dx. L 0 L 0

Biologically the first of these simply measures the gradient while the second measures the deviation from the simple gradient. Show that the heterogeneity or energy integral

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