14.3 Turing instability
Turing’s insight was that diffusion, which smooths a single substance, can destabilise a uniform state of two. Consider \[ u_t = \Delta u + f(u, v), \qquad v_t = d\,\Delta v + g(u, v) , \] with a uniform steady state \((u^*, v^*)\), \(f = g = 0\) there, which is stable without diffusion: the Jacobian \(J = \begin {pmatrix} f_u & f_v \\ g_u & g_v \end {pmatrix}\) at \((u^*, v^*)\) has \(\operatorname {tr}J < 0\) and \(\det J > 0\). Perturbations \(e^{\sigma t + ik\cdot x}\) grow at the rates \(\sigma \) that are eigenvalues of \(J_k = J - |k|^2\operatorname {diag}(1, d)\).
Theorem 14.3.1 (Turing). The uniform state is unstable to some wavenumber \(k\) if and only if \[ df_u + g_v > 2\sqrt {d\det J} . \] It requires \(f_u\) and \(g_v\) of opposite signs; if \(u\) is the species that activates itself (\(f_u > 0\)), it requires \(d > 1\): the inhibitor must diffuse faster. The most unstable wavenumber is near \(|k|^2 = (df_u + g_v)/2d\).
Proof. \(\operatorname {tr}J_k = \operatorname {tr}J - (1 + d)|k|^2 < 0\) for all \(k\), so instability needs a real positive eigenvalue, that is \(\det J_k < 0\). Now \(h(q) = \det J_k = dq^2 - (df_u + g_v)q + \det J\) with \(q = |k|^2 \geq 0\) is a quadratic with \(h(0) = \det J > 0\); it is negative somewhere on \(q > 0\) exactly when its minimum, at \(q = (df_u + g_v)/2d\), is negative and that point is positive, which is the stated inequality. Subtracting \(f_u + g_v < 0\) from \(df_u + g_v > 0\) gives \((d - 1)f_u > 0\), so \(f_u > 0\) forces \(d > 1\), and then \(g_v < -f_u < 0\). □
For the Schnakenberg kinetics \(f = a - u + u^2v\), \(g = b - u^2v\) with \(a = 0.1\), \(b = 0.9\), the steady state is \(u^* = 1\), \(v^* = 0.9\), and \(f_u = 0.8\), \(f_v = 1\), \(g_u = -1.8\), \(g_v = -1\), so \(\operatorname {tr}J = -0.2\), \(\det J = 1\): stable without diffusion. The condition becomes \(0.8d - 1 > 2\sqrt d\), that is \(d > 8.57\) (Problem 14.6.3). Above the threshold a band of wavenumbers grows and the nonlinearity saturates it into a steady periodic pattern (Figure 17). In two dimensions the same mechanism gives spots or stripes, and patterns of this kind have been found in chemical reactors and in the development of digits, hair follicles and the stripes of fish.
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