20) $$ \frac\rm d \phi\rm d t = – \left( \mu\nu + \beta + \frac12 \xi N \right) \fraczN \theta + \left( \beta – \frac12 \xi z – \frac1N\frac\rm d N\rm d t \right) \phi . \\ $$ (5.21)These equations have the symmetric steady-state given by θ = 0 = ϕ and c, z, N satisfying $$ c = \frac\mu\nu z2\mu+\alpha N , \qquad z = \frac2\beta N (2\mu+\alpha N) , , $$ (5.22)from Eqs. 5.17 and 5.19. Note that the steady state value of N will depend upon the initial conditions, it is not determined by Eq. 5.18. This is because

the steady-state equations obtained by setting the time derivatives in

Eqs. 5.17–5.19 are not independent. The difference (Eqs. 5.18 and 5.19) is equal to z/N MEK activation times the sum (Eqs. 5.17 + 5.19). In “Asymptotic Limit 1: β ≪ 1” and “Asymptotic Limit 2: α ∼ ξ ≫ 1” below, so as to discuss the stability of a solution in the two asymptotic regimes β ≪ 1 and α ∼ ξ ≫ 1, we augment the steady-state Eqs. 5.17–5.19 with the condition \(\varrho=2N^2/z\), with \(\varrho\) assumed to be \(\cal O(1)\). The linear stability of θ = 0 = ϕ is given by assuming θ and ϕ are small, yielding the system $$ \displaystyle\frac\rm d\rm d t \left( \beginarrayc \theta \\[2ex] \phi \endarray \right) = \left( \beginarraycc – \left( \displaystyle\frac2\mu cz + \displaystyle\frac\xi z2 + \displaystyle\frac\beta zN + \displaystyle\frac\beta Nz \right) & \left(\displaystyle\frac\beta Nz + \displaystyle\frac1N – \displaystyle\frac\xi N2 \right) \\ – ( \mu \nu + \beta + \displaystyle\frac12 \xi N ) \displaystyle\fraczN & \left( \beta + \mu\nu – \displaystyle\frac2\mu cz \right) \displaystyle\fraczN \endarray \right) \left( \beginarrayc \theta \\[2ex] \phi \endarray \right) . $$ (5.23)An instability of the symmetric solution is indicated by the determinant of this matrix being negative. Substituting Eq. 5.22 into the determinant, yields $$ \mboxdet = \frac \beta \mu \nu ( 4 \beta \mu – \alpha \xi N^2 ) 4\beta\mu + 2 \alpha \beta N + 2 \mu \xi N + 2 \alpha \mu \nu N + \alpha \xi N^2 . $$ (5.24)Hence we find that the symmetric (racemic) state is unstable if \(N > 2 \sqrt \mu\beta / \alpha \xi \), that is, large aggregation rates (α, ξ) and slow grinding (β) are preferable for symmetry-breaking. We consider two specific asymptotic limits of parameter values so as to derive specific results for steady-states and conditions on stability. In both limits, we have that the aggregation rates dominate fragmentation (α ∼ ξ ≫ β), so that the system is strongly biased towards the formation of crystals and the dimer concentrations are small.