Fig. 2.

The approximative solution to the transmission dynamics given in Eq. (18) against the full solution to the dynamics obtained numerically by solving the equivalent system with Eqs. (1) and (2), each represented by the coloured solid and dashed lines, respectively, for a range of initial conditions M(t₀). In the left panel we have fixed a value of $R_0=R_{0†}(z,k)+0.1\simeq 2.18$ and in the right panel with a value of $R_0=R_{0†}(z,k)+0.01\simeq 2.09$. The dotted black horizontal lines corresponds to the value of M at the stable (upper) and unstable (lower) equilibria numerically obtained from Eq. (9). Lastly, the grey region corresponds to values for which $|M-M_{†}(z,k)|<1$ and hence the expansion used to obtain Eq. (18) leads to good agreement with the full numerical solution.