Extended Data Figure 2: Front speeds at critical point.

Sketch illustrating solutions to the boundary value problem in equation (8) for a downstream front near the critical point. a, Eigenvalue s as a function of u_f. u_c is the value of u_f such that q⁻ = q⁺. For this value there are infinitely many possible eigenvalues s, indicated by the thin line. b, c, Phase planes (q, q′) showing solutions for the second order differential equation (8). Downstream fronts are heteroclinic connections from the upper fixed point q⁺ to the lower fixed point q⁰. When u_f = u_c and hence q⁻ = q⁺, the upper fixed point is not hyperbolic and there are infinitely many connections, each corresponding to a value of s. When u > u_c, q⁺ is hyperbolic and there is a unique connection and hence a unique value of s.