Fig. 4.

The average wave speed dependence on the carrying capacity is shown for the stochastic model, as well as the hybrid model for thresholds of Θ = 10,25 and 100 in (a). The dashed–dotted graph of the hybrid model with a threshold of Θ = 100 coincides with the solid line representing the stochastic model. The wave speed of the PDE is c = 2, but the discrete mean field equations can slightly deviate from this value in an Ω-independent way. (b) shows, for a fixed carrying capacity of Ω = 100, explicitly the threshold dependence for several values of Θ, and the solid line is a fit via the function $c=a_1+a_2\mathit{Erf}(a_3⁎\mathit{Log}(\mathrm{\Theta })+a_4)$, where Erf is the error function and we obtained $a_1=1.5$, $a_2=0.13$, $a_3=1.4$, $a_4=-3.0$. The other parameters are h = D = λ = 1, and all stochastic and hybrid results are obtained from averaging 256 different simulations.