Fig. 4.

Expansions transition from pulled to pushed waves at an intermediate strength of Allee effect within the weak regime. (A) Three different growth profiles displaying increasing magnitude of Allee effect (growth profile 0 is purely logistic; 1 and 2 have a weak Allee effect), but with the same low-density growth rate. Unlike pulled waves, the velocities of expansion in the three cases are not the same. In particular, both logistic (0) and the less-severe Allee effect (1) result in pulled waves with the same velocity, given by $2\sqrt{r_{0}D}$. In contrast, condition 2, with a larger but still weak Allee effect, leads to a pushed wave with velocity greater than $2\sqrt{r_{0}D}$. (B) In a generic model of the Allee effect (SI Appendix, Eq. S13), it can be shown that the transition from pulled to pushed waves occurs at a threshold magnitude of Allee effect that is different from the onset of a weak Allee effect. B shows the theoretical prediction for expansion velocity as the magnitude of the Allee effect (difference between maximal and low density per capita growth rate) is increased slowly, keeping the growth rate at low density constant. No change is seen in the dynamics of the expansion when a weak Allee effect is introduced. However, as the magnitude of the Allee effect is increased further, the expansion dynamics undergo a transition at a threshold magnitude of the Allee effect (vertical red line). The red curve is the actual expansion velocity, and the gray line is the velocity of a pulled wave with the same low density growth rate, r₀.