Figure 2.

Effects of excluding freeriders. (a) Between the excluders and freeriders. I: If is smaller than a threshold value , the defectors dominate. II: If is greater than , exclusion leads to bistable competition between the two strategies. With increasing or decreasing the minimal frequency of the excluders outcompeting the defectors decreases. III: If and are sufficiently high and low, the excluders dominate. The parameters are as in figure 1a. (b) In the presence of second-order freeriders. The triangle is as in figure 1b, except that z denotes the excluder frequency and the vertex E corresponds to its homogeneous state. Similarly, the edge EC consists of a continuum of equilibria. Here, we specifically assume and . EC is separated into stable and unstable segments. The coloured area in the interior of triangle is the region in which holds. In fact, given the occasional mutation to a defector, the population's state must converge to the vicinity of the point K₁, because the advantage of the excluders over the cooperators becomes broken when the population's state goes up beyond K₁.