Two interacting populations with self-interactions and incompatible preferences, playing snowdrift games.
The corresponding vector fields (small arrows), sample trajectories (large arrows) and phase diagrams (colored areas) were determined for B0 and C0. The flow lines move away from unstable stationary points (empty circles) and are attracted towards stable stationary points (black circles and solid diagonal line). Saddle points (crosses) are attractive in one direction, but repulsive in another. The representation is the same as in Fig. 2. In particular, the colored areas represent the basins of attraction, i.e. all initial conditions (p(0),q(0)) leading to the same stable fix point [red = (0,0), salmon = (u,0), mustard = (v,1), rainbow colors = (u,v), with 0u, v1]. The model parameters are as follows: (A) |B| = |C| = 1 and f = 0.8, i.e. population 1 is more powerful than population 2, (B) |C| = 2|B| = 2 and f = 1/2, i.e. both populations are equally strong, (C) |C| = 2|B| = 2 and f = 0.8, (D) 2|C| = |B| = 2 and f = 0.8. (A) In the multi-population snowdrift game (MSD), a mixture of cooperative and uncooperative behaviors results in both populations, if |B| = |C|. (B) For |B||C| and equally strong populations, everybody ends up with non-cooperative behavior in each population. (C) For |B||C| and , the weaker population 2 forms a “tacit alliance” with the minority of the stronger population 1 and opposes its majority. (D) Same as (C), but now, all individuals in the weaker population 2 show their own preferred behavior after the occurrence of a “revolutionary” transition, during which the stable stationary solution (the evolutionary equilibrium) changes discontinuously from (u,0) to (v,1).