Simulation results for two interacting populations with self-interactions and incompatible preferences, playing stag hunt games.
The corresponding vector fields (small arrows), sample trajectories (large arrows) and phase diagrams (colored areas) were determined for B0 and C0. 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 (stationary solutions) [yellow = (1,1), blue = (0,1), green = (1,0)]. The dashed diagonal line represents an infinite number of unstable fix points. 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. Due to the asymptotically stable fix points at (1,0) and (0,1), all individuals of both populations finally show the behavior preferred in population 1, when starting in the green area, or the behavior preferred in population 2, when starting in the blue area. This case can be considered to describe the evolution of a shared behavioral norm. Only for similarly strong populations (f1/2) and similar initial fractions p(0) and q(0) of cooperators in both populations (yellow area), both populations will end up with population-specific norms (“subcultures”), corresponding to the asymptotically stable point at (1,1). The route towards the establishment of a shared norm may be quite unexpected, as the flow line starting with the white circle shows: The fraction q(t) of individuals in population 2 who are uncooperative from the viewpoint of population 1 may grow in the beginning, but later on go down dramatically. Therefore, a momentary trend does not allow one to easily predict the final outcome of the struggle between two interest groups.