Fig. 3.

With very disparate growth rates, essentially only the fast-growing strategy changes its abundance until carrying capacity is reached. At carrying capacity, the difference in growth rates becomes ineffective, such that the structure of the payoff matrix, A, determines the dynamics. (a) If all payoffs are the same, $a_{11}=a_{12}=a_{21}=a_{22}=10$, then the dotted line $x_1+x_2=10$ is a continuum of equilibria. Thus, starting with an initial population composition, x₁ remains more or less constant and x₂ adjusts such that carrying capacity is reached—given that x₁ is not too low and x₂ not too high, initially. (b) If the complete symmetry in the payoffs is broken, $a_{11}=9.9$ and $a_{12}=a_{21}=a_{22}=10$, all trajectories move to a slow manifold (bold line) close to $x_1+x_2=10$ relatively quickly. Trajectories are nearly horizontal since x₁ grows much faster than x₂. At this manifold, they slowly converge to the global attractor (0,10), because the payoff configuration favors strategy 2.