Evolutionary games and population dynamics: maintenance of cooperation in public goods games

Abstract

The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behaviour seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the Prisoner's Dilemma and public goods games. Traditional approaches to studying the problem of cooperation assume constant population sizes and thus neglect the ecology of the interacting individuals. Here, we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation. In public goods games, cooperation can gain a foothold if the population density depends on the average population payoff. Decreasing population densities, due to defection leading to small payoffs, results in smaller interaction group sizes in which cooperation can be favoured. This feedback between ecological dynamics and game dynamics can generate stable coexistence of cooperators and defectors in public goods games. However, this mechanism fails for pairwise Prisoner's Dilemma interactions and the population is driven to extinction. Our model represents natural extension of replicator dynamics to populations of varying densities.

Figure 1

Population dynamics of cooperators engaging in public goods interactions in absence of defectors for different death rates d. Cooperators are unable to survive for $d>d_{\text{max}}=(r-1)(N-1)N^{-N/(N-1)}$ and the only stable equilibrium is $x_0=0$. However, for $d<d_{\text{max}}$, the system undergoes a bifurcation and two interior equilibria appear: one stable branch at higher x (solid line) and one unstable branch at lower x (dashed line). Consequently, cooperators thrive at sufficiently high densities but go extinct otherwise, i.e. approach $x_0$. For d=0, the equilibrium $x_0$ becomes unstable and the system converges to x=1. The dynamics is illustrated for N=5 and r=3.

Figure 2

Full analysis of the population dynamics of cooperators and defectors engaging in public goods games in groups of N individuals. Four panels illustrate the different dynamical scenarios. The phase space is spanned by the population density $x+y$ (or $1-z$) and the relative fraction of cooperators $f=x/(x+y)$. The left boundary (z=1) is attracting and consists of a line of stable fixed points (filled circles), which represent states where the population cannot maintain itself and disappears. Conversely, the right boundary, which denotes the maximal population density (z=0), is repelling. Along the bottom boundary, i.e. in absence of cooperators (f=0), population densities decrease and eventually vanish. Finally, along the top boundary, i.e. in absence of defectors (f=1), there are two fixed points that are either both saddle points (open circles) as in (a), (c), (d) or a saddle point and a stable equilibrium (filled circle) as in (b). In addition, there may be an interior focus Q present. Q is stable if it lies to the right of the dashed line, which marks $\tilde{z}=N^{-1/(N-1)}$. (a) Q is stable and cooperators and defectors can coexist. Trajectories spiral towards Q, except for low initial population densities or abundant defection, in which case the population goes extinct. (b) Increasing death rates push the unstable Q upwards until it disappears. This leads to persistence of cooperators and elimination of defectors. The population vanishes for low initial population densities or abundant defection. (c) For smaller multiplication factors r, Q becomes unstable and the population always goes extinct. Trajectories originating in the vicinity of Q approach z=1 in an oscillatory manner with increasing amplitude. (d) When eliminating the unstable focus Q by increasing d, the population continues to go extinct but the oscillatory dynamics has disappeared. In summary, for increasing d, the dynamics changes from the left to the right column, whereas for decreasing r, it changes from the top to the bottom row. The different scenarios are illustrated for N=8 and (a) $r=3,d=0.5$; (b) $r=5,d=1.6$; (c) $r=2.7,d=0.5$; (d) $r=2.1,d=0.5$.