Evolutionary Games and Population Dynamics
by Christoph Hauert, Version 1.0, October 2006.
- » Ecology
The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behavior 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 the problem of cooperation based on the replicator dynamics assume constant (infinite) 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 whenever the population density depends on the average population payoff. Defection decreases the population density, due to small payoffs, resulting in smaller interaction group sizes in which cooperation may be favoured. This feedback between ecological dynamics and game dynamics generates fascinating and rich dynamical behavior, including Hopf bifurcations accompanied by stable and unstable limit cycles. Our model represents natural extension of replicator dynamics to populations of varying densities.
These tutorials complement and illustrate several scientific articles co-authored with Michael Doebeli, Joe Yuichiro Wakano, Miranda Holmes and Martin A. Nowak. It provides interactive Java applets to visualize and experiment with the system's dynamics for parameter settings of your choice.
Further scenarios will be added as research progresses into this fascinating and rather unexplored field of feedback mechanisms between game theoretical interactions and population densities.
In infinite populations where individuals randomly interact in public goods games, cooperators are doomed and readily disappear. In contrast, varying population densities can lead to stable coexistence of cooperators and defectors in public goods games. When increasing the multiplication factor r of the public good the system undergoes a series of bifurcations and the dynamics ranges from extinction, to periodic oscillations and finally stable co-existence.
Spatial 'reaction-diffusion' dynamics promotes cooperation based on different types of pattern formation processes. Individuals can migrate (diffuse) in order to populate new territories. Slow diffusion of cooperators fosters aggregation in highly productive patches (activation), whereas fast diffusion enables defectors to readily locate and exploit these patches (inhibition). These antagonistic forces promote co-existence of cooperators and defectors in static or dynamic patterns, including spatial chaos of ever changing configurations.
In order to combine game dynamics and population dynamics in a replicator equation we assume that x denotes the density of cooperators, y the density of defectors and z=1-x-y the abundance of empty space. Thus, x+y denotes a normalized population density such that for x+y=0 (or z=1) the population has gone extinct. The dynamics of x, y and z is determined by the average payoffs (or fitness) of cooperators fC and defectors fD arising from game theoretical interactions. Cooperators and defectors are assumed to die at a constant rate d and give birth according to a constant baseline birth rate b augmented by their performance fC and fD. In addition, birth events are conditional on the availability of empty space and hence are proportional to z. This leads to the following population dynamic model:
dy/dt = y (z (b+fD)-d)
dz/dt = -dx/dt -dy/dt
This system of equations represents a natural extension of the replicator dynamics. If the population density x+y is kept constant (dz/dt = 0) by adjusting the death rate accordingly, i.e. by setting d = z(b+f/(1-z)), where f = x fC+y fD denotes the mean fitness, the traditional replicator dynamics is recovered (upon normalizing x+y=1). The average payoffs fC and fD are determined by the actual game theoretical interactions under consideration.
This work was first published in
Hauert, Ch., Holmes, M. & Doebeli, M. (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond B 273, 2565-2570. Corrigendum: Proc. R. Soc. Lond B 273, 3131-3132.