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Ecological public goods games: cooperation and bifurcation - PubMed

Ecological public goods games: cooperation and bifurcation

Christoph Hauert et al. Theor Popul Biol. 2008 Mar.

Abstract

The Public Goods Game is one of the most popular models for studying the origin and maintenance of cooperation. In its simplest form, this evolutionary game has two regimes: defection goes to fixation if the multiplication factor r is smaller than the interaction group size N, whereas cooperation goes to fixation if the multiplication factor r is larger than the interaction group size N. Hauert et al. [Hauert, C., Holmes, M., Doebeli, M., 2006a. Evolutionary games and population dynamics: Maintenance of cooperation in public goods games. Proc. R. Soc. Lond. B 273, 2565-2570] have introduced the Ecological Public Goods Game by viewing the payoffs from the evolutionary game as birth rates in a population dynamic model. This results in a feedback between ecological and evolutionary dynamics: if defectors are prevalent, birth rates are low and population densities decline, which leads to smaller interaction groups for the Public Goods game, and hence to dominance of cooperators, with a concomitant increase in birth rates and population densities. This feedback can lead to stable co-existence between cooperators and defectors. Here we provide a detailed analysis of the dynamics of the Ecological Public Goods Game, showing that the model exhibits various types of bifurcations, including supercritical Hopf bifurcations, which result in stable limit cycles, and hence in oscillatory co-existence of cooperators and defectors. These results show that including population dynamics in evolutionary games can have important consequences for the evolutionary dynamics of cooperation.

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Figures

Figure 1
Figure 1

Dynamics of homogeneous defector a and cooperator b populations interacting in ecological public goods games with N = 8, r = 4. The location of stable (solid line) and unstable (dashed line) fixed points are shown for different death rates d and a baseline birthrate of b = 1 (the horizontal dotted line marks d = b). a Defectors cannot survive on their own if d > b and the population goes extinct. However, for d < b defectors persist at an equilibrium density 1 − d/b. b Cooperators can handle much higher death rates due to the fitness benefits from cooperation. However, this requires sufficiently high population densities because otherwise cooperative interactions are too rare. The threshold density is indicated by the unstable equilibrium at low densities. For b > d this threshold disappears and cooperators persist irrespective of their initial density. Note that the equilibrium density of cooperators is always substantially higher than that of defectors because of the higher fitness of cooperators.

Figure 2
Figure 2

Bifurcations and dynamical regimes (separated by dashed lines) in heterogeneous populations of cooperators and defectors interacting in ecological public goods games with N = 8, b = 1 and the multiplication factor r as the bifurcation parameter. The solid line indicates the location of the interior fixed point Q as a function of r. a For b < d (b = 1, d = 1.2) increasing r produces the following dynamical scenarios (see text for details): (a) no Q, extinction; (b) Q unstable node, extinction; (c) Q unstable focus, extinction. A Hopf bifurcation occurs between regimes (c) and (d). If it is supercritical, stable limit cycles appear as r approaches rHopf and cooperators and defectors can co-exist in ever lasting oscillations. (d) Q stable focus, coexistence. If the Hopf bifurcation is subcritical, the basin of attraction of Q is bounded by an unstable limited cycle close to the bifurcation point (r > rHopf). (e) Q stable node, co-existence; (f) no Q, cooperation. For unfavorable initial conditions, the population goes extinct even for (d)–(f). b The dynamics is considerably less rich for b > d (b = 1, d = 0.8) but extinction no longer occurs. The following scenarios are observed for increasing r: (a) no Q, defection; (b) Q stable, co-existence; (c) no Q, cooperation.

Figure 3
Figure 3

Hopf bifurcations and first Lyapunov coefficient l1 as a function of the group size N and the death rate d for different birth rates b. For sufficiently large d, Hopf bifurcations no longer occur. This threshold corresponds to the condition that the interior fixed point Q exists, i.e. q̂ < 1. a In the special case with b = 0 the Hopf bifurcation can be analyzed in detail (see text) and occurs at r=rHopf=N(N1N−1−1). For N < N* ≈ 8.493 the bifurcation is supercritical and stable limit cycles are observed for r slightly smaller than rHopf. Similarly, for N > N* the bifurcation is subcritical as reflected in unstable limit cycles for r slightly above rHopf. b Setting b = b0 = 1 − r/N denotes the minimal b that ensures compatibility with the interpretation of Eq. (6) in terms of per capita birth and death rates. For very small d no Hopf bifurcations occur because d < b holds (c.f. Fig. 2b). The region of supercritical Hopf bifurcations and stable limit cycles is considerably smaller and for N ≥ 7 all bifurcations are subcritical. c Setting b = 1 corresponds to the minimal birthrate that is compatible with all group sizes N. In that case, stable limit cycles no longer occur. Instead, the Hopf bifurcation is always subcritical and accompanied by unstable limit cycles.

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