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The Red King effect: when the slowest runner wins the coevolutionary race - PubMed

  • ️Wed Jan 01 2003

The Red King effect: when the slowest runner wins the coevolutionary race

Carl T Bergstrom et al. Proc Natl Acad Sci U S A. 2003.

Abstract

Mutualisms provide benefits to those who participate in them. As a mutualism evolves, how will these benefits come to be allocated among the participants? We approach this question by using evolutionary game theory and explore the ways in which the coevolutionary process determines the allocation of benefits in mutualistic interactions. Motivated by the Red Queen theory, which states that coevolutionary processes favor rapid rates of evolution, we pay particular attention to the role of evolutionary rates in the establishment of mutualism and the partitioning of benefits among mutualist partners. We find that, contrary to the Red Queen, in mutualism evolution the slowly evolving species is likely to gain a disproportionate share of the benefits. Moreover, population structure serves to magnify the advantage to the slower species.

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Figures

Figure 1
Figure 1

Evolutionary trajectories for the mutualism game, determined numerically from Eq. 1. Trajectories above the diagonal (red) lead to the equilibrium favoring species 1; trajectories below the diagonal (blue) lead to the equilibrium favoring species 2. The diagonal (yellow) is the separatrix between the two domains of attraction. The vertical and horizontal lines (gray) show the places at which the change in strategy frequency switches direction, for species 1 and 2, respectively. The game parameters determine the positions of these lines.

Figure 2
Figure 2

The effect of evolutionary rate on domains of attraction when k = 1.5. (Left) Species 1 evolves eight times slower than species 2. (Center) Equal rates of evolution. (Right) Species 2 evolves eight times slower. In this game, the slower-evolving species has the larger domain of attraction around its favored equilibrium.

Figure 3
Figure 3

Summary of the local dynamics. Upper right quadrant (shaded): slower-evolving species reaches its favored equilibrium. Lower left quadrant: faster-evolving species reaches its favored equilibrium. Upper left and lower right quadrants: evolutionary rates do not affect outcome.

Figure 4
Figure 4

Domains of attraction for the equilibrium favoring the fast evolver (species 1, red) and the slow evolver (species 2, blue) in the local dynamics (Left) and the global dynamics (Right). Domains of attraction are equal in size under the local dynamics, but the domain of attraction around the equilibrium favored by the slow evolver increases in the global dynamics. Parameters: species 1 evolves eight times faster than species 2, and k = 1. In the global dynamics, each species' carrying capacity at its favored equilibrium is four times that at the disfavored equilibrium and each new patch is founded by nine individuals. Local domains of attraction are computed as in the section on the local dynamics of mutualism. Global domains of attraction are found by exact numerical solution. Given the starting strategy frequencies for each species, the fraction of patches of each type (i.e., with each possible distribution of founders) has a bivariate binomial distribution. Patch types above the local separatrix go to equilibrium 1 and those below the separatrix go to equilibrium 2, and from this we find the fraction of patches reaching each equilibrium at the end of the season. This gives the new global strategy frequencies; we then apply the mapping shown in Fig. 6 to determine whether the global system will ultimately converge to the equilibrium favoring species 1 or to that favoring species 2.

Figure 5
Figure 5

Summary of the global dynamics. Each subpopulation ends with either species 1 playing Generous and species 2 playing Selfish, or vice versa. The fraction of subpopulations in each state (upper horizontal line) at the end of one season determines the expected frequencies in each new subpopulation at the start of the subsequent season. Carrying capacity ratios are α = 4 and β = 4.

Figure 6
Figure 6

Global dynamics. The solid curve maps the fraction of patches at the equilibrium favored by slowly evolving species 2 in one generation to the fraction of such patches in the next generation. The dashed line has slope 1, for reference. The unstable equilibrium occurs at the intersection of these curves, ≈0.273. Parameters are as in Fig. 4. The position of this equilibrium does not depend strongly on the number of founders in each patch, because the advantage to the slowly evolving species does not derive from stochastic variation in patch composition.

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