COMMENTARY Limits of Hamilton ’ s rule

The evolution of cooperation is a fundamental problem in evolutionary biology. Over the last decades a wealth of models and mechanisms have been proposed for explaining how cooperators can thrive under Darwinian selection. At the same time, discussions of the conceptual connections between the different approaches have often been neglected. The synthesis proposed by Lehmann & Keller (2006) is therefore a welcome contribution to the literature on the evolution of cooperation. Their framework for understanding the evolution of cooperative traits is based on the fitness gradient. A particular mechanism is said to favour cooperation if it generates a positive fitness gradient towards higher values of the cooperative trait. The proposed framework is based on an extension of Hamilton’s rule that is obtained by adjusting and reinterpreting costs, benefits and genetic relatedness. While such an approach may be useful in many circumstances, we would like to point out that if selection on cooperation is frequency-dependent, the classification given by Lehmann & Keller (2006) is not applicable in an interesting class of evolutionary scenarios. When fitness gradients are determined entirely by processes that are not affected by the current state of the population, they remain constant as long as the environment stays the same, and hence evolution can only come to a halt due to exhaustion of genetic variation. However, when selection is frequency-dependent, fitness gradients depend on the current state of the population and hence change as the population evolves. In this case, the evolutionary dynamics generally converges to points in phenotype space where fitness gradients are zero. Such points are called singular points in the framework of adaptive dynamics (Dieckmann & Law, 1996; Metz et al., 1996; Geritz et al., 1998). After convergence to a singular point, the further development of the evolutionary process is determined by the second derivative of the fitness function (since the first derivative, i.e. the fitness gradient, is zero at the singular point). In particular, if the second derivative is positive, the singular point represents a fitness minimum, and evolutionary branching, that is, a splitting of the evolving population into two diverging phenotypic clusters, is a possible outcome. It has been shown that such convergence to fitness minima and subsequent evolutionary branching is a generic outcome of frequency-dependent selection in many different types of models (e.g. Metz et al., 1996; Geritz et al., 1998; Doebeli & Dieckmann, 2000; Kisdi & Gyllenberg, 2005). If evolutionary branching occurs in cooperative traits, this would imply that if the population is at the evolutionary branching point (i.e. at the singular point at which the fitness function has a minimum), mutants with both more cooperative and less cooperative traits can invade. Therefore, the classification suggested by Lehmann & Keller (2006) cannot be applied in such situations. Their classification applies to directional scenarios, in which a particular mechanism is either conducive to the evolution of cooperation, in which case more cooperative mutants can invade and less cooperative mutants cannot, or it is not conducive to cooperation, in which case more cooperative mutants cannot invade while less cooperative mutants can. In contrast, mechanisms that generate evolutionary branching are conducive to both more and less cooperation at the same time, and hence do not appear to be captured in the framework of Lehmann & Keller (2006). To illustrate the evolutionary branching in cooperative traits, we present two examples from opposite ends of the spectrum of mechanisms envisaged by Lehmann & Keller (2006). In the first example, cooperative investments yield direct benefits to the cooperator, but there are no iterated interactions and no kin selection [this corresponds to the category ‘direct benefits’ in Table 3 of Lehmann & Keller (2006)]. In the second example there is only kin selection, but no direct benefits and no iterated interactions [corresponding to the category ‘kin selection’ in Table 3 of Lehmann & Keller (2006)]. The first example is taken from Doebeli et al. (2004). Consider a situation where cooperative investments x have costs, but yield a benefit to both the individual making the investment and to others it is interacting with. Then, when an individual with trait x plays against a y-individual, the payoff to x is P(x,y) 1⁄4 B(x + y) ) C(x), where B and C are monotonically increasing benefit and cost functions with B(0) 1⁄4 C(0) 1⁄4 0. Thus, the payoff to x is the benefit obtained from the sum of the investments x + y, minus the cost of x, reflecting the fact that x not only benefits from the partner’s investment y, but also from its own investment. The evolutionary dynamics of the trait x is determined by the selection gradient D(x) 1⁄4 B¢(2x) ) C¢(x) (Doebeli et al., 2004) and the cooperative trait increases as long as D(x) > 0. It is important to note that this condition is formulated in terms of derivatives of the benefit and cost functions, and not in terms of the absolute payoff P. Thus, contrary to what seems to be implied in Lehmann & Keller (2006), whether individuals receive net direct benefits from the act of cooperation, i.e. positive payoffs P, is not the determinant of whether cooperation is favoured. Correspondence: Michael Doebeli, Departments of Zoology and Mathematics, University of British Columbia, University Boulevard, Vancouver, BC, V6T 1Z4, Canada. e-mail: [email protected]