The future of a mutation‐limited tool‐box

Abstract

Waxman and Gavrilets give a timely overview of a line of research which has been given the brand-name Adaptive Dynamics (AD). Regardless of whether the branding was appropriate, AD has been influential. My goal here is not to refute points made by the authors, but rather to try to give insight into the need for developing AD approximations, and to discuss their use and limitations. In his book on selection in age-structured populations, Charlesworth (1994, p. 11) states that there is no easily applicable method to model evolution in arbitrarily structured populations. By combining a few assumptions, there now is such a method, or at least an approximate theory, and that method is known as AD. The main assumptions that have led to a workable approximation are that (1) population sizes are large; (2) populations are composed of types that mutate; and (3) mutation rates are low enough that evolutionary and ecological timescales are separated. The third assumption allows us to consider invasion of mutants and their effects on the population dynamics as separate processes: mutant types appear and perturb the composition of the population slightly. Then, after each appearance of a successful mutant, the ecological dynamics return to a stationary state, for example, a new equilibrium density. Changes in the properties of these attractors over time constitute the evolutionary dynamics of the system and usually involve some change in the composition of the population. In most AD models, that means that the set of common types in the population is modified by invasion and/or elimination. Note that the population composition is defined in terms of types. These are entities that mutate, but nothing more is specified about them. For this reason, Metz et al. (1996) claimed that AD is a theory of nearly faithful replicators. These could be allelic types, or organismal phenotypes, or any other entity that can be assumed to mutate only rarely. Much of the hassle over whether AD applies to genetic models is superfluous: it is more likely that the dynamics of alleles satisfies the assumption of mutation-limited evolution than organismal phenotypes do. The group of researchers connected to the Metz et al. (1996) paper embodies a large corpus of knowledge of theoretical population dynamics, and AD is their major venture into evolutionary dynamics. They noticed a general property of population dynamics which they called the conditional linearity principle (Metz & de Roos, 1992): if the time series of all variables representing an ecological environment are given, then the population dynamics of an organism living in that environment becomes linear in the population densities. That means all individuals seem to live independently, because any interactions among them are conditioned out and subsumed in the ecological environment. Feedbacks from the population to itself therefore disappear and the dynamics becomes easier to analyse. This principle is maximally exploited in their approach to evolutionary dynamics. When rare mutants appear, they experience the environment created by the common types in the population and conditional linearity applies. Their evolutionary success is derived from their longterm growth rate in that environment, which has been called invasion fitness (Metz et al., 1992). On an evolutionary time scale, most mutants will fail to perturb and change the resident attractor significantly, because drift causes their extinction. Even when they have a selective advantage, the probability that they will actually grow to appreciable frequency is usually small, because they start growing from a population of a single individual (references in Waxman and Gavrilets, Ewens, 1969). Waxman and Gavrilets are right when they conclude that assuming small to very small mutational effects excludes the larger mutational effects that are most likely to invade successfully and become established in a population. Such an assumption might therefore produce a biased view of mutation-limited evolution. I do not agree with Waxman and Gavrilets that fitness is primarily a population genetic concept. It is a (Darwinian) demographic concept which derives from explicit population dynamics and ecology. It should not be an instantaneous fitness used to iterate the population dynamics, but a more long-term measure, that takes into account the effects of environmental variation and population structure.