Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective

Abstract

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock–scissor–paper configurations. Our concept differs from that of Heino et al. (TREE 13:367–370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.’s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock–scissor–paper configurations. Since in ‘realistic’ (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.

Towards a mechanistic foundation of evolutionary theory

“…General conditions under which a fitness measure can be found that is optimized by an underlying evolutionary process have been studied by (Metz and Geritz, 2016;Metz et al, 2008), whose results confirm that the existence of a constant fitness function that would capture the evolutionary dynamics requires very special assumptions. Importantly, it is possible to derive a more general, non-constant fitness concept from evolutionary birth-death processes: invasion fitness (Metz et al, 1992), obtained from underlying birth-death processes in the limit of large populations sizes and rare mutations, captures not only the fitness landscape at a given point in time, but also the dynamics of the fitness landscape, i.e., how the fitness landscape changes as a result of evolutionary dynamics.…”

“…In stochastic implementations, birth and death rates need to be updated after every single birth and death event, because these events lead to changes in population composition. This generates a basic feedback loop that has the potential to generate complexity in the evolutionary process (this feedback loop is traditionally referred to as frequency dependence, (Heino et al, 1998;Metz and Geritz, 2016). In addition, the abiotic environment may of course also depend on time, both through biotic effects such as nutrient depletion, and through fluctuations in the physical and chemical environment.…”

Eco‐evolutionary feedbacks—Theoretical models and perspectives

“…minimum) is reached. Such simplistic EEFs have been termed frequency independence in the broad sense by Metz and Geritz (). Overall, recent models have become more elaborate.…”

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

“…Like reproductive value, the concept of fitness is both fundamental to evolutionary theory and difficult to formalize in a fully consistent way Akçay and Van Cleve, 2016;Lehmann et al, 2016;Metz and Geritz, 2016;Doebeli et al, 2017). Here, we have defined fitness as the RV-weighted contribution of a genetic site to the next time-step, which can be decomposed as the sum of the RV-weighted survival probability, v g −d g (x) = v g (1 − d g (x)), and the RV-weighted birth rate,b g (x).…”