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.
Highlights
The concept of frequency dependence comes from population genetics
In general mechanistically-based eco-evolutionary models only allow the classical equations of population genetics to be extracted in the exceptional cases that a population model can be collapsed to the simple population model of the genetics textbooks, or when all phenotypes are closely similar so that selection is very weak
As the usefulness of a definition depends on how it performs, we review in Sect. 5 the main ideas of adaptive dynamics as the minimal formalised arena in which this performance may be tested
Summary
The concept of frequency dependence comes from population genetics. Textbooks on this subject are filled mainly with discrete time models without generation overlap and with fitnesses that depend only on the genotype and nothing else, determined by multiplying survival to adulthood with adult reproductive output. (In the case of biparental reproduction we shall measure the latter as the number of contributed alleles, i.e., half the number of offspring.1) Starting from this conveniently simple ecological scenario they immediately focus on gene frequencies, thereby hiding the unrealistic consequence that the population either grows to infinity or declines to zero (except for a few exceptional fitness configurations that make the long term population average of the fitnesses precisely equal to one). The introduction of more ecological realism, e.g., by considering the empirically ubiquitous cyclically driven size structured populations in continuous time kept in check by the delay in reaching reproductive size when food becomes scarce (e.g., De Roos and Persson 2013), torpedoes the simple fitness concept of population genetics, and therewith the classical concept of frequency dependence The latter term is often felt to be heuristically useful in more general evolutionary discourse, and has gotten reified in the mind of the community, in an unscientifically vague manner. In general mechanistically-based eco-evolutionary models only allow the classical equations of population genetics to be extracted in the exceptional cases that a population model can be collapsed to the simple population model of the genetics textbooks, or when all phenotypes are closely similar so that selection is very weak In the latter case the extracted equations for the, slow, genetic dynamics turn out to be those for small additive frequency independent fitness differences.
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