- Research article
- Open Access
The role of sexual preferences in intrasexual female competition
- Alicia M Frame^{1}Email author
https://doi.org/10.1186/1471-2148-12-218
© Frame; licensee BioMed Central Ltd. 2012
- Received: 31 January 2012
- Accepted: 19 October 2012
- Published: 14 November 2012
Abstract
Background
While search costs have long been understood to affect the evolution of female preference, other costs associated with mating have been the focus of much less attention. Here I consider a novel mate choice cost: female-female intrasexual competition, that is, when females compete with each other for mates. This competition results in cost to female fecundity, such as a reduction in fertility due to decreased direct benefits, sperm limitation, or time and resources spent competing for a mate. I asked if female-female competition affects the evolution of preferences, and further, if the presence of multiple, different, preferences in a population can reduce competitive costs.
Results
Using population genetic models of preference and trait evolution, I found that intrasexual competition leads to direct selection against female preferences, and restricts the parameter space under which preference may evolve. I also examined how multiple, different, preferences affected preference evolution with female intrasexual competition.
Conclusions
Multiple preferences primarily serve to increase competitive costs and decrease the range of parameters under which preferences may evolve.
Keywords
- Sexual selection
- Mate choice
- Female preference
- Competition
- Population genetics
Background
Costs associated with female preferences are often assumed to be directly related to the act of searching for a preferred mate: 1) time spent searching for a mate, 2) the potential for a choosy female to go unmated, or 3) an increased risk of predation [1]. These previously considered costs are viability costs, where the female’s chances of survival and successful mating are affected; here I present an argument for the role of fertility costs and their effect on preference evolution. There is great potential for costs involved in mate choice to be derived from female-female intrasexual competition as well. In general, these costs have not been widely studied or taken into account as potential selective forces driving (or preventing) female preference evolution [2].
In resource-based polygyny, males provide females with resources such as parental care, defense, or territories in which to raise their young. In such scenarios, the cost of competing for a desired male is clear cut: it is well accepted that males may only support a limited number of females, and increasing beyond that threshold leads to decreased female reproductive fitness [3]. Even in systems where resource limitations are less obvious, reduction in parental efforts can lead to decreased female fitness. For example, in dendrobatid frogs, brood sizes decreased significantly after multiple matings due to decreased male parental effort [4]. Similarly, in polygynous tree swallows, females mated with polygynous males had reduced fitness because of decreased parental care [5].
Even in polygynous species where males offer little to females, females may still incur costs simply by waiting to mate with a preferred male, by competing with other females for a preferred male’s attention, or by suffering reduced fecundity from male sperm depletion. In lekking birds, dominant females monopolizing preferred males time can lead to delayed breeding and decreased reproductive fitness [6]. Females may also respond to competition for males with direct aggression, potentially injuring competitors [7, 8]. Sperm depletion and exhaustion, due to males mating multiply, may be costly to females as well [9]. Sperm exhaustion has been tied to reduced reproductive fitness for females in insects [10], fish [11], and crustaceans [12–14]. Although these costs are small compared to those suffered by females mating in resource based polygyny, they are all associated with significant decreases in reproductive fitness.
In all of these situations, females are likely to experience a cost for preferring ‘popular’ males, i.e. those who have many mates. In fact, when females suffer fitness reductions from mating with sperm depleted males, if they can accurately assess the number of mates a male has, they choose males with fewer mates [12]. In general, however, it may be difficult for females to ascertain whether they are likely to suffer competitive costs: for example, in systems where males have large or overlapping territories, females have little or no information about additional mates; in systems where males provide resources that cannot easily be quantified, the female may have no information about these costs whatsoever.
Without direct knowledge, what can females do to avoid costly competition? One possibility is that multiple preferences may aid in alleviating or preventing competition. Indeed, many of the species discussed previously as examples of costly female competition have multiple male traits and preferences as well (guppies: [15]; tree swallows: [16]; Great Snipe: [17]). If females have differing preferences, and if males display differing traits, then competition could be reduced. For example, if females of some species may prefer complex song, long tails, or both, and males may have one or both of those traits; females choosing mates with high quality plumage may reduce their cost of competition because they are not competing with those who choose males with a complex song.
Empiricists have found cases of repeatable variability in genetically determined female preferences [18, 19]. In such scenarios, females appear to be selecting mates based on multiple independent male traits. Marchetti [19] found evidence that female yellow browed leaf warblers based their choice of mate on several male characters, and although females preferred high quality males, different females used different traits to distinguish between these males [20]. Not only demonstrated multiple preferences in female guppies, but demonstrated that they are heritable and genetically independent. The genetic assumptions of my model are built upon these findings.
Although there is ample empirical evidence of intrasexual mate competition in females, to my knowledge it has not been incorporated into evolutionary models. Fawcett and Johnstone [21] considered the potential for female competition to alter mate choice from a game theoretic point of view, and showed that female competition could alter mating decisions. However, their model ignored genetics and focused primarily on alternative strategies, which is problematic because linkage disequilibrium between genes is a powerful evolutionary force. I chose to use a population genetic model which explicitly considers distinct genotypes and the potential for non-random association between loci (linkage disequilibrium) to evolve via assortative mating, leading to indirect selection on preference and traits.
Here, I argue that competition alone, regardless of the type of trait possessed by males, will impact preference evolution. To address these issues, I first model the evolution of a single female preference in a system with costly intrasexual competition for mates, to determine when preferences may still evolve and the strength of selection acting on preference. Then, I consider whether or not the presence of an additional female preference alleviates competitive costs, and how selection on preferences changes with the introduction of an additional preference. When discussing multiple preferences, I am referring to multiple preferences controlled by independent loci: females may have no preferences, a single preference, or both. As novel preferences evolve to fixation, the result is that the majority of females possess both preferences.
Model specification and results
I model mate choice with costly female competition for mates using a population genetic model with haploid loci and discrete non-overlapping generations, based on previous models of sexual selection via female choice [22]. The model assumes polygyny; all females mate, but males have variable mating success.
For each model, I begin by describing the life cycle in terms of birth, mating, fertility selection, and zygote formation. Using these equations, I can then calculate the strength of direct selection on preference using the notation of Barton and Turelli [23].
One preference, one trait (two locus model)
Female preference and male traits are controlled by two haploid loci, each with two alleles: the preferences locus, P, controls female preference, and the trait locus, T, controls male traits. Uppercase letters indicate the presence of a preference or trait, lowercase letters indicate the absence. These two loci yield four genotypes: PT, Pt, pT, and pt. I denote their frequencies as x_{1}, x_{2}, x_{3}, and x_{4}; X_{T} is used to denote the frequency of the male trait allele (x_{1} + x_{3}), and X_{P} is used to denote the frequency of the female preference allele (x_{1} + x_{2}).
Females choose mates based on their preferences. A female without the preference allele (a p female) will mate randomly with respect to male genotype, whereas a female with the preference allele (a P female) is α times more likely to mate with a male possessing the trait allele, given that she has evaluated one of each type.
Mating table for one preference/one trait model
Males | |||||
---|---|---|---|---|---|
x1 | x2 | x3 | x4 | ||
Females | x1 | $\frac{\text{\alpha}\left({\text{x}}_{1}{\text{x}}_{1}\right)}{Z}$ | $\frac{\left({\text{x}}_{1}{\text{x}}_{2}\right)}{Z}$ | $\frac{\text{\alpha}\left({\text{x}}_{1}\text{x}3\right)}{Z}$ | $\frac{\left({\text{x}}_{1}{\text{x}}_{4}\right)}{Z}$ |
x2 | $\frac{\text{\alpha}\left({\text{x}}_{2}{\text{x}}_{1}\right)}{Z}$ | $\frac{\left({\text{x}}_{2}{\text{x}}_{2}\right)}{Z}$ | $\frac{\text{\alpha}\left({\text{x}}_{2}{\text{x}}_{3}\right)}{Z}$ | $\frac{\left({\text{x}}_{2}{\text{x}}_{4}\right)}{Z}$ | |
x3 | x_{3}x_{1} | x_{3}x_{2} | x_{3}x_{3} | x_{3}x_{4} | |
x4 | x_{4}x_{1} | x_{4}x_{2} | x_{4}x_{3} | x_{4}x_{4} | |
Z = α(x_{1} + x_{3}) + x_{2} + x_{4} |
Recombination follows sexual selection and fertility selection; recombination rates are assumed to be 0.5 between all loci for simplicity (free recombination).
Here, Δp is the sum of direct selection and indirect selection. For any two loci X and Y, a_{X,0}C_{ XY } measures how the frequency of an allele at locus Y changes due to the selection at locus X (a_{X,0}) and the genetic association between locus X and Y (C_{ XY }). Thus, change in preference is driven by direct selection on preferences, a_{P,0}C_{ PP }, as well as indirect selection via the linkage disequilibrium between preference and trait, a_{T,0}C_{ PT } (from [23], eq 16).
This represents direct selection on locus P_{i}, favoring preference, with strength a_{(P,0)}, multiplied by the genetic variance at the P_{i} locus, C_{ PP }.
Where P is the frequency of the preference allele, T is the frequency of the trait allele, and D_{ P,T } is the linkage disequilibrium between preference and trait.
Because 1≥T≥0, and linkage between preference and trait is greater than or equal to 0, the right hand side of (7) is always positive, and, in turn, (6) is always negative.
Two preferences, two traits (four locus model)
Having shown that a single preference is selected against when females compete, I now consider whether or not a second preference is sufficient to alleviate competition, leading to direct selection for preferences.
In this model, there are an additional two loci: two preference loci, P_{1} and P_{2}, control female preference, and two trait loci, T_{1} and T_{2}, control male display traits. These four loci yield 2^{4}= 16 genotypes: P_{1}P_{2}T_{1}T_{2}, P_{1}P_{2}T_{1}t_{2}, P_{1}P_{2}t_{1}T_{2}, P_{1}P_{2}t_{1}t_{2}, P_{1}p_{2}T_{1}T_{2}, P_{1}p_{2}T_{1}t_{2}, and so on. I denote their frequencies by x_{1}, x_{2}, …, x_{16}.
Mating table for two preference/two trait model
Males | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Females | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{9} | x_{10} | x_{11} | x_{12} | x_{13} | x_{14} | x_{15} | x_{16} | |
x _{ 1 } | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{1}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{2}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{3}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{4}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{5}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{6}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{7}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{8}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{9}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{10}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{11}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{12}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{13}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{14}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{15}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{16}\right)}{{Z}_{1}}$ | |
X_{2} | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{1}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{2}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{3}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{4}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{5}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{6}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{7}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{8}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{9}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{10}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{11}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{12}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{13}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{14}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{15}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{2}{x}_{16}\right)}{{Z}_{1}}$ | |
X_{3} | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{1}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{2}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{3}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{4}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{5}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{6}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{7}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{8}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{9}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{10}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{11}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{12}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{13}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{14}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{15}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{3}{x}_{16}\right)}{{Z}_{1}}$ | |
X_{4} | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{1}\right)}{{Z}_{4}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{2}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{3}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{4}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{5}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{6}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{7}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{8}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{9}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{10}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{11}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{12}\right)}{{Z}_{1}}$ | $\frac{\alpha p\phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{13}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{14}\right)}{{Z}_{1}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{4}{x}_{15}\right)}{{Z}_{1}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{1}{x}_{16}\right)}{{Z}_{1}}$ | |
x_{5} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{3}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{4}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{5}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{6}\right)}{{Z}_{2}}$ | $\frac{\left({x}_{5}{x}_{7}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{8}\right)}{{Z}_{2}}$ | $\frac{\alpha \left({x}_{5}{x}_{9}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{10}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{11}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{12}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{13}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{14}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{15}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{5}{x}_{16}\right)}{{Z}_{2}}$ | |
x_{6} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{3}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{4}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{5}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{6}\right)}{{Z}_{2}}$ | $\frac{\left({x}_{6}{x}_{7}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{8}\right)}{{Z}_{2}}$ | $\frac{\alpha \left({x}_{6}{x}_{9}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{10}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{11}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{12}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{13}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{14}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{15}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{6}{x}_{16}\right)}{{Z}_{2}}$ | |
x_{7} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{3}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{4}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{5}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{6}\right)}{{Z}_{2}}$ | $\frac{\left({x}_{7}{x}_{7}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{8}\right)}{{Z}_{2}}$ | $\frac{\alpha \left({x}_{7}{x}_{9}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{10}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{11}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{12}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{13}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{14}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{15}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{7}{x}_{16}\right)}{{Z}_{2}}$ | |
x_{8} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{2}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{3}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{4}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{5}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{6}\right)}{{Z}_{2}}$ | $\frac{\left({x}_{8}{x}_{7}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{8}\right)}{{Z}_{2}}$ | $\frac{\alpha \left({x}_{8}{x}_{9}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{10}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{11}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{12}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{13}\right)}{{Z}_{2}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{14}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{15}\right)}{{Z}_{2}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{8}{x}_{16}\right)}{{Z}_{2}}$ | |
x_{9} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{3}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{4}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{5}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{6}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{9}{x}_{7}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{8}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{9}{x}_{9}\right)}{{Z}_{3}}$ | $\frac{\left({x}_{9}{x}_{10}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{9}{x}_{11}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{12}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{13}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{14}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{9}{x}_{15}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{9}{x}_{16}\right)}{{Z}_{3}}$ | |
x_{10} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{3}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{4}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{5}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{6}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{10}{x}_{7}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{8}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{10}{x}_{9}\right)}{{Z}_{3}}$ | $\frac{\left({x}_{10}{x}_{10}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{10}{x}_{11}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{12}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{13}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{14}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{10}{x}_{15}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{10}{x}_{16}\right)}{{Z}_{3}}$ | |
x_{11} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{3}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{4}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{5}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{6}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{11}{x}_{7}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{8}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{11}{x}_{9}\right)}{{Z}_{3}}$ | $\frac{\left({x}_{11}{x}_{10}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{11}{x}_{11}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{12}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{13}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{14}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{11}{x}_{15}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{11}{x}_{16}\right)}{{Z}_{3}}$ | |
x_{12} | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{2}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{3}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{4}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{5}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{6}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{12}{x}_{7}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{8}\right)}{{Z}_{3}}$ | $\frac{\alpha \left({x}_{12}{x}_{9}\right)}{{Z}_{3}}$ | $\frac{\left({x}_{12}{x}_{10}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{12}{x}_{11}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{12}\right)}{{Z}_{3}}$ | $\frac{\alpha \phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{13}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{14}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\alpha \left({x}_{12}{x}_{15}\right)}{{Z}_{3}}$ | $\frac{\phantom{\rule{0.25em}{0ex}}\left({x}_{12}{x}_{16}\right)}{{Z}_{3}}$ | |
x_{13} | (x_{13}x_{1}) | (x_{13}x_{2} | (x_{13}x_{3}) | (x_{13}x_{4}) | (x_{13}x_{5}) | (x_{13}x_{6}) | (x_{13}x_{7}) | (x_{13}x_{8}) | (x_{13}x_{9}) | (x_{13}x_{10}) | (x_{13}x_{11}) | (x_{13}x_{12}) | (x_{13}x_{13}) | (x_{13}x_{14}) | (x_{13}x_{15}) | (x_{13}x_{16}) | |
x_{14} | (x_{14}x_{1}) | (x_{14}x_{2}) | (x_{14}x_{3}) | (x_{14}x_{4}) | (x_{14}x_{5}) | (x_{14}x_{6}) | (x_{14}x_{7}) | (x_{14}x_{8}) | (x_{14}x_{9}) | (x_{14}x_{10}) | (x_{14}x_{11}) | (x_{14}x_{12}) | (x_{14}x_{13}) | (x_{14}x_{14}) | (x_{14}x_{15}) | (x_{14}x_{16}) | |
x_{15} | (x_{15}x_{1}) | (x_{15}x_{2}) | (x_{15}x_{3}) | (x_{15}x_{4}) | (x_{15}x_{5}) | (x_{15}x_{6}) | (x_{15}x_{7}) | (x_{15}x_{8}) | (x_{15}x_{9}) | (x_{15}x_{10}) | (x_{15}x_{11}) | (x_{15}x_{12}) | (x_{15}x_{13}) | (x_{15}x_{14}) | (x_{15}x_{15}) | (x_{15}x_{16}) | |
x_{16} | (x_{16}x_{1}) | (x_{16}x_{2}) | (x_{16}x_{3}) | (x_{16}x_{4}) | (x_{16}x_{5}) | (x_{16}x_{6}) | (x_{16}x_{7}) | (x_{16}x_{8}) | (x_{16}x_{9}) | (x_{16}x_{10}) | (x_{16}x_{11}) | (x_{16}x_{12}) | (x_{16}x_{13}) | (x_{16}x_{14}) | (x_{16}x_{15}) | (x_{16}x_{16}) | |
Z_{1} = αp(x_{1} + x_{5} + x_{9} + x_{13}) + α(x_{2} + x_{3} + x_{6} + x_{7} + x_{10} + x_{11} + x_{14} + x_{15}) + x_{4} + x_{8} + x_{12} + x_{16} | |||||||||||||||||
Z_{2} = α(x_{1} + x_{2} + x_{5} + x_{6} + x_{9} + x_{10} + x_{13} + x_{14}) + x_{3} + x_{4} + x_{7} + x_{8} + x_{11} + x_{12} + x_{15} + x_{16} | |||||||||||||||||
Z_{3} = α(x_{1} + x_{3} + x_{5} + x_{7} + x_{9} + x_{11} + x_{13} + x_{15}) + x_{2} + x_{4} + x_{6} + x_{8} + x_{10} + x_{12} + x_{14} + x_{16} |
I first confirmed that multiple preferences evolved in the face of costly competition. Multiple preferences evolve but require stronger preference strengths (i.e. greater α) to reach fixation than preferences evolving in the absence of costly competition (Figure 1, gray line). Interestingly, the strength of preference necessary to overcome the costs of choice is lower when multiple preferences are present versus a single preference. With simulations alone, however, it is impossible to determine if this is due to a decrease in competitive costs or an increase in indirect selection driven by stronger joint preferences by females with both preferences for males with both traits.
Again, as expected, selection on preference is a function of trait frequency. Without cost, or when both traits are fixed, selection on preference is 0. Under all other conditions, as before, selection is negative. Because of the complexity of (8), proving that it is always negative is not feasible; I used numerical simulations to verify that with two preferences and traits, a_{P1,0}≤0.
Comparing figures two and three, it is clear that the presence of a second preference alters the strength of direct selection, but does not lead to direct selection for multiple preferences. In general, it appears that the presence of a second preference does decrease costs, but only when preferences are common. When preferences are rare, the presence of a second preference can increase competitive costs drastically by leading to female with two preferences having very strong preferences for rare two-trait males; this in turn would lead to fierce competition. Thus, a second preference would not directly reduce competitive costs when introduced at a low frequency. When preferences are already at a high frequency, there is a benefit to having multiple preferences (see Figure 3), but here I focused on low initial frequencies as an evolutionarily realistic scenario.
Indirect selection
Simulation studies
- 1
female preference for arbitrary male traits,
- 2
female preference for male traits favored by natural selection,
- 3
female preference for condition dependent traits, and
For each scenario, I simulated the evolution of two preferences introduced simultaneously to the evolution of two preferences introduced successively (i.e. the second preference is only introduced after the first one is at equilibrium). I performed numerical simulations in Matlab; equilibrium conditions were found by running recursion equations for genotype frequency, as described above, until trait and preference alleles reached equilibrium. The results presented below are derived from genotype frequencies at equilibrium, which I defined as when the percentage change in genotype frequencies between successive generations was less than 10^{-16}.
Female preferences for arbitrary male traits
I also considered the role of the cost function itself (as defined in equation 2) in determining the conditions under which preference may evolve. Numerically, I simulated a convex cost function and a concave cost function, and compared the parameters under which preferences could evolve. As one might expect, a concave cost function expanded the parameter space where preferences evolved while a convex function further restricted the space where preferences could evolve. Regardless of the shape of the cost function, as long as fertility was reduced in some way due to competition, the parameter space where preferences could evolve was restricted.
Female preferences for male traits favored by natural selection
Where i∈4,8,12,16. The x_{ i }^{ η } values in (4) replace the x_{ i } values in (1).
Condition dependent male traits
For this scenario, I added a fifth locus C, which denotes an individual’s condition. Individuals with c are considered low condition; those with C are high condition, and thus favored by natural selection. The result is 16⋅2=32 genotypes. I included directional mutation from C to c in order to maintain variation in condition.
Values for k _{ ij } for condition dependent mate choice; i represents the female preference genotype, j represents male trait genotype
Condition dependent mate choice | ||||
---|---|---|---|---|
T_{1}T_{2}C (x_{1}, x_{9},x_{17},x_{25}) | T_{1}t_{2} C (x_{3}, x_{11},x_{19},x_{27}) | t_{1}T_{2} C (x_{5}, x_{13},x_{21},x_{29}) | t_{1}t_{2}C(x_{7}, x_{15},x_{23},x_{31}), all c genotypes(x_{2},x_{4},x_{6}, . . .,x_{32}) | |
P_{1}P_{2} (x_{1}, x_{2},x_{3},x_{4}, x_{5}, x_{6},x_{7},x_{8}) | 1.5 | 1 | 1 | 0 |
P_{1}p_{2} (x_{9}, x_{10},x_{11},x_{12}, x_{13}, x_{14},x_{15,}x_{16}) | 1 | 1 | 0 | 0 |
p_{1}P_{2} (x_{17}, x_{18},x_{19},x_{20}, x_{21}, x_{22},x_{23},x_{24}) | 1 | 0 | 1 | 0 |
p_{1}p_{2} (x_{25}, x_{26},x_{27},x_{28}, x_{29}, x_{30},x_{31},x_{32}) | 0 | 0 | 0 | 0 |
- 1
evolution of preference along with condition, where preference and condition are introduced at low frequency simultaneously and allowed to evolve together, and
- 2
evolution of preference in a system where the condition allele is at mutation selection balance (mutation rate for c is 0.005).
By examining both the evolution of condition allele with preference, and the introduction of preference into a high condition population, I can better distinguish the interaction between multiple preferences and condition evolution.
Discussion
The results from my models indicate that intrasexual competition is costly and, when present, direct selection acts against preference evolution. Multiple preferences change the shape of the cost curve but fail to alleviate costly competition when introduced at a low frequency; direct selection still acts against female preference when multiple preferences are present. This is not to say that intrasexual competition entirely prevents preference evolution; simulation results indicated that preferences may still evolve if they are sufficiently strong enough to overcome natural selection, and that the multiple preferences evolving simultaneously may reduce (but not eliminate) direct selection. Although multiple preferences do not lead to direct (i.e. natural) selection for preference evolution, their presence is likely to increase the strength of indirect selection on preference and trait evolution, creating strong joint preferences in females with both preferences for males with both traits; this leads to a decrease in the initial preference strength required for evolution.
In general, these results are consistent with other models, where costs associated with mate choice have been shown to prevent or restrict the evolution of multiple female preferences (Kirkpatrick, 1985); [24, 25]. Kirkpatrick’s (1985) model of the sexy son hypothesis showed that handicap traits, which only lower fitness, do not spread. Models explicitly considering multiple male traits with costly female preference, in terms of search costs/viability selection, also found that female preferences did not evolve due to high joint costs to preference [24, 25]. In these models, if it was more costly for a female to search for and find a mate with multiple preferred traits rather than a male with a single trait, then multiple preferences could not evolve. Similarly, in my model, having multiple preferences served to increase competitive costs when male traits were rare.
My model supports the idea that intrasexual competition is likely to be a significant cost acting against the evolution of female preferences. There are many examples of intrasexual competition: direct aggression between females [7, 8], reduced fecundity due to decreased male parental efforts (Summers 1990), [5], as well as decreased fecundity from male sperm depletion [11], (Royer and McNeil 1997), [9, 12–14]. Yet, in the majority of these species, female preferences have evolved regardless – including multiple preferences. In my models, competitive costs are not an insurmountable obstacle; although multiple preferences fail to alleviate competition, they don’t appear to be significantly more costly than a single preference, and in fact serve to increase indirect selection on preferences (see Figure 1; the minimum α required for preference evolution is lower for multiple preferences).
Multiple preferences may in fact serve to alleviate competition, just not in the way modeled here. One possibility is that if individual females have different preferences, controlled by a single locus, instead of multiple preferences controlled by multiple loci, competition could be averted. However, this scenario is unlikely: in most species with multiple preferences, these preferences appear to be controlled by independent genes Brooks and Coulridge, [18, 19]. As my model has shown, if preferences are controlled by independent loci, after several generations, many individuals have both preferences leading to increased competition, not avoidance.
Perhaps multiple preferences may not indirectly prevent competition, but instead involve preferences for traits which indicate how many times a male has mated. One study showed that female cockroaches discriminated against males that had mated multiple times, and were able to detect cues on males derived from previous mates, in addition to traits indicating male quality [12]. However, it is difficult to imagine how common the ability to detect prior matings is, and there is only one such example in the literature. Another possibility is that females could evolve multiple preferences and switch between preferences when they sense competition for a desired male. This would require knowledge about population wide preference frequencies, but would be possible in lekking species or animals that live in social groups.
Conclusions
When multiple preferences are present, indirect selection on female preference evolution is much stronger. Perhaps instead of relieving competition, multiple preferences allow female choice to evolve by jointly increasing the strength of indirect selection to the point where many weak preferences can overcome natural selection against competition.
Appendix
Appendix 1. a_{p,0} equations for a single preference and trait, two locus model
where X_{ p } represents the presence of preference alleles in females; X_{ p }=1 if a female has allele P, and 0 if she does not. Likewise, X_{ t* }=1 if a male has allele T, and 0 if he does not. z_{ i } is the normalization for sexual selection (as described in equation 2). ϕ_{ i } is the fertility selection against male genotype i (see equation 3 in the text). For example, for an x_{1} individual (PT), X_{ p }=1 and X_{ t* }=1, and ${X}_{p},{X}_{{t}^{*}}=\frac{{\varphi}_{4}}{{Z}_{2}}\text{.}$
Equation (A1) can be used to calculate the a terms present in equation (4) in the text. To calculate the as, the fitness equation for a model (here, A1) is set equal to a generic equation for fitness in terms of as and Cs, and a function of the Xs. Terms are then matched to solve for a in the model under consideration. This procedure is described fully in appendix B of Kirkpatrick and Servedio.
Appendix 2. a_{p,0} equations for a single preference and trait, four locus model
As in Appendix 1, X_{ Pi } represents the presence of preference alleles in females, where X_{ P1 }=0 if a female has preference i, and 0 if she does not. Likewise, X_{ Ti }=0 if a male has trait i, and 0 if he does not. Z_{ i } is the normalization for sexual selection (Z_{1}, Z_{2}, and Z_{3} are described in Table 2; Z_{4}=1). ϕ_{ i } is the fertility selection against male genotype i (see equation 3 in the text). As with female preference, there are only four unique male genotype combinations such that ϕ_{ 1 } is the discount for T1T2 males, ϕ_{ 2 } is for T1t2 males, ϕ_{ 3 } is for t_{1}T_{2} males, and ϕ_{ 4 } is the discount for t_{1}t_{2} males.
As in Appendix 1, equation (A2) is used to calculate the a terms present in equation (5) in the text. Because of the complexity of equation (A2), I applied a weak selection approximation to get a shorter, analytically tractable expression for a_{P,0}: I assumed that costs were low, preferences weak, and linkage disequilibrium small (confirmed via simulations), and performed a taylor series expansion of a_{P,0}. This method yielded equation (8), a considerably shorter expression for direct selection on preferences. To confirm the validity of the weak selection approximation, I compared it to the original expression and confirmed that, as α, γ, and D_{i,j} decreased, the two expressions converged. For the sake of comparison to (6), the equation used in Figure 3 is the original formulation of a_{P,0}, not the weak selection approximation.
Author contributions
AMF conducted all of the research presented in this paper and wrote the manuscript. She has read and approved the final manuscript.
Declarations
Acknowledgements
I would like to that Maria Servedio, Joel Adamson, Sumit Dhole, Alex Mills, and Jonathan Rowell for their helpful suggestions. This study was supported by NSF grant DEB0614166. The University of North Carolina at Chapel Hill’s libraries provided support for this open access publication.
Authors’ Affiliations
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