Effects of female preference intensity on the permissiveness of sexual trait polymorphisms

Abstract Recent developments in sexual selection theory suggest that on their own, mate preferences can promote the maintenance of sexual trait diversity. However, how mate preferences constrain the permissiveness of sexual trait diversity in different environmental regimes remains an open question. Here, we examine how a range of mate choice parameters affect the permissiveness of sexual trait polymorphism under several selection regimes. We use the null model of sexual selection and show that environments with strong assortative mating significantly increase the permissiveness of sexual trait polymorphism. We show that for a given change in mate choice parameters, the permissiveness of polymorphism changes more in environments with strong natural selection on sexual traits than in environments with weak selection. Sets of nearly stable polymorphic populations with weak assortative mating are more likely to show accidental divergence in sexual traits than sets of populations with strong assortative mating. The permissiveness of sexual trait polymorphism critically depends upon particular combinations of natural selection and mate choice parameters.

equilibrium line imply meaning, honesty, and design in intersexual signals (Fuller, Houle, & Travis, 2005;Prum, 2010). Direct selection on sexual traits only changes the slope of equilibrium line. However, the stable line of equilibria disappears and collapses to a single equilibrium point when mate preferences are under even weak directional selection (Grafen, 1990;Kirkpatrick & Ryan, 1991;Prum, 2010). Although direct selection on mate preferences drastically alters the stability of the equilibrium line, the Fisherian runaway process can still continue even when mate preferences are under strong selection (Hall, Kirkpatrick, & West, 2000).
Kirkpatrick's haploid version of the null model of sexual selection (Kirkpatrick, 1982) and subsequent models expanding Kirkpatrick's original model (Bulmer, 1989;Seger, 1985;Seger & Trivers, 1986;Takahasi, 1997) analyze the effects of mate preference strengths on the shape and stability of the equilibrium line. In contrast, here, we investigate what can happen away from the line. In this paper, we use Kirkpatrick's model and focus on how mate preferences affect the size and shape of the attraction basin around the stable line of equilibria.
We use the dynamical systems theory approach (Beisner, Haydon, & Cuddington, 2003;Meyer, 2015) and use the size and shape of attraction basins as measures of permissiveness of polymorphism as a way to estimate the possible effects of random perturbations of populations away from the line. An attraction basin is the set of all starting male-female allele frequency combinations from which populations in a given environment evolve to a set of polymorphic equilibria within the basin. The size and shape of an attraction basin is a measure of the permissiveness of sexual trait polymorphism in a given set of environmental conditions; the larger the basin, it is the less likely a given perturbation will draw a population outside the zone of attraction and lose its polymorphism. Permissiveness of sexual trait polymorphism can be defined as the capacity of the system to allow the polymorphism to be maintained in potentially variable conditions. We examine how mate choice parameters, other selective forces independent of mate choice (natural selection), and their interactions, affect the permissiveness of sexual trait polymorphism. It is important to note that diploid models involving the Fisher process can show differences in evolutionary dynamics compared with the haploid version (Greenspoon & Otto, 2009 (Kirkpatrick, 1982) second model is that our model does not include viability selection on sexual traits, whereas Kirkpatrick's model does include it. The literature on sexual selection utilizes three important preference functions: fixed relative preference, best of N males, and absolute preference function. Kirkpatrick's original models use the fixed relative preference function. Here, we also use the fixed relative preference framework used by Gavrilets (Gavrilets, 2004) which has a slightly different way of parametrization than the Kirkpatrick's original model. Apart from these modifications, our basic models (model 1 and 2) are identical with Kirkpatrick's model (Kirkpatrick, 1982). Previous work has concentrated on the structure and stability of the line of equilibrium. Here, we explore the effects of the size and shape of the zone of attraction around the equilibrium line in order to assess effects of random fluctuations away from the line.
Consider a haploid population exhibiting polymorphism in both sexual traits and in mating preferences for the sexual traits. Assume locus T controls male traits and an unlinked locus P controls female preference for the male traits. Let each locus have two alleles which correspond to different phenotypes, T 1 , T 2 for different sexual traits in males and P 1 , P 2 for different female preferences. Let m 1 , m 2 , m 3, and m 4 be the frequencies of T 1 P 1 , T 1 P 2 , T 2 P 1, and T 2 P 2, respectively, in males and f 1 , f 2 , f 3, and f 4 in females. For every combination of starting frequencies and zygote frequencies, m i = f j and ∑m i = ∑ f j = 1. Note that there is no cost associated with mate preferences and sexual traits.
Let the relative preference of a P 1 female for T 1 males be 1 and her preference for T 2 males be 1α 1 . Similarly, let the preference of P 2 females for T 2 males be 1 and her preference for T 1 males be 1α 2 , where α 1 and α 2 are mate choice parameters (discrimination coefficients) measuring the strength of preference. If α 1 = α 2 = 0, there is no choice with respect to male traits and α 1 = α 2 = 1 means both females only chose their preferred traits (complete positive assortative mating).
Recurrence equations for zygote frequencies in the next generation are Here, (1a) T 1 frequencies were computed numerically by iterating the equations for 30,000 generations for all combinations of male-female starting frequencies and α 1 and α 2 using MATLAB 2015b. We found 30,000 generations to be more than sufficient time for the populations to attain a stable equilibrium for the entire range of α 1 and α 2 .
For a given constant α 1 and α 2 , joint initial male-female allele frequencies that will maintain sexual trait polymorphism at equilibrium form a zone in the joint frequency state space with two distinct boundaries ( Figure 1a). We will refer to the central zone as the polymorphic zone and boundaries as the upper (U) and lower (L) boundaries by where they intersect the axis of P 1 starting frequencies ( Figure 1a).
To determine the polymorphic zone and boundaries, we first computed the equilibrium T 1 frequency (T 1 frequency after 30,000 generations) for all possible combinations of starting frequencies of T 1 and P 1 for a given constant α 1 and α 2 . The polymorphic zone is an attraction basin for polymorphic equilibria; it represents all joint T 1 and P 1 starting frequencies that eventually produce the equilibrium T 1 frequency between 0.001 and 0.999 (0.001 ≤ T 1 (equilibrium frequency) ≤0.999). To compute U (the boundary separating the polymorphic zone and the attraction basin for T 1 fixation), we identified unique threshold starting frequencies of P 1 for the entire range of T 1 starting frequencies such that any change in starting frequency of P 1 above the threshold will result in T 1 fixation, that is, T 1(equilibrium frequency) >0.999. Similarly, to compute L (the boundary separating the polymorphic zone and the attraction basin for T 2 fixation), we identified the threshold starting frequencies of P 1 such that any change in the starting frequency of P 1 below this threshold will result in T 1 loss, that is, T 1(equilibrium frequency) <0.001. U and L separate very different evolutionary outcomes. Populations with joint male-female allele frequencies starting anywhere inside the central zone (within U and L) retain sexual trait polymorphisms.
Populations with joint frequencies starting anywhere outside the central zone lose sexual trait polymorphism (either T 1 is fixed or it is lost).
The area of the attraction basin is a measure of the permissiveness of sexual trait polymorphism in a given set of environmental conditions. If the area of the polymorphic zone is small, then the permissiveness is low and if the area is large, then the permissiveness is high. If random factors change allele frequencies, then low permissiveness implies a greater chance that a changed allele frequency combination will cross a boundary (leading to loss or fixation) than the same change under conditions of high permissiveness. In addition, changes in mate choice parameters (α 1 and α 2 ) alter the size and shape of the polymorphic zone, affecting the permissiveness ( Figure 1). We explored the effects of α 1 and α 2 on the area of the polymorphic zone.

| Effects of choice parameters on the permissiveness of sexual trait polymorphism
Different combinations of α 1 and α 2 alter the position, shape, and size of the polymorphic zone (Figures 1b and c). When preferences are nearly equal and weak, the polymorphic zone remains narrow; the system has low permissiveness. As α 1 = α 2 = α increases, the zone boundaries (U and L) gradually move apart and the polymor-

| Model 2: Sexual Trait (T) under directional viability selection
The standard model of intersexual selection (Kirkpatrick, 1982;Prum, 2010) normally assumes directional viability selection on male traits in addition to mate choice. For example, viability selection could be caused by the physical environment. Our model 2 is same as Kirkpatrick's second model in (Kirkpatrick, 1982). Here, we explored the range of permissiveness when male traits are under directional viability selection independent of the preference trait P.
Let T 2 males have a disadvantage such that the T 2 trait viability is 1-s relative to T 1 males; s is the viability selection coefficient (0 ≤ s ≤ 1). Let viability selection on males occur before selective mating; this alters the frequencies of males available for mating. Aside from this modification, the model is the same as Model 1 (a true null model without viability selection on male traits). New gamete frequencies available for mating in males are Zygote frequencies in the next generation can be obtained by substi- , and m ′ 4 for m 1 , m 2 , m 3, and m 4 in Equations 1a to 1d. Note that m 1 , m 2 , m 3, and m 4 are the frequencies of T 1 P 1 , T 1 P 2 , T 2 P 1, and T 2 P 2, respectively, in males.
T 1 frequencies were computed numerically by iterating Equations 1a to 1d after substituting new gamete frequencies for all combinations of male-female starting frequencies and α 1 and α 2 using MATLAB 2015b. Five thousand generations were more than sufficient for populations to reach a stable equilibrium.

| Joint effects of mate choice and natural selection parameters on the permissiveness of sexual trait polymorphism
The strengths of s, α 1, and α 2 have interacting effects and this determines the size, shape, and position of the polymorphic zone ( Figure 1d shows changes in the zone as a function of s for α 1 = α 2 = 0.6). Figure 2 shows how the permissiveness changes as a function of α mean under different viability selection regimes.
For a constant α mean , s increases the difference between the maximum and minimum values of permissiveness and thus, effectively increases the permissiveness range. This suggests that for the same change in α 1 and/or α 2 , the permissiveness of polymorphism changes more in environments with strong s than weak s (each dot in Figure 2a-i represents the area of the polymorphic zone for the unique combination of α 1 and α 2 ).

| D ISCUSS I ON
Classical theory suggests that on its own, selective mating should reduce the variance in sexually selected traits (Kirkpatrick & Ryan, 1991;Pomiankowski & Møller, 1995), yet many species show variation in these traits (Brooks & Endler, 2001;Gray & McKinnon, 2007;Pomiankowski & Møller, 1995;Wellenreuther et al., 2014). Recent developments in sexual selection theory suggest that on their own, mate preferences can promote the maintenance of sexual trait diversity and promote coexistence (M'Gonigle et al., 2012). However, how mate preferences constrain the maintenance of sexual trait diversity in different environmental regimes remains an open question. Our study shows that the permissiveness of sexual trait polymorphism increases in environments with strong selective mating; the risk of loss of sexual trait diversity is significantly lower when preferential mating is strong compared to when it is weak. Now we discuss a potential mechanism which can produce these results.
Selective mating produces an overall negative frequencydependent effect which makes the line of polymorphic equilibria a stable attractor (Seger, 1985). For a constant frequency of the preference allele in populations, male traits exhibit higher fitness relative to the other trait when lower in frequency (see figure 1b in Seger, 1985). Thus, populations starting with a higher frequency of male alleles require a higher threshold frequency of corresponding female alleles to continue the Fisher process and lead populations to fixation. This is the reason that U and L in Figure 1a are curved and not horizontal straight lines. Strong assortative mating amplifies the negative frequency-dependent effect; note how Figure 1b shows that U and L remain straight horizontal lines when α is weak but become more curved as α becomes strong. This can potentially increase the size of attraction basin and make polymorphic attractors (stable line of polymorphic equilibria) significantly more robust to random factors when assortative mating is strong than when it is weak. Gavrilets (Gavrilets, 2004) used a hybrid deficiency index (I) to measure the potential for reproductive isolation in a model identical to our Model 1. He found that hybrids are maintained in populations even if both females show strong mating preferences. Hybrids are eliminated only when mating preferences are extremely strong (but not completely assortative). For example when α 1 = α 2 = α > 0.9, see figure 9.5 in Gavrilets (2004). Our results show that strong mating preferences make sexual trait polymorphisms more permissive. Thus, for strong α, polymorphic populations sitting on or close to the stable line may not necessarily develop reproductive isolation and can remain polymorphic for long periods.
In summary, strong assortative mating significantly increases the permissiveness of sexual trait polymorphism under a broad range of environmental regimes. These results suggest that early stages of population divergence by accident could stall especially with strong mating preferences, and further parametric changes may need to occur before complete divergence.

ACK N OWLED G M ENTS
We thank anonymous reviewers for comments and suggestions on the manuscript. AP is supported by the higher degree by research (HDR) scholarship provided by Deakin University and JAE by ARC grants DP110101421 and DP150102817.

CO N FLI C T O F I NTE R E S T
None declared.

AUTH O R CO NTR I B UTI O N S
Models were designed and analyzed by AP. The manuscript was prepared by AP and was revised by JAE.