COMPLEX EFFECTS OF ASSORTATIVE MATING
ON ADAPTATION TO ENVIRON-MENTAL
CHANGE IN THE PRESENCE OF SEX-SPECIFIC
SELECTION

Claire Godineau, Ophélie Ronce, Céline Devaux

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Claire Godineau, Ophélie Ronce, Céline Devaux. COMPLEX EFFECTS OF ASSORTATIVE MATING ON ADAPTATION TO ENVIRON-MENTAL CHANGE IN THE PRESENCE OF SEX-SPECIFIC SELECTION. Evolution - International Journal of Organic Evolution, inPress. hal-05134709

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COMPLEX EFFECTS OF ASSORTATIVE MATING ON ADAPTATION TO ENVIRON-
MENTAL CHANGE IN THE PRESENCE OF SEX-SPECIFIC SELECTION

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Claire Godineau ¹,², Ophélie Ronce ¹, Céline Devaux ¹,³

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1. ISEM, Univ Montpellier, CNRS, IRD, Montpellier, France

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2. Department of Biology, University of Florida, Gainesville, Florida, USA

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3. Institut de Recherche en Biologie Végétale, Université de Montréal, Montréal, Québec,
Canada

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Corresponding author: Claire Godineau - clairegodineau@gmail.com

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Author contributions: CG, CD and OR conceptualized the project. CG derived the
analytical results, wrote the codes for the analytical model and the individual-based
simulations with valuable input from CD and OR. CG ran the simulations and CD
produced the figures. The manuscript was written by CG and CD, and edited and
reviewed by OR.

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Funding: CG was supported by a grant to Matthieu Alfaro (project MICHEL ‘Math-ématiques pour Individus affrontant des CHangements d’Environnements Latents’) from Montpellier University of Excellence (MUSE, an ANR ‘Investissements d’avenir’ program ANR-16IDEX-0006). A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant’s open access conditions.

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Acknowledgments: We thank the reviewers and the associate editor for their helpful comments on the manuscript.

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Data availability: code and dataset are available in a Zenodo repository: https://doi.org/10.5281/ze

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Conflict of interest statement: None
## 27 ABSTRACT

Previous theory suggests that fast adaptation of body size and phenology to climate warming could be facilitated by assortative mating. We here test whether this still holds when natural selection is sex-specific and assortative mating is driven by different mechanisms. We model a trait with identical distribution for females and males, but different optima across sexes and time. Predictions derived from the infinitesimal model of inheritance are confirmed by individual-based simulations. We find that female maladaptation depends on a sexual-conflict mismatch in a constant environment, and an adaptive lag generated by the environmental change. By strengthening the effect of natural selection on females, assortative mating reduces the sexual-conflict mismatch compared to random mating. However, it can either increase or decrease the adaptive lag, depending on the relative strength of natural selection on females versus males. Conditions under which assortative mating is beneficial depend on whether the environmental change and the sexual conflict displace the mean phenotype in the same direction, the strength of the sexual conflict and the assortment, and the mechanism that drives assortative mating. Associated with sex-specific selection, assortative mating does not always facilitate adaptation to environmental change, and its effects depend on how it is modeled.

## 45 Introduction

Fast evolution is key for the long-term persistence of populations facing gradually changing environments, such as contemporary climate change (Hoffmann and Sgrò, 2011; Lynch et al., 1991; Merilä and Hendry, 2014). Empirical data show that phenological traits, such as flowering, breeding or migration times, and body size have evolved in response to changes in current climate (Blanckenhorn, 2015; Bonnet et al., 2017; Des Roches et al., 2020; Franks et al., 2007; Hamann et al., 2018; Merilä and Hendry, 2014; Smith et al., 2021). These traits also share several features: they are constrained by temperature, frequently involved in assortative mating, and often thought to be under sexual conflict induced by sex-specific selection. For instance, flowering times shift earlier in the season as temperature increases (Parmesan and Yohe, 2003) and/or pollinators become active sooner (Duchenne et al., 2020; Elzinga
et al., 2007), affecting both male and female reproductive success. Mating is assortative as only plants that flower at the same time can cross-fertilize (Devaux and Lande, 2008; Weis et al., 2005, 2014). Male competition most likely favors early-flowering males (Bertin and Newman, 1993; Forrest, 2014), although for females, early flowering and its associated risk of frost can lead to failure of seed maturation (Chuine and Beaubien, 2001; Neuner et al., 2013), generating a sexual conflict for optimal reproductive time. In threespine sticklebacks, increased temperature selects for decreased body size (Des Roches et al., 2020; Smith et al., 2021), and females and males mate assortatively for body size (Conte and Schluter, 2013; McKinnon et al., 2004). Observed sexual dimorphism in body shape, including some traits related to body size, suggests that females and males experience different selection (Leinonen et al., 2011). Here, we propose to examine how assortative mating and sex-specific selection jointly affect evolutionary responses to a gradually changing environment.

Past models have contributed to understanding the mechanisms that facilitate adaptation to climate change. These models often assume selection on some quantitative phenotype (such as phenology or body size) with a gradually moving phenotypic optimum (Bürger, 1999; Bürger and Lynch, 1995; Charlesworth, 1993; Kopp and Matuszewski, 2014; Lande and Shannon, 1996; Lynch et al., 1991). As the optimum starts moving, the population mean phenotype lags behind the optimum, but eventually reaches an equilibrium at which its rate of change equals that of the optimum change, and at which the adaptive lag is thus constant. This lag causes a fitness load, reduces population growth, and can impede population persistence (Bürger and Lynch, 1995; Lande and Shannon, 1996). It is predicted to be larger with faster optimum change, weaker selection, and lower genetic variance (Bürger and Lynch, 1995; Lynch et al., 1991). While previous models considered randomly mating populations, Godineau et al. (2021) recently showed that this adaptive lag has the same expression for populations under assortative mating, but is expected to be reduced compared to populations under random mating, because assortative mating maintains more genetic variance. This model thus partly explains why traits that show fast evolution in response to climate change are often under assortative mating. This model, however, ignores sex-specific selection on these traits.
Selection is sex-specific whenever optima or strengths of selection differ between fe-
males and males. Meta-analyses suggest that at least 17% of linear selection gradients
differ in direction between females and males (Cox and Calsbeek, 2009), and that the
strength of sex-specific selection depends on microclimatic variables (de Lisle et al.,
2018). Sex-specific selection is considered to be at the evolutionary origin, via dif-
ferent mechanisms, of sexual dimorphism, for example in body size (Glucksmann,
1974; Pincheira-Donoso et al., 2021), migration dates (Morbey and Ydenberg, 2001;
Pedersen et al., 2019; Rubolini et al., 2004), and flowering and breeding times (Bertin
and Newman, 1993; Glucksmann, 1974). Sex-specific selection induces intralocus
conflicts when genes shared by both sexes prevent each sex from maximizing its fit-
ness (Bonduriansky and Chenoweth, 2009; Parker and Partridge, 1998; Pennell et al.,
2016). As the evolutionary interests of the sexes diverge, the sexual conflict leads to a
fitness load in the population, preventing the population from reaching its maximum
fitness value (Connallon and Hall, 2016; Ekrem and Kokko, 2023; Lande, 1980). The
demographic consequences of sexual conflict depend on its intensity, and the contri-
bution of each sex to the population growth.

This sexual conflict can further be affected by environmental change. In a model
without assortative mating, Connallon and Hall (2016) found that sexual conflict is
modified by how female and male optima shift through time. For example, they found
that large adaptive lags can make selection gradients more aligned between the sexes
and make one sex closer to its optimum compared to a constant environment. Sex-
ual selection generated by assortative mating also interacts with sexual conflict by
modifying the trade-off between female and male fitness. Under assortative mating,
males are selected to resemble the most common and/or successful females, which
displaces the population mean phenotype for males towards the female optimum, and
thus away from the one favored by natural selection for males (Lande, 1980). Over
time, this selection regime makes the mean population phenotype evolve towards an
evolutionary optimum, at which the natural selection gradient averaged over the sexes
balances the sexual selection gradient (Lande, 1980). Stronger assortative mating, or
stronger natural selection on females versus males, associated with perfect genetic
correlation between female and male traits, shifts the evolutionary optimum closer to
the female optimum (Lande, 1980). If females limit reproduction (as in many natu-
ral populations; Crowley, 2000; Harts et al., 2014), any environmental change or any
sexual selection that brings the evolutionary optimum closer to the female optimum
should decrease the female fitness load, and thus the population extinction risk.

In addition to modifying sexual conflicts, assortative mating also modifies genetic re-
sponses through its antagonistic effects on genetic variance: it decreases genetic vari-
ance in fathers and thus genetic polymorphism (Godineau et al., 2021; Kirkpatrick
and Nuismer, 2004), and it increases genetic variance by creating positive associa-
tions among allelic effects (Crow and Kimura, 1970). The effects of assortative mating
on sexual conflict and genetic variance could depend on how assortative mating oper-
ates and is modeled. Assortative mating is usually modeled by a preference function of
females for males (e.g. Sachdeva and Barton, 2017); in this case, the mating probabil-
ity of a male is proportional to its phenotypic resemblance to the females. Assortative
mating by preference is likely for traits related to body size. Assortative mating is also
modeled for phenological traits such as flowering time, breeding time, or migration
time. In this case, a female can only mate with a male that is simultaneously sexually
mature or simultaneously present at the same migration site. The mating probabil-
ity of a male depends on the frequency of the other males at that same time, and
varies over time (Fox, 2003; Weis et al., 2005). We here aim to test whether assorta-
tive mating by time maintains more genetic diversity than does assortative mating by
female preference (Fox, 2003; Weis et al., 2005) and thus increases female adaptation
to environmental change.

In this paper, we examine the effect of assortative mating and intralocus sexual con-
flict on evolutionary responses to a gradually changing environment. We modify sev-
eral components of the analytical model from Godineau et al. (2021) for the evolution
of a quantitative trait based on the infinitesimal model of trait inheritance. We inves-
tigate the effects of sexual conflict, due to sex-specific natural selection on a single
trait, and sexual selection, due to assortative mating, on adaptation by measuring the
mismatch between the population mean breeding value and the female optimum. We
express the mismatch at equilibrium as a function of genetic variance, and separately,
the genetic variance as a function of genic variance (i.e., genetic polymorphism) under
assortative mating by female preference or by time, and under random mating as a reference case. We validate our analytical results using an individual-based model in which all components of genetic variance can evolve.

## Methods

We develop a series of models and progressively relax their assumptions to disentangle the effects of sex-specific natural selection on adaptation to environmental change under (i) assortative mating by female preference for males resembling themselves, (ii) assortative mating by time, which is obligate for breeding times as only individuals sexually active at the same time can mate, and (iii) random mating. We develop an analytical model (based on the model in Supplementary Information S1 of Godineau et al., 2021) using the infinitesimal model of trait inheritance to understand the deterministic changes over time of a unique trait under the aforementioned mating patterns, assuming stabilizing selection around some sex-specific optimal values. The optimal trait values that maximize female and male fitness differ, and the strength of natural selection on the trait can also differ between females and males, generating an intralocus sexual conflict (Bonduriansky and Chenoweth, 2009; Pennell et al., 2016). Both female and male optima are either constant or shift over time at the same and constant speed $k$. The infinitesimal model assumes a large number of loci with small additive effects with free recombination, and a given and constant genic variance (i.e., the genetic variance at linkage and Hardy-Weinberg equilibrium; Barton et al., 2017); we also assume an isolated population of infinite size. We use this model to predict, at equilibrium and for each mating pattern, (i) the population mean breeding value and its mismatch to the female optimum as a function of genetic variance, and separately (ii) genetic variance as a function of genic variance. We then build an individual-based model assuming a finite and constant population size and a limited number of loci, to test for the robustness of the analytical predictions to the evolution of both genic and genetic variances.

We describe the life cycle of an annual species with separate sexes, although this model can apply to hermaphroditic species with different selection functions for females and males. We model a unique trait that affects the female and male fitness
components (see Section 1.1 below). Natural selection, mating, and recombination occur successively within each of the annual non-overlapping generations. We assume that male availability does not limit female reproduction, so that the population growth rate depends solely on the number of females in the population ("female demographic dominance"; Crowley, 2000; Harts et al., 2014). When female and male optima differ, no single phenotypic value can maximize both female and male fitness. As a result, the position of the population mean phenotype at equilibrium relative to the female or the male optimum determines female and male maladaptation, respectively. We focus here on female maladaptation only, measured by the mismatch between the population mean breeding value and the female optimum, although the model also predicts male maladaptation.

## 1. Analytical model

We derive predictions for the deterministic changes across generations in the population mean breeding value $\bar{g}$ and genetic variance $V_g$ of a single quantitative trait for females ($f$) and males ($m$). The phenotype of an individual is the sum of its breeding value and its environmental effect, $z = g + e$. At birth, breeding values are distributed as a Gaussian with mean $\bar{g}$ and variance $V_g$; random environmental effects are distributed as a Gaussian with a null mean $\bar{e}$ and constant variance $V_e$. At birth, female and male phenotypes are thus distributed as a Gaussian with mean $\bar{z}_f = \bar{z}_m = \bar{g}$ and variance $V_z = V_{zf} = V_{zm} = V_g + V_e$. We derive, separately for females and males, the changes from one generation to the next in the bivariate Gaussian distribution of breeding and phenotypic values due to (i) natural selection on females and males, (ii) sexual selection generated by assortative mating between mates, and (iii) recombination and segregation. We use an infinitesimal model of trait inheritance, assuming that, conditional on the breeding value of the mother and the father, the distribution of breeding values of the offspring in any family is distributed as a Gaussian with mean value equal to the average breeding value over the two parents, and a variance equal to half the genic variance $V_{LE}$. Here genic variance is assumed constant, so genetic variance can evolve only through changes in associations among allelic effects. All derivations are provided in Supplementary Information A.
## 1.1 Sex-specific natural selection

Selection is sex-specific such that fitness for an individual of sex j is given by $W_j(z_j) = \exp\left(-\frac{(z_j - \theta_j)^2}{2\omega_j^2}\right)$, and declines as a Gaussian function of the distance between the phenotype $z_j$ and the sex-specific optimum $\theta_j$, with the width of selection defined by $\omega_j$. This fitness function integrates selection on viability and fecundity (see Engen et al., 2011 for how the optimal phenotype maximizing fitness can be derived from the different optima for each life history component), and it can result from natural and/or sexual selection arising from other processes than assortative mating. In species with separate sexes, this sex-specific fitness arises when the optimal conditions for viability and/or fecundity differ between sexes. In hermaphroditic species, it arises when the fecundity optima differ between sexes, since both sexes share the same viability optimum. The distance between the female and male optima, $\delta_\theta = \theta_f - \theta_m$, measures the strength of sexual conflict in a constant environment. In a constant environment, optima are constant through time, while in a changing environment, female and male optima change through time at the same constant speed k (Table 1). As a result, fitness functions are year-specific, although, for clarity, year-dependence is omitted in the notations below.

## 1.2 Mating patterns

Under random mating, females can mate with any male in the population, and individuals (females and males) compete with all others of the same sex to produce offspring. The proportion of offspring born from a given mother with phenotype $z_f$ and that are sired by a father of phenotype $z_m$, denoted $w_{mR}(z_m|z_f)$, depends solely on the male fitness:

$$w_{mR}(z_m|z_f) = \frac{W_m(z_m)}{\iint p_m(z_m, g_m)W_m(z_m)dg_mdz_m}, \quad (1)$$

with $p_m(z_m, g_m)$ the joint distribution of phenotypic and breeding values of males before natural selection and reproduction.

Under assortative mating by female preference, the proportion of offspring sired by a father with a given mother increases with the male fitness, and the similarity between
237 the male phenotype $z_m$ and the female phenotype $z_f$:

$$w_{mP}(z_m|z_f) = \frac{W_m(z_m)A_P(z_m|z_f)}{\iint p_m(z_m, g_m)W_m(z_m)A_P(z_m|z_f)dg_mdz_m}, \quad (2)$$

238 with $A_P(z_m|z_f)$ the preference function of a female for males of phenotype $z_m$ centered on its own phenotype $z_f$ and with a range $\alpha^2$ (Table 1):

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$$A_P(z_m|z_f) = \exp\left(\frac{-(z_f - z_m)^2}{2\alpha^2}\right). \quad (3)$$

240 The preference range is constant across females and years. The strength of stabilizing selection applying to male phenotypes to match the female phenotypes increases with the strength of the assortment, quantified by $1/\alpha^2$.

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243 Under assortative mating by time, and unlike assortative mating by female preference, competition for access to a given female varies over time within a given year, because the quantity of male gametes fluctuates within a year. The proportion of offspring sired by a father with phenotype $z_m$ with a specific mother with phenotype $z_f$ depends on the male fitness, the overlap between the female and male phenologies, and the overlap between the male phenology and the phenologies of the other competing males:

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$$w_{mT}(z_m|z_f) = \int \phi_f(t|z_f) \frac{W_m(z_m)\phi_m(t|z_m)}{\iint p_m(z_m, g_m)W_m(z_m)\phi_m(t|z_m)dg_mdz_m} dt \quad (4)$$

249 with $\phi_f(t|z_f) = \frac{1}{\sqrt{2\Pi\alpha^2}} \exp\left(\frac{-(z_f - t)^2}{2\alpha^2}\right)$ (respectively $\phi_m(t|z_m)$) the probability density function describing the probability that gametes for female $z_f$ (respectively male $z_m$) are produced at time $t$ within the year.

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## 2. Individual-based model

253 The analytical predictions at equilibrium for the population mean breeding value, its mismatch to the female optimum, and for genetic variance are compared to the outcomes of simulations from an individual-based model (modified from Godineau et al. 2021). This model is built with the same assumptions as the analytical model except that (i) population size is finite and constant ($N = 1000$), (ii) the number of loci determining the quantitative trait is limited ($L = 5$), (iii) both genic $V_{LE}$ and genetic

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variances $V_g$ for the trait can evolve across generations, (iv) individuals contribute through female and male functions and therefore are hermaphroditic, (v) mutations per allele for all loci occur at rate $\mu = U/(2L)$, $U = 0.1$ being the genomic mutation rate, and are sampled in a Normal distribution of null expectation and variance equal to $V_m/U$ with $V_m = 0.04$ the mutational variance introduced each generation, and (vi) segregation of alleles is explicitly modeled.

## 2.1 Matings

Regardless of the mating pattern, $N$ females are sampled with replacement in the population according to their fitness. Under random mating, $N$ males are sampled with replacement in the population according to their fitness, and then randomly paired with the females. Under assortative mating by female preference, males are sampled according to the product of their relative fitness and the preference function of the females (eqn. 3). Under assortative mating by time, males are sampled according to the proportion of their gametes produced at time $t$ ($\phi_f(t|z_f)$) and their relative fitness.

## 2.2 Metrics recorded

As for the analytical model, we record the mismatch between the population mean breeding value and the female optimum. The genetic variance $V_g$ is estimated before selection as:

$$V_g = \frac{1}{N-1} \sum_{j=1}^{N} \sum_{l=1}^{L} (a_{fjl} + a_{mjl}) - \bar{g})^2 \quad (5)$$

with $a_{fjl}$ (resp. $a_{mjl}$) the maternal (resp. paternal) allelic effect at locus $l$ for individual $j$ and $\bar{g}$ the population mean breeding value of the trait.

The genetic variance is estimated as:

$$V_{LE} = \frac{1}{N-1} \sum_{j=1}^{N} \sum_{l=1}^{L} (a_{fjl} - \bar{a}_{fl})^2 + (a_{mjl} - \bar{a}_{ml})^2 \quad (6)$$

with $\bar{a}_{fl}$ (resp. $\bar{a}_{ml}$) the population mean maternal (resp. paternal) allelic effect at locus $l$.
## 2.3 Simulations and parameter choice

Simulations last for 3000 generations, which ensures that populations have reached equilibrium (the average change in genetic variance per generation over the last 1000 generations is within an interval [-0.008; 0.008]). In a constant environment, simulations are initialized with allelic effects sampled in a Normal distribution of null expectation and variance equal to 4 (100 times the mutational variance). In a changing environment, simulations start with a population at mutation-random genetic drift-selection equilibrium under a constant environment. The mismatch between the population mean breeding value and the female optimum, as well as genetic and genetic variances, are averaged over the last 1000 generations to discard expected fluctuations due to random genetic drift. Simulations are replicated 10 times per parameter set. We use the same parameter values in the analytical and individual-based models. This parameter range encompasses a diversity of combinations of the main parameters of the models, namely, the strength of natural selection on females relative to that on males, the strength of assortment, the strength of sexual conflict and the speed of the environmental change. Parameters were chosen after inspection of their combined effects on female maladaptation using the analytical model only, while keeping constant the harmonic mean of the width of the female and male functions (i.e maintaining the overall strength of natural selection constant, unless explicitly stated; Table 1).

## Results

The analytical model predicts female maladaptation (the total mismatch between the population mean breeding value and the female optimum) as a function of genetic variance (see Supplementary Information A). We show that female maladaptation has the same expression under assortative mating (A) by female preference (P) or by time (T):

$$M_P = M_T = M_A = \frac{S_{nm}(1-S_s)(\theta_f - \theta_m)}{2\bar{S}_A} + \frac{k}{h^2\bar{S}_A} \quad (7)$$

with $\bar{S}_A = \frac{S_{nf}(1+S_s)+S_{nm}(1-S_s)}{2}$ the strength of selection on the trait under assortative mating,
S<sub>nf</sub> = V<sub>z</sub>/V<sub>z</sub>+ω<sub>f</sub><sup>2</sup>, respectively S<sub>nm</sub> = V<sub>z</sub>/V<sub>z</sub>+ω<sub>m</sub><sup>2</sup>, the strength of natural selection on females and males,

S<sub>s</sub> = (V<sub>z</sub>(1-S<sub>nm</sub>)/V<sub>z</sub>(1-S<sub>nm</sub>)+α<sup>2</sup>) the strength of sexual selection, which increases with the strength of the assortment, 1/α<sup>2</sup>, and h<sup>2</sup> the heritability of the trait (Table 1).

Note that sexual selection is defined here specifically as the additional stabilizing selection on the trait that arises from assortative mating (Kirkpatrick and Nuismer, 2004).

If other forms of sexual selection also acted on the trait, they would be incorporated into the sex-specific fitness function.

Under random mating (R), for which sexual selection is null, female maladaptation simplifies to:

$$M_R = \frac{S_{nm}(\theta_f - \theta_m)}{2\bar{S}_R} + \frac{k}{h^2\bar{S}_R}, \quad (8)$$

with $\bar{S}_R = \frac{S_{nf}+S_{nm}}{2}$ the strength of selection on the trait under random mating.

For any mating pattern, the first component of female maladaptation corresponds to a sexual-conflict mismatch, M<sub>0</sub>, due to the sexual conflict in a constant environment that moves the evolutionary optimum (i.e., the phenotypic value that maximizes the population fitness) away from the female optimum. This sexual-conflict mismatch is under assortative mating:

$$M_{0A} = \frac{S_{nm}(1-S_s)(\theta_f - \theta_m)}{2\bar{S}_A}, \quad (9a)$$

and under random mating:

$$M_{0R} = \frac{S_{nm}(\theta_f - \theta_m)}{2\bar{S}_R}. \quad (9b)$$

Also for any mating pattern, the second component of female maladaptation corresponds to an adaptive lag, M<sub>k</sub>, which prevents the population from reaching the evolutionary optimum. This adaptive lag is under assortative mating:
$$M_{kA} = \frac{k}{h^2 \bar{S}_A}, \qquad (10a)$$

and under random mating:

$$M_{kR} = \frac{k}{h^2 \bar{S}_R}. \qquad (10b)$$

Both components of female maladaptation depend on the strength of selection on the trait ($\bar{S}_A$ and $\bar{S}_R$, eqns. 7 and 8). Note that $\bar{S}_A$ under assortative mating corresponds to the mean of natural selection on females and males weighted by sexual selection, such that increased sexual selection has the same effect as increased natural selection on females and decreased natural selection on males.

## 1. Female maladaptation due to sexual conflict

Female maladaptation in a constant environment (eqns. 9a and 9b) increases with a stronger sexual conflict (i.e., larger difference between the sex-specific optima), weaker natural selection on females relative to the total strength of selection on the trait, and weaker sexual selection (Fig. 1 C, F, and I). This sexual-conflict mismatch has the sign of $\delta_{\theta} = \theta_{f} - \theta_{m}$, and the population mean breeding value always lies between the female and male optima (Fig. 2). If we ignore potential differences in genetic variance between mating patterns, the sexual-conflict mismatch is predicted to be smaller under assortative mating, compared to random mating, because sexual selection strengthens the effect of natural selection on females, and weakens the impact of natural selection on males, thus shifting the population mean breeding value towards the female optimum. When the strength of sexual selection is maximal ($S_s = 1$), the effect of selection on males vanishes, and female adaptation is predicted to be perfect, with the population mean breeding value matching the female optimum (eqn. 9a; Fig. 1C, F, and I).

## 2. Female maladaptation due to environmental change

For all mating patterns and all else being equal, the adaptive lag, $M_k$, of the mean population breeding value to the evolutionary optimum (i.e., the population mean phenotype in a constant environment) is predicted to have the same sign as the environmental change, to scale with it, and to be independent of the positions of the optima (eqns. 10a and 10b). This lag under assortative mating equals that under
random mating only if natural selection is equally strong in both sexes (dotted lines in Fig. 1A and B; eqns. 10a and 10b). Sexual selection and natural selection on females interact to determine the size (i.e., absolute value) of the adaptive lag caused by the environmental change. When natural selection is stronger on females than on males, selection on the trait tends to be stronger under assortative than random mating, as $\bar{S}_A > \bar{S}_R$ if genetic variance is the same for the two mating patterns. Under this condition, the lag is smaller under assortative mating than under random mating, regardless of the direction of the environmental change. Conversely, when natural selection is weaker on females than on males, the adaptive lag is larger under assortative compared to random mating and larger for stronger assortative mating (Fig. 1A and B; eqns. 10a and 10b).

### 3. Total female maladaptation in a changing environment

Total female maladaptation under environmental change results from various combinations of the signs and sizes of its two components caused by the sexual conflict and the moving optima (eqns. 7 and 8; Fig. 1D, E, G, H, J, K). These relationships are simply reversed when the positions of the female and male optima are also reversed, and we therefore only present results with $\delta_{\theta} = \theta_{f} - \theta_{m} > 0$, i.e., when natural selection favors larger female than male trait values. All else being equal, increased sexual conflict (difference between the optima, $\delta_{\theta}$) increases the effect on female maladaptation of the relative strength of natural selection on females, and its interaction with the strength of sexual selection (compare lines in Fig 1; see also eqns. 7 and 8). When natural selection favors larger female than male trait values and the environmental change favors increasing values of the optima ($\delta_{\theta} > 0$ and $k > 0$), both the sexual conflict and the environmental change move the population mean breeding value away from the female optimum (Figs. 1D, G, J and 2A, B). In contrast, when natural selection favors larger female than male trait values, but the environmental change favors decreasing values of the optima ($\delta_{\theta} > 0$ and $k < 0$), the sexual-conflict mismatch and the adaptive lag have opposite signs and thus compensate for each other (Fig. 2C-D). In such case, the adaptive lag can shift the population mean phenotype closer to the female optimum than would be expected based on the sexual conflict mismatch alone (Fig. 2C). However, if the population mean breeding value is already close to the female optimum in a constant environment, the environmental change can make the
population mean breeding value overshoot the female optimum (Fig. 2D). This scenario can result in better (Fig. 2F) or worse (Fig. 2D and E) female adaptation under assortative compared to random mating, depending on the strength of female versus male natural selection, the strength of sexual selection and the speed of the environmental change (Fig. 1 E, H, K; see also eqns. 7 and 8). We thus now identify general conditions under which assortative mating, compared to random mating, facilitates female adaptation to a changing environment. We begin by finding the conditions under which environmental change reduces or cancels the sexual-conflict mismatch, and then compare these conditions between the two mating patterns. Then, we describe the conditions under which assortative mating, compared to random mating, reduces female maladaptation (i.e., $|M_A| < |M_R|$).

## 3.1 Conditions for which female maladaptation is reduced by environmental change

We define the optimal speed of the environmental change, $\tilde{k}$, that cancels the mismatch to the female optimum, i.e., when the adaptive lag exactly compensates for the sexual-conflict mismatch; this optimal speed is under assortative mating:

$$ \tilde{k}_A = \frac{1}{2} h^2 S_{nm} (1 - S_s) (\theta_m - \theta_f), \qquad (11a) $$

which simplifies under random mating (for $S_s = 0$) to:

$$ \tilde{k}_R = \frac{1}{2} h^2 S_{nm} (\theta_m - \theta_f). \qquad (11b) $$

When selection favors larger female than male trait values ($\theta_f > \theta_m$), the optimal speeds of the environmental change leading to perfect female adaptation are negative (Fig. 3). Faster optimal rates of environmental change are required to compensate for the sexual-conflict mismatch existing in a constant environment with stronger natural selection on males than on females (Fig. 3; see also eqns. 11a and 11b), weaker sexual selection (compare Figs. S1A and Fig. 3), or stronger sexual conflict (compare Figs. S1B and Fig. 3). Indeed, these parameters increase female maladaptation in a constant environment, and a substantial displacement of the population mean breeding value is necessary to reach perfect female adaptation. Conversely, when the population mean breeding value in a constant environment is already close to the female optimum, i.e., when sexual selection is strong, natural selection on females
is strong, or sexual conflict is weak, only a slow environmental change can cancel female maladaptation (Figs. 3 and S1). If we assume the same genetic variance for both mating patterns, $\tilde{k}_A = (1-S_s)\tilde{k}_R$ thus, perfect female adaptation is reached under assortative mating for slower environmental changes than under random mating.

## 3.2 Conditions for which female maladaptation is equal under random and assortative mating

Within [$\tilde{k}_R, \tilde{k}_A$] there is a critical speed of the environmental change for which $M_A = -M_R$, i.e., the mismatches have the same size but are of opposite signs. When mating patterns share the same genetic variance, this critical speed is defined for any strength of natural selection and sexual selection as:

$$k_o^* = h^2 S_{nm} (\theta_m - \theta_f) \frac{S_{nf} + S_{nm}(1-S_s)}{S_{nf}(2+S_s) + S_{nm}(2-S_s)} \quad (12a)$$

Outside the boundaries defined by [$\tilde{k}_R, \tilde{k}_A$], the mismatches have the same signs for random and assortative mating. When the strength of selection is different on females and males, the critical speed of the environmental change allowing $M_A = M_R$ is defined by:

$$k_s^* = h^2 S_{nm} (\theta_f - \theta_m) \frac{S_{nf}}{S_{nm} - S_{nf}} \quad (12b)$$

When selection is equally strong on females and males, there is no speed of the environmental change that can make the size of female maladaptation equal for the two mating patterns.

Critical speeds, $k_s^*$ and $k_o^*$, increase with the strength of natural selection on females and the difference between the optima, while only $k_o^*$ depends on the strength of sexual selection. These critical speeds define intervals within or outside which assortative mating facilitates female adaptation (Figs. 3 and S1). At $k_s^*$ and $k_o^*$, the benefit of assortative mating on the sexual-conflict mismatch is canceled by the larger adaptive lag that it produces compared to random mating, and adaptation of populations is the same as for random mating (but the sizes of the sexual-conflict mismatch and the adaptive lag differ between mating patterns).
### 3.3 Conditions for which assortative mating facilitates female adaptation

For an equal strength of natural selection on females and males (gray dashed line in Fig. 3), adaptation is better under assortative compared to random mating for any speed of the environmental change greater than $k_o^*$ (purple line in Fig. 3). For weaker natural selection on females than males ($S_{nf} < S_{nm}$, left of the gray dashed line in Fig. 3), assortative mating facilitates female adaptation for $k \in [k_s^*, k_o^*]$. Within this range, the benefit of assortative mating on the sexual-conflict mismatch is not canceled by the larger adaptive lag that assortative mating produces compared to random mating (Fig. 2C). For stronger natural selection on females than males ($S_{nf} > S_{nm}$, right of the gray dashed line in Fig. 3), assortative mating facilitates adaptation for $k$ larger than $k_o^*$ and smaller than $k_s^*$. When the environmental change favors increasing values of the optima, both the sexual-conflict mismatch and the adaptive lag are smaller under assortative compared to random mating (Fig. 2B). When the environmental change favors fast decreasing values of the optima ($k < k_s^* < 0$), female maladaptation is also smaller under assortative compared to random mating for the same reasons, but in that case, the population mean breeding value overshoots the female optimum (Fig. 2F). Increased strength of sexual selection shrinks the range of speeds of the environmental change in which assortative mating facilitates female adaptation (compare Figs. 3 and S1A) but increases the benefit of assortative mating within this range (shown as color intensity in figures), as it decreases both components of the mismatch (Fig. 1 and 2). Increased difference between the optima increases both the benefit of assortative mating and the range of speeds of the environmental change in which assortative mating helps adaptation (Fig. 3 and S1B).

### 4. Predicted and evolved genetic variance at equilibrium

The above results predict female maladaptation as a function of genetic variance but do not consider potential differences in the evolved genetic variance between mating patterns. Here, we show that different amounts of genetic variance $V_g$ are expected at equilibrium (see Supplementary Information A for derivations) under random mating, assortative mating by female preference, and by time. However, in each case, genetic variance can be expressed similarly as a function of genetic variance, $V_{LE}$:

$$V_g^* = \frac{V_{LE}}{1 + h^2 \tilde{S}}, \qquad (13)$$
with $\tilde{S}$ a compound measure of the stabilizing effect of natural and sexual selection on the trait, which is for assortative mating by female preference:

$$ \tilde{S}_P = \frac{S_{nf}(1 + S_s)^2 + S_{nm}(1 - S_s) - S_s(1 + S_s)}{2}, \qquad (14a) $$

for assortative mating by time:

$$ \tilde{S}_T = \frac{S_{nf}(1 + S_s)^2 + S_{nm}(1 - S_s^2) - 2S_s}{2}, \qquad (14b) $$

and for random mating:

$$ \tilde{S}_R = \frac{S_{nf} + S_{nm}}{2}. \qquad (14c) $$

For all mating patterns, genetic variance is predicted to increase with increased genetic variance and decreased compound measure of the stabilizing selection. Under random mating, genetic variance is systematically predicted to be lower than genetic variance as $\tilde{S}_R > 0$, and genetic variance decreases with the strength of total natural selection, $\tilde{S}_R$ (eqn. 14c, and compare green curves below the grey line between Fig. 4A-C and Fig. 4D-F). For assortative mating, whether by female preference or time, genetic variance is expected to increase with increased strength of sexual selection in the parameter range explored (eqn. 14b and 14a; compare orange and red lines Fig. 4A-C with Fig. S2A-C). Still, it can be smaller or larger than genetic variance, depending on complex and non-intuitive interactions between the strength of natural and sexual selection (eqns. 14a and 14b, see Supplementary Information B). For example, genetic variance tends to be higher than genetic variance (Fig. 4) except when sexual selection is weak, as observed under random mating (red and orange curves below the grey line in Fig. S2A-C). If everything else is kept equal, including $V_{LE}$, $\tilde{S}_P > \tilde{S}_T$, and thus the analytical model predicts that assortative mating by time maintains more genetic variance than does assortative mating by female preference (eqns. 14a and 14b; also compare orange with red curves in Figs. 4 and S2).

For the explored combinations of values for natural selection, sexual selection, sexual conflict, and speed of the environmental change, and using the amount of $V_{LE}$ measured in the simulations, the analytical predictions at equilibrium for genetic variance
V<sub>g</sub> deviate on average by a few percents from those obtained in the simulations unless
natural selection is weak (Figs. 4 and S2). This good fit of the predictions and the
simulations suggests that, conditional on knowing V<sub>LE</sub>, the analytical model predicts
fairly well the genetic variance, including the size of the associations created among
allelic effects under assortative mating by female preference and by time.

The simulations show that the amount of genic variance V<sub>LE</sub> maintained at equilib-
rium in the populations is increased by a changing, compared to a constant environ-
ment (compare up or down triangles with squares in Figs. 4 and S2), weaker natural
selection (compare Fig. 4A-C with Fig. 4D-F), weaker assortative mating (compare
Fig. 4A-C with Fig. S2A-C), and stronger sexual conflict (compare Fig. 4A-C with Fig.
S2D-F). Also, the amount of V<sub>LE</sub> at equilibrium is affected by the type of assortative
mating, with V<sub>LE</sub> systematically smaller for assortative mating by female preference
than by time (compare orange and red symbols in Figs. 4 and S2).

The evolution of genetic variance depends on the evolution of V<sub>LE</sub>, as it scales with it,
and on the potential of assortative mating to create associations among allelic effects,
which is encompassed into the compound measure of the stabilizing effect of natural
and sexual selection (eqn. 13). The genic and genetic variance that have evolved un-
der assortative mating by female preference are always the lowest, highlighting that
associations among allelic effects created by this mating pattern do not compensate
for the loss of genetic polymorphism (Figs. 4 and S2). As predicted by the analytical
model, genetic variance is always lower than genic variance under random mating
(Figs. 4 and S2), and under assortative mating when sexual selection is weak (Fig. S2
A-C). Genetic variance can be larger under assortative mating by time in the explored
parameter range than under random mating. The conditions for which this happens
depend on whether the positive associations among allelic effects created under assor-
tative mating, which increase with genic variance and the strength of sexual selection
but decrease with the strength of natural selection on females, can compensate for
the loss of genic variance compared to random mating (Figs. 4 and S2).

**5. Effect of the evolution of the genetic variance on female maladaptation**

The analytical model predicts that the size of female maladaptation depends on genetic
variance, V<sub>g</sub>, through its effect on the strength of natural selection on females, S<sub>nf</sub>,
and males, S<sub>nm</sub>, the strength of sexual selection, S<sub>s</sub>, and the heritability of the trait,
h² (eqns. 7 and 8). In the presence of sexual conflict, the relation predicted between female maladaptation and genetic variance for assortative mating, whether by time or preference, differs from that for random mating. This relation is not linear, not always monotonic (Figs. 5 and S6), and depends on the direction and speed of the environmental change (compare Fig. 5D-F with A-C or G-I), the strength of sexual selection (compare panels in Fig. 5 with Fig. S3), the total strength of natural selection (compare panels in Fig. 5 with Fig. S4) and the strength of the sexual conflict (compare panels in Fig. 5 with Fig. S5). Therefore, increased genetic variance at equilibrium does not necessarily decrease female maladaptation (e.g., Fig. 5G) but there can be a value of genetic variance that minimizes female maladaptation; this value strongly depends on the direction of the environmental change and the mating pattern (Fig. 5).

For the parameter range explored, in a changing environment, when selection favors larger female than male trait values ($δ_θ > 0$) and the environmental change favors increasing values for the optima ($k > 0$), for all mating patterns, increasing genetic variance is expected to decrease both the sexual-conflict mismatch (eqn 9; Fig. 5 D-F; long-dashed lines on Fig. S6) and the adaptive lag (eqn 10; dotted lines on Fig. S6), and thus to decrease female maladaptation (Fig. 5A-C). However, there is no value of genetic variance that cancels female maladaptation. When selection favors larger female than male trait values ($δ_θ > 0$) but the environmental change favors decreasing values for the optima ($k < 0$), the effect of increasing genetic variance on female maladaptation depends on many parameters, which define the compensation level between the positive sexual-conflict mismatch and the negative adaptive lag (Figs. 5G-I and S3-5 G-I). In that case, the relation of female maladaptation with genetic variance depends on the sizes of both components. These sizes decrease with increasing genetic variance, but with different curvatures and signs (long- and short-dashed lines in Fig. S6; eqns. 9 and 10). When the size of the adaptive lag remains smaller than that of the sexual-conflict mismatch, the population mean breeding value lies between the female and male optima (the mismatch is positive, Fig. S6). When the size of the adaptive lag is larger than that of the sexual-conflict mismatch, the population mean breeding value overshoots the female optimum and lies outside the range of the two optima. There is thus an optimal value of genetic variance that can cancel female
maladaptation (Fig. S6; eqns. 7 and 8).

We wish here to address the consequences on female maladaptation of the different levels of genetic variance that evolved under the different mating patterns. Using the amount of genetic variance $V_g$ evolved in the simulated populations, and for the explored combinations of parameters, the predictions of female maladaptation from the analytical model match well the simulations (Fig. 5 and S3-5). Deviations are larger under assortative than random mating, most probably because effective population size is expected to be smaller, and thus random genetic drift is expected to be stronger. However, when predictions and simulations deviate substantially, the mismatches are actually small (Figs. 5 and S3-5). The populations in the simulations never evolve towards the optimal values of genetic variance that would minimize or cancel female maladaptation when $\delta_{\theta} > 0$ and $k < 0$. In a few cases, female adaptation is better under assortative mating by female preference than by time, showing that lower evolved genetic variance does not always confer worse adaptation (Figs. 5G and S3G-H). Similarly, increased genetic variance under random mating rarely translates into better adaptation compared to assortative mating (but see Fig. 5I).

If the differences in genetic variance between the three mating patterns do affect quantitative prediction of female maladaptation, in the explored scenarios, they do not affect the conclusions previously mentioned. When we predict a larger mismatch to the female optimum under random than assortative mating while assuming the same genetic variance, we also obtain a similar pattern in the simulations in which genetic variance evolves (Figs. 5 and S3-5). For example, we predict and measure in the simulations that female maladaptation is larger under random mating than under assortative mating when the environmental change is $k = -0.2$ and natural selection on females is stronger than on males (Fig. 5G).

## Discussion

We investigate here how the combination of sexual selection and sexual conflict affects the mismatch between the population mean breeding value and the female optimum, and the genetic variance of a trait in a gradually changing environment. We find that the type of assortative mating does not affect the predicted relation between adaptation and genetic variance, but does affect the relation between genetic variance and
genetic polymorphism. We show that a sexual-conflict mismatch in a constant envi-
ronment, and an adaptive lag add up and sometimes compensate for each other to
determine adaptation. We define domains in which each mating type facilitates adap-
tation compared to the others; these domains depend on complex relations among
all components investigated: the strength and the type of assortment, the strength
of the sexual conflict, and the speed of the environmental change. Below, we discuss
the mechanisms by which assortative mating, compared to random mating, affects
the components of female maladaptation, and the components of genetic variance.

Combined effects of assortative mating and sex-specific natural selection on the
components of female maladaptation

Female adaptation depends on the balance between (i) a sexual-conflict mismatch in a
constant environment caused by sex-specific natural selection, and (ii) an adaptive lag
caused by the shift of the optima over time. The sexual conflict generates maladapta-
tion for both females and males, as their trait cannot evolve independently and cannot
reach their respective optima (Lande, 1980). The population mean phenotype, how-
ever, evolves towards its evolutionary optimum, at which the population mean fitness
is maximized. Sexual selection caused by assortative mating further increases male
maladaptation, but decreases female maladaptation. Sexual selection indeed oper-
ates on males to resemble the most common and/or successful females, shifting the
evolutionary optimum towards the female optimum compared to a population under
random mating (Lande, 1980). The lag induced by the environmental change across
generations is independent of the sexual conflict, and thus has the same expression
as the adaptive lag of Bürger and Lynch (1995) under random mating, and Godineau
et al. (2021) under assortative mating.

When the adaptive lag and the sexual-conflict mismatch both push the population
mean breeding value away from the female optimum, assortative mating facilitates fe-
male adaptation when natural selection is stronger on females than on males, or when
the environmental change is not too fast. This scenario corresponds, for example, to a
situation in which climate change favors earlier breeding times in both sexes each year
and with the female optimum being earlier than the male optimum; or to a situation
where climate change favors smaller individuals in both sexes, with females being
smaller than males. Increased assortative mating - for example caused by smaller
within-individual trait variation or larger inter-individual trait variation (Devaux and Lande, 2010) - would shift the mean trait value closer to the female optimum.

When the adaptive lag is opposite in direction to the sexual-conflict mismatch, the environmental change can decrease female maladaptation compared to a constant environment, or conversely, the sexual conflict can reduce the female maladaptation caused by the environmental shift. If the environment changes slowly, assortative mating facilitates female adaptation, for any value of natural selection on females, as its large beneficial effect in a constant environment, compared to random mating, cannot be canceled by its detrimental effect on the adaptive lag. If the environment changes quickly, assortative mating allows better adaptation than does random mating, but only if natural selection is stronger on females than on males. In that case, the population mean breeding value may or may not overshoot the female optimum, depending on the amount of evolved genetic variance. This corresponds, for example, to plant or animal species undergoing climate change that favors earlier or smaller individuals in both sexes and for which the male optimal breeding time is earlier than that of females, or, to animal species, for which the male optimal size is smaller than that of females. The female phenotypes in the constant environment are too early or too small relative to the female optimum because of conflicting selection with males, but are pre-adapted to climate change.

### Combined effects of assortative mating and sex-specific natural selection on the components of genetic variance

In line with previous studies (Godineau et al., 2021; Kirkpatrick and Nuismer, 2004), our results illustrate again the antagonistic effects of assortative mating on genetic variance. Sexual selection generated by assortative mating decreases genetic polymorphism, by shrinking, compared to random mating, the pool of males available for mating with the fittest females. Sexual selection also inflates genetic variance by creating positive associations among allelic effects, especially when natural selection is weak. Analytical predictions here show new effects of assortative mating on the inflation of genetic variance, compared to random mating: for a given amount of genetic polymorphism, the inflation is larger when natural selection is weaker on females than on males. Furthermore, for the same amount of genetic variance, assortative mating by time has a greater opportunity to create associations among allelic effects than does
649 assortative mating by preference (eqn. 13 with $V_{g,P} < V_{g,T}$).

650 Simulations show that genic and genetic variances are affected by the type of assortative mating. Our individual-based model therefore confirms that the frequency-dependence of mating probability under assortative mating by time maintains more 651 genetic polymorphism than does assortative mating by female preference (Fox, 2003; 652 Weis et al., 2005). Indeed, males under assortative mating by time compete with different temporal pools of males, which protects the genetic polymorphism. Genetic 653 variance in simulations also depends on the speed of the environmental change, consistent with previous studies under random (Bürger and Lynch, 1995) or assortative 654 mating (Godineau et al., 2021). 655
656
657 mating (Godineau et al., 2021).

658 **Effects of genetic variance on adaptation**

660 Classical theory of adaptation to directional selection (Kopp and Matuszewski, 2014; 661 Lande and Shannon, 1996) predicts that increasing genetic variance increases pop- 662 ulation adaptation. We find the same pattern when the sexual-conflict mismatch 663 and the adaptive lag push the population mean breeding value in the same direction, 664 but not when they push it in opposite directions. In that case, and under certain 665 conditions, we predict that an optimal amount of genetic variance can cancel female 666 maladaptation, but this optimum is never reached in the simulated populations. 667

667 Simulations also show that female maladaptation under the two types of assortative 668 mating follows the same line of predicted mismatch as a function of genetic variance.

669 In other words, all else being equal, female maladaptation under assortative mating 670 is simply explained by the amount of genetic variance maintained in the populations.

671 Assortative mating by time facilitates female adaptation in most cases because it main-
672 tains larger genetic variance than does assortative mating by female preference for a
673 given strength of assortment. One exception to this pattern occurs when the increase
674 in genetic variance considerably reduces the sexual-conflict mismatch, and the adap-
675 tive lag is opposite and strong enough to move the population mean breeding value
676 away from the female optimum.

677 **Limitations and implications of the model**

678 The good fit between the predictions from the analytical model and the simulations
679 suggests that, conditional on knowing the genic variance, the strong assumptions
680 made by the infinitesimal model - a large number of loci, constant genic variance, no
681

demographic stochasticity or inbreeding - have a limited impact on the quality of its
predictions.

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We choose to present results for female maladaptation only, as done in many previous
theoretical and empirical studies (Connallon and Hall, 2016; Harts et al., 2014 but
see, Austen and Weis, 2016; Fraser et al., 2019; Tonnabel et al., 2019 for estimates
of male fitness components in plants or animals), because females usually contribute
more to population growth rate than do males (Harts et al., 2014). However, males do
contribute to the level of female maladaptation as their correlated response to sexual
and natural selection shifts the population mean phenotype away from the female
optimum. Any condition that reduces female maladaptation, however, increases male
maladaptation, which could impact population growth rates in cases of biparental
care or when males are rare and some females do not mate (Rankin and Kokko, 2007).

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Male maladaptation would be particularly high when the population mean phenotype overshoots the female optimum, for example, when the sexual-conflict mismatch and the adaptive lag are opposite and genetic variance is limited.

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We choose to model a trait with identical distribution in females and males, thus
focusing on the effects of intralocus sexual conflict and sexual selection mechanisms
on maladaptation without allowing for the evolution of the sexual conflict. Under a
constant environment, sexual conflict can be reduced or even canceled by reducing
the genetic correlation of the trait between sexes, or by increasing sex-specific trait
expression (Bonduriansky and Chenoweth, 2009; Lande, 1980; Poissant et al., 2010;
Rhen, 2000, Pennell et al., 2016) a form of phenotypic plasticity that is common for size
in many animals (Cox and Calsbeek, 2009; Pincheira-Donoso et al., 2021) or breeding
time in plants and animals (Bertin and Newman, 1993; Forrest, 2014; Glucksmann,
1974; Pedersen et al., 2019; Rubolini et al., 2004; Wang et al., 2021).

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Our model also assumes that the optimal phenotypic values for all fitness components
(female and male fecundity and viability) shift at the same rate; when it is not the case,
female maladaptation is not constant at equilibrium (see Cotto et al., 2019 for a sce-
nario in which optima for viability and fecundity shift at different rates, or Connallon
and Hall, 2016 in which the female and male optima shift at different rates). The
model of Connallon and Hall (2016) for random mating shows that if the female op-
timum shift is slower than the shift in the male optimum, but in the same direction,

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the genetic correlation between the male and female traits actually helps the females track their optimum.

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Our results show that understanding the genetic responses of traits requires estimates of the fitness landscape for each sex, each fitness component and across time. Variation in selection gradients is now reported (Morrissey and Hadfield, 2012; Siepielski et al., 2009), but it can be attributed to fluctuations in the fitness landscapes themselves and/or fluctuations in the phenotypic distribution (Chevin and Haller, 2014; de Villemereuil et al., 2020). Empirical data have also shown antagonistic selection gradients between females and males in natural populations and for several fitness components (Cox and Calsbeek, 2009; de Lisle et al., 2018), but are rarely used to infer the relative positions of female and male optima or their rates of change. Our results also show that sexual conflict can pre-adapt populations to climate change, and that increasing genetic variance in the populations, e.g. with assisted gene flow, can help them reach optimal values of genetic variance that cancel female maladaptation. Our model has a clear heuristic value: it identifies the processes, including the mating pattern, that drive adaptation to a changing environment, and guides future empirical studies of the effects of sex-specific natural selection, sexual selection, and mating pattern on adaptation.

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Figure 1: Sexual-conflict mismatch ($M_0$, eqns. 9a and b, first column of panels), adaptive lag ($M_k$, eqns. 10a and b, first row of panels), and their sum defining total female maladaptation at equilibrium (eqns. 7 and 8, panels D, E, G, H, J, K) shown as a function of the strength of sexual selection ($S_s$, with $\alpha^2$ from 0 to 1,000) for several scenarios of the speed of the environmental change ($k = 0.2$ and $k = -0.2$) and the strength of the sexual conflict ($\delta_\theta = \theta_f - \theta_m = 1, 3$, or 6). The strength of natural selection on females ($S_{nf}$) is indicated by a gradient of line widths and colors, with thinner orange lines representing larger values among $\omega_f^2 = 17, 24, 32, 47, 62$, and dashed black lines corresponding to $S_{nf} = S_{nm}$. Note that the harmonic mean of the width of the functions for natural selection on females and males, $\frac{2\omega_f^2\omega_m^2}{\omega_f^2+\omega_m^2}$, is fixed at 32. There would be no difference in the adaptive lag ($M_k$) under random mating for $S_s = 0$ across different $S_{nf}$ values, if total natural selection could be kept constant. Other parameters are: $V_g = 5$ and $V_e = 4$, resulting in $h^2 = 0.556$ (Table 1).

Figure 2: Partitioning of female maladaptation at equilibrium into the sexual-conflict mismatch ($M_0$, eqns. 9a and b, low-opacity arrows) and the adaptive lag ($M_k$, eqns. 10a and b, high-opacity arrows) shown for random mating (thin green arrows, $\alpha^2 \to \infty$), and for two strengths of assortment ($\alpha^2 = 46$, medium orange arrows; and $\alpha^2 = 6$, large orange arrows). The positions of the population mean breeding values at equilibrium (eqns. S42 and S57) are indicated by solid lines matching the corresponding color and thickness, while the female (pink) and male (blue) optima are represented by dashed lines. Six combinations of the speed of the environmental change ($k = 0.7, k = -0.2$, and $k = -0.7$) and the strength of natural selection on females ($S_{nf} = 0.1$ and $S_{nf} = 0.31$) are shown. Note that (i) the harmonic mean of the width of the functions for natural selection on females and males, $\frac{2\omega_f^2\omega_m^2}{\omega_f^2+\omega_m^2}$, is fixed at 32; (ii) arrows are slightly displaced for readability; (iii) lags to the moving female optimum are indicated at arbitrary positions along the X-axis, representing time once equilibrium has been reached; (iv) the arrowhead can be larger than the magnitude of the mismatch. Other parameters are: $\delta_\theta = \theta_f - \theta_m = 6$, $V_g = 5$ and $V_e = 4$, resulting in $h^2 = 0.556$ (Table 1).
Figure 5: Female maladaptation — total mismatch to the female optimum, *M* — for random mating (green), assortative mating by female preference (orange), or assortative mating by time (red), as a function of genetic variance (*V*<sub>g</sub>), predicted from the infinitesimal model (green and orange lines; eqns. 7 and 8) and evolved in the individual-based model, in which *V*<sub>LE</sub> evolves (green, orange, and red symbols, respectively). Several combinations are shown for the speed of environmental change, with squares for *k* = 0, upward triangles for *k* = 0.2, and downward triangles for *k* = −0.2, and for the strength of natural selection on females relative to males, with the harmonic mean of selection widths <2*ω*<sub>*f*</sub><sup>2</sup>*ω*<sub>*m*</sub><sup>2</sup> / (*ω*<sub>*f*</sub><sup>2</sup>+*ω*<sub>*m*</sub><sup>2</sup>) fixed at 32. Other parameters are: *δ*<sub>θ</sub> = *θ*<sub>*f*</sub> − *θ*<sub>*m*</sub> = 6, *V*<sub>*e*</sub> = 4, and *α*<sup>2</sup> = 6 (Table 1).
<table><caption>Table 1: Description of parameters and values explored in the analytical and individual-based models</caption><thead><tr><th>parameter</th><th>description</th><th>standard values</th><th>explored values</th></tr></thead><tbody><tr><td>θ<sub>f</sub></td><td>optimal value for females</td><td>3</td><td>1.5; 0.5</td></tr><tr><td>θ<sub>m</sub></td><td>optimal value for males</td><td>-3</td><td>-1.5; -0.5</td></tr><tr><td>δ<sub>θ</sub></td><td>difference between the optimal values for females and males, which partly determines the strength of the sexual conflict</td><td>6</td><td>3; 1</td></tr><tr><td>ω<sup>2</sup><sub>f</sub></td><td>width of the natural selection function on females</td><td>20; 32; 80</td><td>17; 24; 32; 47; 62</td></tr><tr><td>ω<sup>2</sup><sub>m</sub></td><td>width of the natural selection function on males</td><td>80; 32; 20</td><td>272; 48; 32; 24.3; 21.6</td></tr><tr><td>2ω<sup>2</sup><sub>f</sub>ω<sup>2</sup><sub>m</sub> / (ω<sup>2</sup><sub>f</sub>+ω<sup>2</sup><sub>m</sub>)</td><td>harmonic mean of the width of the natural selection functions, used to control for constant total natural selection</td><td>32</td><td>128; 256; 512</td></tr><tr><td>k</td><td>speed of the environmental change for both female and male optima</td><td>-0.2; 0; 0.2</td><td>-0.5; 0.5; 0.7</td></tr><tr><td>α<sup>2</sup></td><td>strength of assortment (by female preference or time) controlling the strength of sexual selection</td><td>6</td><td>16; 46</td></tr><tr><td>V<sub>g</sub></td><td>genetic variance</td><td>5</td><td></td></tr><tr><td>V<sub>LE</sub></td><td>genic variance</td><td>4</td><td></td></tr><tr><td>V<sub>e</sub></td><td>environmental variance</td><td>4</td><td></td></tr><tr><td>V<sub>m</sub></td><td>mutational variance</td><td>0.04</td><td></td></tr><tr><td>L</td><td>number of loci determining the quantitative trait</td><td>5</td><td></td></tr><tr><td>N</td><td>number of individuals</td><td>1000</td><td></td></tr><tr><td>h<sup>2</sup></td><td>heritability of the trait</td><td></td><td></td></tr><tr><td>S<sub>nf</sub></td><td>strength of natural selection on females</td><td></td><td></td></tr><tr><td>S<sub>nm</sub></td><td>strength of natural selection on males</td><td></td><td></td></tr><tr><td>S<sub>s</sub></td><td>strength of sexual selection</td><td></td><td></td></tr><tr><td>M</td><td>total female maladaptation</td><td></td><td></td></tr><tr><td>M<sub>0</sub></td><td>female maladaptation due to the sexual conflict</td><td></td><td></td></tr><tr><td>M<sub>k</sub></td><td>female maladaptation due to the environmental change</td><td></td><td></td></tr><tr><td>S&#773;</td><td>strength of selection</td><td></td><td></td></tr><tr><td>S&#771;</td><td>compound measure of the stabilizing selection</td><td></td><td></td></tr><tr><td>k&#771;</td><td>optimal speed of the environmental change</td><td></td><td></td></tr><tr><td>k<sup>*</sup><sub>o</sub></td><td>critical speed of the environmental change for which mismatch under assortative and random mating have the same size but are of opposite signs</td><td></td><td></td></tr><tr><td>k<sup>*</sup><sub>s</sub></td><td>critical speed of the environmental change for which mismatch under assortative and random mating have the same size and same sign</td><td></td><td></td></tr></tbody></table>
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