Keywords: Mimicry, Heterospecific mating, Evolutionary Divergence, Imperfect mimicry, Quantitative genetics, Phylogenetic signal. 1

Mutualistic interactions between defended species represent a striking case of evolutionary convergence in sympatry, driven by the increased protection against predators brought by mimicry in warning traits. However, such convergence is often limited: sympatric defended species frequently display different or imperfectly similar warning traits. The phylogenetic distance between sympatric species may indeed prevent evolution towards the exact same signal. Moreover, warning traits are also involved in mate recognition, so that trait convergence might result in heterospecific courtship and mating. Here, we develop a mathematical model to investigate the strength and direction of evolution of warning trait in defended species with different ancestral traits. Specifically, we determine the effect of phenotypic distances between ancestral trait states of sympatric defended species and of costs of heterospecific sexual interactions on imperfect mimicry and trait divergence. Our analytical results confirm that reproductive interference and historical constraints limit the convergence of warning traits, leading to either complete divergence, or imperfect mimicry. Our model reveals that imperfect mimicry evolves only when ancestral trait values differ between species due to historical constraints and highlights the importance of female and predator discriminations in the evolution of such imperfect mimicry. Our study thus provides new predictions on how reproductive interference interacts with historical constraints and may promote the emergence of novel warning traits, enhancing mimetic diversity.

Introduction

Mutualistic interactions frequently drive convergent evolution of different traits in sympatric species. For example, avian vocal resemblance has been suggested to allow the formation of mixed-species flocks where all individuals from the different species may benefit from a reduced predation risk and an increase of their foraging [START_REF] Goodale | Vocal mimicry by a passerine bird attracts other species involved in mixed-species flocks[END_REF]; likewise trait similarity between sympatric species may promote pollinator attraction in nectar-rewarding flowers [START_REF] Benitez-Vieyra | How to look like a mallow: evidence of floral mimicry between turneraceae and malvaceae[END_REF][START_REF] Schemske | Floral convergence and pollinator sharing in two bee-pollinated tropical herbs[END_REF]. In aposematic species, warning traits are associated with defenses against predators, such as venom or disgusting taste. Because predators eventually learn such associations, convergence in warning trait among defended species facing similar communities of predators is often observed (Müllerian mimicry, see [START_REF] Sherratt | The evolution of müllerian mimicry[END_REF] for a review).

Mimicry is certainly the best documented case of mutualistic interactions driving trait evolution in sympatric species and is observed in a wide range of organisms including plants [START_REF] Lev-Yadun | Müllerian mimicry in aposematic spiny plants[END_REF], mollusks [START_REF] Cortesi | Conspicuousness is correlated with toxicity in marine opisthobranchs[END_REF], vertebrates [START_REF] Sanders | Evidence for a müllerian mimetic radiation in asian pitvipers[END_REF][START_REF] Springer | Mimetic relationships involving fishes of the family blenniidae[END_REF] and insects [START_REF] Mallet | Why are there so many mimicry rings? correlations between habitat, behaviour and mimicry in heliconius butterflies[END_REF]. Field experiments report the intense selection exerted by predators favoring warning trait convergence in sympatry [START_REF] Arias | Crossing fitness valleys: empirical estimation of a fitness landscape associated with polymorphic mimicry[END_REF][START_REF] Benson | Natural selection for mullerian mimicry in heliconius erato in costa rica[END_REF][START_REF] Chouteau | Warning signals are under positive frequencydependent selection in nature[END_REF][START_REF] Kapan | Three-butterfly system provides a field test of müllerian mimicry[END_REF][START_REF] Mallet | Strong natural selection in a warning-color hybrid zone[END_REF]. Surprisingly, despite such intense selection, many sympatric defended species exhibit only imperfect resemblance (e.g. [START_REF] Savage | The colouration of the venomous coral snakes (family elapidae) and their mimics (families aniliidae and colubridae)[END_REF]) or even different warning traits (e.g. [START_REF] Beccaloni | Ecology, natural history and behaviour of ithomiine butterflies and their mimics in ecuador (lepidoptera: Nymphalidae: Ithomiinae)[END_REF]) [START_REF] Briolat | Diversity in warning coloration: selective paradox or the norm?[END_REF].

The level of trait convergence between sympatric species may vary depending on their level of phylogenetic relatedness. For instance, the significant phylogenetic signal observed on the warning trait of mimetic butterflies of the tribe Ithomiini [START_REF] Chazot | Mutualistic mimicry and filtering by altitude shape the structure of andean butterfly communities[END_REF][START_REF] Elias | Mutualistic interactions drive ecological niche convergence in a diverse butterfly community[END_REF] suggests that historical constraints may limit the convergent evolution of warning traits. Such historical constraints are expected to be more different between distantly related species than closely-related ones, because closely-related species are expected to share similar ancestral trait values. These historical constraints may also imply differences between clades in both (1) the developmental pathway involved in the variation of warning traits or (2) the selective trade-offs between warning signals and other traits. For example, evolutionary history of different species influences their diet and since diet can influence the warning trait [START_REF] Grill | Effects of a larval antipredator response and larval diet on adult phenotype in an aposematic ladybird beetle[END_REF][START_REF] Ojala | Life-history constraints and warning signal expression in an arctiid moth[END_REF], this may lead to different species-specific trade-offs limiting convergence between defended species. Historical constraints thus not only determine ancestral trait values, but also the evolvability of the traits in different species. Theoretical studies suggest that ancestral trait states may play a key role in the evolution of warning traits, because the convergence of trait can be facilitated by an initial resemblance between species [START_REF] Balogh | Müllerian mimicry: an examination of fisher's theory of gradual evolutionary change[END_REF][START_REF] Franks | Batesian mimics influence mimicry ring evolution[END_REF][START_REF] Franks | The evolution of multicomponent mimicry[END_REF]. The initial resemblance between species, in the eyes of predators, depends on predator discrimination capacities, that then determines the strength of selection promoting convergence of warning traits. Other theoretical studies highlight that the level of standing genetic and phenotypic variance within species strongly influences convergence between species [START_REF] Ruxton | Evolutionary implications of the form of predator generalization for aposematic signals and mimicry in prey[END_REF]. The balance between (1) the shared predation pressure faced by individuals from different sympatric species and (2) the historical constraints within each species may thus strongly shape the level of evolutionary convergence in warning traits. This balance may also modify the direction of evolution of traits within the different defended species living in sympatry. While convergence [START_REF] Sherratt | The evolution of müllerian mimicry[END_REF] usually assumes a joint evolution of traits in several sympatric species toward resemblance (e.g. [START_REF] Flanagan | Historical demography of müllerian mimicry in the neotropical heliconius butterflies[END_REF]; [START_REF] Symula | Molecular phylogenetic evidence for a mimetic radiation in peruvian poison frogs supports a müllerian mimicry hypothesis[END_REF]), resemblance might also emerge from advergence, whereby trait evolution occurs in a given species (i.e. the 'mimic' species), leading to high similarity to the ancestral trait displayed in another species (i.e. the 'model' species) (see [START_REF] Dalziell | Mimicry for all modalities[END_REF] for the terminology).

Moreover, the convergence of warning traits in different species may entail costs due to behavioral interference, thereby limiting positive selection on trait resemblance. Warning traits are indeed frequently involved in species recognition [START_REF] Jiggins | Reproductive isolation caused by colour pattern mimicry[END_REF][START_REF] Kronforst | Linkage of butterfly mate preference and wing color preference cue at the genomic location of wingless[END_REF][START_REF] Merrill | Divergent warning patterns contribute to assortative mating between incipient heliconius species[END_REF][START_REF] Naisbit | Disruptive sexual selection against hybrids contributes to speciation between Heliconius cydno and Heliconius melpomene[END_REF], leading to increased risk of confusion in mimetic species during sexual interactions. Such risk might be even higher between closely-related species, which are more likely to share multiple similar traits because of common ancestry. Species sharing sim-ilar warning traits may thus be exposed to substantial reproductive interference incurring fitness costs during mate acquisition due to interspecific interactions, including heterospecific courtship and mating as well as heterospecific male rivalry [START_REF] Gröning | Reproductive interference between animal species[END_REF]. Empirical examples of such reproductive interferences in Müllerian mimetic systems have been reported in the literature [START_REF] Estrada | Interspecific sexual attraction because of convergence in warning colouration: is there a conflict between natural and sexual selection in mimetic species?[END_REF][START_REF] Vasconcellos-Neto | Interspecific hybridization in mechanitis butterflies (ithomiinae): a novel pathway for the breakdown of isolating mechanisms[END_REF]. However, empirical studies precisely estimating the level of reproductive interference in sympatric species are scarce. Pheromone differences between mimetic species have been documented to limit the rate of erroneous mating (see [START_REF] Darragh | Male sex pheromone components in Heliconius butterflies released by the androconia affect female choice[END_REF]; [START_REF] González-Rojas | Chemical signals act as the main reproductive barrier between sister and mimetic Heliconius butterflies[END_REF] for empirical examples in Heliconius butterflies). However, the pheromones of day-flying butterflies usually act as shortdistance cues that may be perceived only during courtship [START_REF] Mérot | Beyond magic traits: Multimodal mating cues in heliconius butterflies[END_REF]. Females deceived by the color pattern of the heterospecific males may have already spent time and energy or may need to deploy substantial efforts to avoid heterospecific mating. Therefore, females may still suffer from costs associated to reproductive interference, even if females refuse mating with heterospecific males. When females are courted by heterospecific males displaying their preferred cue before being rejected, this also results in increased costs associated with mate searching in males (i.e. signal jamming in [START_REF] Gröning | Reproductive interference between animal species[END_REF]).

Reproductive interference can generate reproductive character displacement [START_REF] Gröning | Reproductive interference between animal species[END_REF][START_REF] Kyogoku | Reproductive interference: ecological and evolutionary consequences of interspecific promiscuity[END_REF], whereby reproductive traits are more dissimilar between species in sympatric than in allopatric populations [START_REF] Brown | Character displacement[END_REF]. Such reproductive character displacement may thus impair convergence driven by mutualistic interactions. Theoretical studies have investigated how the evolution of female preferences may promote reproductive character displacement in males [START_REF] Mcpeek | The evolution of female mating preferences: Differentiation from species with promiscious males can promotes speciation[END_REF][START_REF] Yamaguchi | Reproductive character displacement by the evolution of female mate choice[END_REF]: reproductive interference costs are predicted to favor divergence between female preference and trait displayed by heterospecifics, because this reduces mating attempts with heterospecifics, and therefore promotes the divergence of reproductive traits between conspecific and heterospecific males through sexual selection. Female discrimination then determines the level of divergence between female preference and trait displayed by heterospecifics necessary to limit the cost of reproductive interference [START_REF] Mcpeek | The evolution of female mating preferences: Differentiation from species with promiscious males can promotes speciation[END_REF][START_REF] Yamaguchi | Reproductive character displacement by the evolution of female mate choice[END_REF]. Numerical simulations assuming two discrete warning traits and fixed warning trait-based assortative mating show that reproductive interference may impair the convergence of warning traits [START_REF] Boussens-Dumon | Sex, competition and mimicry: an eco-evolutionary model reveals how ecological interactions shape the evolution of phenotypes in sympatry[END_REF]. Nevertheless, understanding the impact of reproductive interference on the evolution of warning trait requires to specifically explore the evolution of female preference towards this trait. Moreover, the outcomes of these antagonistic selective forces might range from trait divergence to full convergence, through limited convergence and cannot be investigated in models assuming only discrete and well-differentiated warning traits, calling for a theoretical framework providing general expectations on the gradual evolution of convergent traits.

Here, we thus investigate the selective pressure limiting the convergence of traits involved in mimetic interactions, by building a mathematical model that describes the evolution of quantitative traits in two sympatric species engaged in mimetic interaction. We specifically study the evolution of (1) the quantitative trait t involved in mimetic interaction, displayed in both males and females and (2) the preference p, which value indicates the male trait value preferred by the female. We assume that individuals from different species gain protection from predators, by sharing similar warning trait values with other defended individuals living in the same environment, whatever species they belong to. However, trait similarity between species generates fitness costs for females via reproductive interference [START_REF] Mcpeek | The evolution of female mating preferences: Differentiation from species with promiscious males can promotes speciation[END_REF][START_REF] Yamaguchi | Reproductive character displacement by the evolution of female mate choice[END_REF]. We neglect fitness costs of reproductive interference acting on males [START_REF] Mcpeek | The evolution of female mating preferences: Differentiation from species with promiscious males can promotes speciation[END_REF][START_REF] Yamaguchi | Reproductive character displacement by the evolution of female mate choice[END_REF], reflecting the asymmetrical investment in reproduction between sexes observed in numerous species [START_REF] Trivers | Parental investment and sexual selection[END_REF]. We assume that a parameter c RI modulates the strength of reproductive interference. The strength of reproductive interference depends on the degree of similarity between species. Because the selective forces acting on warning traits strongly depend on the sensitivity of both females and predators, we test the effect of their discrimination capacity on convergent evolution. We then investigate the interactions between these opposed selective forces with the effect of historical constraints, re-flecting evolutionary history, by assuming different ancestral trait values in the two interacting species, as well as stabilizing selection promoting these ancestral values within each species. Using weak selection approximation [START_REF] Barton | Natural and sexual selection on many loci[END_REF][START_REF] Kirkpatrick | General models of multilocus evolution[END_REF], we obtain equations describing the evolution of the mean trait and mean preference values in both species.

We then use analytical results and numerical analyses to investigate the effect of reproductive interference on the convergence of trait, depending on different ecological factors.

Methods

We consider two sympatric species, called species 1 and 2. In species i for i ∈ {1, 2}, males and females display a warning trait t i . We assume that only females express a mating preference for males. The value of female preference p i indicates the value in male trait triggering the highest attraction of the female. We investigate the evolution of the warning trait and preference within each species, influenced by both natural selection and mate choice.

Model structure

We assume constant population size in the two species and balanced sex ratio. We consider discrete and non-overlapping generations. The offspring in each new generation are produced by sexual reproduction between males and females from the previous generation, following a Wright-Fisher Model (Fisher, 1930;[START_REF] Wright | Evolution in mendelian populations[END_REF]. Following the framework from [START_REF] Barton | Natural and sexual selection on many loci[END_REF]; [START_REF] Kirkpatrick | General models of multilocus evolution[END_REF], in species i for i ∈ {1, 2}, we assume that the distribution of traits in the offspring depend on the so-called group absolute fitness W i (t m , t f , p f ) of the parental generation. This absolute group fitness accounts for the trait values displayed by the males (t m ) and the females (t f ), as well as on the preference of the females (p f ) producing the offspring generation. This group absolute fitness W i thus describes the effect of selection acting on viability and fecundity, as well as the sexual selection due to mate preference, on the evolution of trait and preference in the population, as detailed below.

Life cycle

Because warning traits are involved in survival and mate choice, the group absolute fitness is a function of male and female traits and female preference. Following [START_REF] Pomiankowski | Evolution of multiple sexual preferences by fisher's runaway process of sexual selection[END_REF] (see Equation A1), the group absolute fitness associated with each pair is assumed to be given by the product of different fitness terms and of one term describing mate preference. For i ∈ {1, 2}, the group absolute fitness associated with a pair consisting of a male with trait t m and a female with trait t f and preference p f is assumed to be given by:

W i (t m , t f , p f ) = W i r (t m , p f )W i h (t m )W i h (t f )W i pred (t m )W i pred (t f )W i RI (p f ), (1)
where W i r , W i h , W i pred and W i RI describes respectively the action of mate preference, historical constraints, predation and reproduction.

Mate preference

In each species i ∈ {1, 2}, the contribution to the next generation of a mating between a male with trait t i and a female with preference p i due to mate preference is assumed to be given by

W i r (t, p) = exp -a i (p i -t i ) 2 , ( 2
)
where female discrimination a i > 0, assumed constant among conspecific females, quantifies how much females of species i reject males with a non-preferred trait value.

Ancestral trait value

Phenotypic evolution in both species away from their ancestral trait is limited by historical constraints, specific to each species. The phenotypic evolution thus strongly depends on ancestral trait values in both species t a1 , t a2 , as well as on the stabilizing selection promoting this ancestral trait value t ai within each species. The strength of the stabilizing selection within each species i depends on the coefficient s i . The fitness component due to historical constrainsts is thus as-sumed to be given by:

W i h (t i ) = exp -s i (t i -t ai ) 2 .
(3)

Predation depending on the level of mimicry of trait t.

Within each species, the evolution of the trait t, expressed by males and females from a species, is strongly influenced by the trait displayed in the other species. Müllerian mimicry indeed generates positive density-dependent selection [START_REF] Benson | Natural selection for mullerian mimicry in heliconius erato in costa rica[END_REF][START_REF] Chouteau | Warning signals are under positive frequencydependent selection in nature[END_REF][START_REF] Mallet | Strong natural selection in a warning-color hybrid zone[END_REF], due to predator learning. This density-dependence is non linear and is often modeled as an hyperbolic decrease in mortality (see [START_REF] Joron | The evolution of a müllerian mimic in a spatially distributed community[END_REF][START_REF] Llaurens | The effect of dominance on polymorphism in müllerian mimicry[END_REF] for example). The impact of predation on the fitness of an individual displaying the trait value t is assumed to be given by:

W pred (t) = 1 - d D(t) , (4)
where d ∈ (0, 1) the basal predation rate, and D(t) is the level of protection of an individual with trait t increasing with the density of resembling individuals:

D(t) = 2 ∑ i=1
protection gained by resemblance with individuals of species i

τ λ i n i f i (τ) exp -b(t -τ) 2 dτ, (5)
where for each i ∈ {1, 2}, f i is the distribution of traits, and n i is the density of individuals, in species i. The density-dependence is modulated by the individual levels of defense λ 1 and λ 2 , assumed constant among individuals of species 1 and 2, respectively, shaping predator deterrence: the higher the defense, the quicker predators learn. The protection gained against predators then depends on the level of resemblance among defended prey, as perceived by predators, and on the number of individuals sharing similar trait values. Due to the positive density-dependent selection, λ i n i is then the population defense level in species i. Cost induced by reproductive interference.

Because heterospecific males may resemble conspecific males, females will suffer from reproductive interference generated by erroneous mating attempts with heterospecific males (see [START_REF] Gröning | Reproductive interference between animal species[END_REF] for a review of reproductive interference costs). The risk of heterospecific mating depends on the relative densities of heterospecific and conspecific males. We assume a balanced sex-ratio within each species i.e. the density of males in species i is n i /2, for i ∈ {1, 2}.

However, we also consider the capacity of females to recognize conspecific males using alternative cues (pheromones for example). In the model, the investment of females in interspecific mating interaction is captured by the strength of reproductive interference c RI ∈ [0, 1]. This cost of reproductive interference incurred to females can be reduced when female choice is also based on alternative cues differing between mimetic species. Using Equation ( 1b) in [START_REF] Yamaguchi | Reproductive character displacement by the evolution of female mate choice[END_REF] the fitness of a female of species i ∈ {1, 2} with preference p i is modulated by:

W i RI (p i ) = (6)
t probability of encountering a conspecific male with trait t

n i n i + n j f i (t)
probability of accepting a conspecific male with trait t

exp -a i (p i -t) 2 dt t n i n i +n j f i (t) exp [-a i (p i -t) 2 ] dt + t n j n i + n j f j (t)
probability of encountering an heterospecific male with trait t c RI exp -a i (p it) 2 probability of accepting an heterospecific male with trait t dt , with j ∈ {1, 2} with j ̸ = i.

Approximation of the evolutionary dynamics.

We assume that in each species the trait and preference are quantitative traits, with an autosomal polygenic basis, and additive effects. We assume that male and female traits have the same genetic basis.

We assume weak natural and sexual selective pressures [START_REF] Iwasa | The evolution of costly mate preferences ii. the "handicap" principle[END_REF][START_REF] Pomiankowski | Evolution of multiple sexual preferences by fisher's runaway process of sexual selection[END_REF] implying that the variance of trait and preference is small relative to the curvature of the fitness function in each species (see Supplement 1). Using the Price's theorem (see [START_REF] Rice | Evolutionary theory: mathematical and conceptual foundations[END_REF] for instance), we can approximate the change in the mean values of traits (t 1 , t 2 ) and preferences (p 1 , p 2 ) in both species, after the natural and sexual selection respectively, by:

   ∆t i ∆p i    = 1 2    G t i C t i p i C t i p i G p i       β t i β p i    , (7)
where for i ∈ {1, 2} G t i and G p i are the additive genetic variances of t i and p i and C t i p i is the additive genetic covariance between t i and p i . β t i and β p i describe the selective forces acting on the trait t i and the preference p i respectively and are given by:

β t i := ∂ ∂t m ln W i (t m , t f , p f ) + ∂ ∂t f ln W i (t m , t f , p f ) (t m ,t f ,p f )=(t,t,p) , ( 8
)
β p i := ∂ ∂p f ln W i (t m , t f , p f ) (t m ,t f ,p f )=(t,t,p) . ( 9
)
Under a weak selection hypothesis, genetic correlations generated by selection and nonrandom mating quickly reach equilibrium [START_REF] Nagylaki | The evolution of multilocus systems under weak selection[END_REF] and can thus be approximated by their equilibrium values.

Following [START_REF] Iwasa | The evolution of costly mate preferences ii. the "handicap" principle[END_REF], we assume that for each i ∈ {1, 2}, G t i and G p i are positive constants maintained by an equilibrium between selection and recurrent mutations. Under weak selection, for each i ∈ {1, 2}, the genetic covariance between t i and p i can be approximated by (see Supplement 4):

C t i p i = a i G t i G p i . ( 10
)
When the variance of trait and preference is small relative to the curvature of the fitness function, for each i ∈ {1, 2}, C t i p i is small in comparison with G t i and G p i , allowing us to approximate the change in the mean values of trait and preference in each species i ∈ {1, 2} by: Analytical expression of the mean trait and preference values.

   ∆t i ∆p i    = 1 2    G t i β t i G p i β p i    . ( 11
)
Assuming weak female and predator discriminations, the mean trait and preference values both converge to the equilibrium values t * 1 and p * 1 with

t * 1 = 2bdλ 2 n 2 (1+λ 1 n 1 +λ 2 n 2 ) 2 t 2 + 2s 1 t a1 -a 1 c RI n 2 n 1 t 2 2bdλ 2 n 2 (1+λ 1 n 1 +λ 2 n 2 ) 2 + 2s 1 -a 1 c RI n 2 n 1 , (12)
and

p * 1 = t * 1 + c RI n 2 n 1 (t * 1 -t 2 ), (13)
when

a 1 n 2 n 1 c RI < 4bdλ 2 n 2 (1 + λ 1 n 1 + λ 2 n 2 ) 2 + 4s 1 , (14)
see Supplement 5.2.

These analytical expressions allow us to predict on the level of resemblance between the trait displayed in species 1 and the fixed trait exhibited in the model species 2 (|t * Supplement 5.2).

Reproductive interference limits mimicry.

Selection exerted by predators favors the advergence of trait in species 1 toward the fixed trait value exhibited in the model species 2. When ( 14) is verified, the level of advergence toward t 2 is given by:

|t * 1 -t * 2 | = 2s 1 2bdλ 2 n 2 (1+λ 1 n 1 +λ 2 n 2 ) 2 + 2s 1 -a 1 c RI n 2 n 1 |t a1 -t 2 |. ( 15
)
Hence, if we assume no reproductive interference (c RI = 0), we have |t *

1 -t 2 | < |t a -t 2 |, implying
that the trait displayed in species 1 gets closer to the trait displayed in species 2. We then observe different evolutionary outcomes ranging from (a) mimicry to (b) imperfect mimicry, see Figures 1(a)

and 1(b). Mimicry in species 1 becomes nearly perfect (|t

* 1 -t 2 | close to zero) when the strength of selection due to predation (2bdλ 2 n 2 /(1 + λ 1 n 1 + λ 2 n 2 ) 2
) is large enough, as compared to the historical constraints limiting the evolution of the trait in species 1 (s 1 ) (outcome (a) mimicry see Figure 1(a)).

However, assuming reproductive interference between females from species 1 and males from species 2 impairs advergence. When reproductive interference is non null but has a limited strength, satisfying Inequality (14), Equation (15) implies that ∂|t * 1t 2 |/∂c RI > 0 (see Supplement 5.4.1). Reproductive interference thus increases the distance between the traits displayed in both species, leading to imperfect mimicry in species 1. Reproductive interference promotes the evolution of preference in the opposite direction of the trait displayed by heterospecific males (13).

Because female preference generates sexual selection on male traits, reproductive interference promotes phenotypic divergence between both species (see Equation ( 13)). Thus reproductive interference limits Müllerian mimicry.

However, when the cost associated with reproductive interference crosses a threshold and ( 14) is verified, i.e. when When inequality ( 14) is not satisfied, the phenotypic distance between both species becomes very large. This very large divergence is biologically unrealistic but suggests that reproductive interference can promote phenotypic divergence between defended species living in sympatry.

n 1 a 1 n 2 4bdλ 2 n 2 (1 + λ 1 n 1 + λ 2 n 2 ) 2 ≤ c RI < n 1 a 1 n 2 4bdλ 2 n 2 (1 + λ 1 n 1 + λ 1 n 1 ) 2 + 4s 1 , (16)
This unrealistic divergence stems from the weak female discrimination (a 1 = O(ε)) assumed:

since females have low discrimination (because a 1 is low), females almost always accept heterospecific males, except when the difference between female preference in species 1 and the trait displayed in species 2 is very high. Reproductive interference promotes female preference that limits fitness costs due to reproductive interference, and therefore promotes a large distance between females preference value in species 1 and the value of the trait displayed in species 2.

Relaxing the weak female and predator discriminations hypothesis, i.e. assuming that a 1 = O(1)

and b = O(1), confirms that reproductive interference limits mimicry in species 1 (see Figure 2). However, in this case, when a strong divergence is favored, this divergence becomes high but stays of order O(1). Indeed, as female discrimination is high, this divergence strongly reduces fitness cost due to reproductive interference. Therefore, stabilizing historical constraints on the trait becomes more important than reproductive interference, thereby preventing very large divergence. Figure 2 shows that numerical simulations with parameter values matching weak female and predator discriminations provide similar predictions as the analytical approximation obtained under the same hypotheses.

Historical constraints allow for the evolution of imperfect mimicry

Our previous results highlight that reproductive interference limits the convergence of warning traits. However, the effect of reproductive interference on trait divergence strongly depends on the historical constraints (s 1 ), generating a stabilizing selection promoting the ancestral trait value (t a1 ). Strong historical constraints promote the display of the ancestral trait in both species (see discriminations. We assume (a) s 1 = 0.5εε ′ , t a1 = 0, t a2 = 1, (b) s 1 = 0, t a1 = 0, t a2 = 1, and (c) s 1 = 0.5εε ′ , t a1 = t a2 = 1 with ε = 0.01. We also assume:

G t 1 = G p 1 = 0.01, a 1 = ε′ , b = ε′ , d = 2ε, λ 1 = 0.1 , λ 2 = 0.1, n 1 = 10, n 2 = 20.
Analytical approximation curves are obtained with ε′ = 0.01.

Equation 15 and Figure 1(d) for an illustration when species 2 is fixed). The effect of historical constraints on the level of trait divergence depends then on the distance between the ancestral trait values in species 1 and 2. When predator pressure exceeds reproductive interference, historical constraints limit the convergence of trait between both species ancestrally displaying different traits (see Supplement 5.4.4). By contrast, when reproductive interference exceeds predator pressure and promotes warning trait displacement, historical constraints may limit the divergence of trait between both species (see Supplement 5.4.4). Assuming historical constraints (s 1 > 0)

and when the ancestral trait values of the two species differ (t a1 ̸ = t a2 ), an increase in strength of reproductive interference leads to a progressive increase in the phenotypic distance between both species until the phenotypic distance between both species becomes very large (see purple curve in Figure 2(a)).

Surprisingly, in the absence of historical constraints (s 1 = 0) or when the ancestral trait values are the same in both species (t a1 = t a2 ), t 1 is either equal to t 2 when ( 14) is verified, or is very large, when ( 14) is not verified (see purple curve in Figures 2(b) and 2(c)). Therefore an increase in the strength of reproductive interference (c RI ) has no effect on the phenotypic distance between both species, as long as this strength remains below a threshold. This effect is also observed assuming strong female and predator discriminations (see green curves in Figures 2(b) and 2(c)).

However, when the strength of reproductive interference (c RI ) is greater than this threshold, assuming weak female and predator discriminations, the phenotypic distance between both species becomes instantaneously very large. A similar trend is observed when female and predator discriminations are strong: the phenotypic distance is null when the strength of reproductive interference remains below a threshold, but it quickly increases to a high value when the strength of reproductive interference crosses the threshold (see green curves in Figures 2(b) and 2(c)).

Our results highlight that historical constraints promoting ancestral traits strongly modulate the effect of reproductive interference on the convergence of warning traits. Surprisingly, drastic divergence might be promoted by a strong strength of reproductive interference, even when the ancestral phenotypes are the same in the two interacting species.

Overall, our analytical results reveal the mechanisms underlying trait and preference evolution. However, these analytical results are obtained under restrictive hypotheses: we assumed fixed trait and preference in species 2 and weak female and predator discrimination. To relax those hypotheses, we then study the joint evolution of traits and preferences in both species in the following sections. We verified that all results obtained in previous sections are maintained when traits and preferences jointly evolve in the two sympatric species (see Supplements 6.3 and 6.4).

Joint evolution of mimicry between two interacting species

In this section, we now focus on the general case where traits and preferences co-evolve in both species.

Higher discrimination in female than in predator does not always favor the convergence of warning traits between two interacting species.

The joint evolution of the traits in both species is shaped by two antagonistic evolutionary forces, generated by reproductive interference and Müllerian mimicry, respectively. Reproductive interference indirectly limits mimicry by impacting females' preference. Therefore, female discrimination a 1 and a 2 may be a key feature to understand the evolution of the trait within each species.

The selection exerted by predation also depends on predator discrimination. Assuming a fixed level of reproductive interference (c RI = 0.002), we thus investigate the impact of the strength of female discrimination, assumed equal in both species a 1 = a 2 = a, and of predator discrimination coefficient b on the evolution of the warning trait.

When female and predator discriminations are low (a and b approximately lower than 3), higher predator than female discrimination favors the convergence of warning traits. Indeed, when female and predator discriminations are low, selection due to predation and reproductive interference is limited and increases with female and predator discriminations, respectively.

Females are not discriminant (a is low) and tend to accept all encountered males, including heterospecific males, whatever the direction of their preference. The difference in fitness cost due to reproductive interference between females with different preferences is then low, leading to low divergent selection generated by reproductive interference. With a higher level of female discrimination, fitness cost due to reproductive interference depends more on the direction of preference, leading to higher selection caused by reproductive interference. A similar reasoning on the difference in fitness cost due to predation between individuals displaying different traits explain that selection promoting convergence, due to predation, increases with the strength of predator discrimination. Therefore, higher predator than female discrimination entails higher selection due to predation than selection due to reproductive interference and promotes mimicry.

By contrast, with higher female discrimination (a approximately greater than 3) and lower predator discrimination (b approximately lower than 3), mimicry becomes more likely (Figure 3).

(|t * 1 -t * 2 | = |t a1 -t a2 |). We assume: G t 1 = G p 1 = G t 2 = G p 2 = 0.01, c RI = 0.002, d = 0.02, λ 1 = λ 2 = 0.1, n 1 = n 2 = 20, s 1 = s 2 = 0.005, t a1 = 0, t a2 = 1.
species 2. Convergence of warning traits is thus more likely to happen when species 1 is weakly defended (λ 1 n 1 small) (see Figure 4(c)). The more species 1 is defended, i.e. the greater λ 1 n 1 is, the closer its mean trait value is to the ancestral trait value t a1 (see Figure 4(a)). Such increase in the defense level of species 1 also impacts the evolution of trait in the sympatric species 2 (see Figure 4(b)): when the individual defense level in species 1 (λ 1 ) is below a threshold, the more individuals from species 1 are protected, the more the mean trait value in species 2 moves away from its ancestral trait (t a2 ). Surprisingly, above this threshold, the better protected species 1 is, the closer the mean trait value in species 2 gets to its ancestral trait value (t a2 ). As the mean trait value in species 1 becomes very close to the ancestral trait value t a1 , trait values in species 2 leading to protection from heterospecific matings necessitate a great departure from the ancestral trait t a2 . Nevertheless, historical constraints still prevent the trait in species 2 to evolve too far away from its ancestral trait value t a2 .

When assuming positive strength of reproductive interference (c RI > 0), advergence in species 1 toward the trait displayed in species 2 is observed when the individual defense level in species 1 is low (λ 1 approximately lower than 0.1) and when the density in species 1 is sufficiently large (n 1 approximately greater than 2). In this case, the population defense level in species 1 (λ 1 n 1 ) is low, the protection gained by positive frequency-dependent selection within species is low, and the advergence toward species 2 is thus strongly promoted. Surprisingly, advergence is impaired for similar values of defense level, when the density of species 1 is low (n 1 approximately lower than 2). When the density of species 1 is low, females pay higher fitness costs due to reproductive interference, because they encounter more often heterospecific than conspecific males.

Altogether, our results suggest that advergence of the warning traits is likely to happen for low level of individual defense in species 1 (i.e. Batesian (λ 1 = 0) or quasi-Batesian (λ 1 > 0 but small) mimicry) and when the density of species 1 is high.

The trait value of species 1 does not always converge toward the trait value initially displayed in species 2 (t a2 ). On the contrary, individuals of species 2 can mimic individuals of species 1 (see blue zone in Figure 4(e)), when the defense level of individuals of species 1 is high and when species 1 is rare. Because individuals from both species are well defended (high λ 1 and λ 2 ), individuals of both species benefit from looking similar. However, when species 1 is rarer, this leads to an increased cost of reproductive interference in species 1, inhibiting convergence towards the ancestral trait value displayed in the alternative species (t a2 ) (see Figure 4(d)). Since predation pressure promotes convergence of traits of both species, the mean trait in species 2 becomes closer to species 1 ancestral trait value t a1 . Surprisingly assuming weak female and predator discriminations, such advergence also happens when individuals of species 2 are more

G t 1 = G p 1 = G t 2 = G p 2 = 0.01, a 1 = a 2 = 1, d = 0.05, b = 1, λ 2 = 0.1, n 2 = 10, s 1 = s 2 = 0.005, t a1 = 0, t a2 = 1.
defended than individuals from species 1, i.e. λ 2 n 2 > λ 1 n 1 (see blue zones in Figures S22 (c) and (d) below the yellow solid line). By contrast, when the density in species 1 exceeds the density in species 2, individuals from both species exhibit traits close to their ancestral traits. Both species are well-protected and then gain little from mimicry. Because unbalanced relative density ratio leads to strong cost of reproductive interference in the scarcest species limiting mimicry, mimicry is more likely to be observed between species of similar density (see Figure 4(f)).

Our results highlight that reproductive interference impacts the evolution of warning traits, and may even reverse the expected model/mimic relationship, depending on the relative abundances and individual defense levels of sympatric species.

Discussion

Reproductive interference alone cannot explain imperfect mimicry

Our results show that reproductive interference and historical constraints promoting ancestral traits can generate a continuous range of phenotypic distances from quasi perfect mimicry to warning trait displacement. Our study suggests that reproductive interference alone is unlikely to promote imperfect mimicry, in contradiction with previous predictions [START_REF] Pfennig | Competition and the evolution of imperfect mimicry[END_REF]. When sympatric species share the same ancestral trait, or in absence of historical constraints, we indeed observe either perfect mimicry or strong trait divergence, depending on the strength of reproductive interference. In our model, imperfect mimicry is observed only when stabilizing selection due to historical constraints promotes different ancestral traits in both species. The contrasted historical constraints undergone by the different species may thus play an important role in imperfect mimicry. These different constraints may be strongly correlated with the phylogenetic distances between species: closely-related species are likely to share similar genetic bases and developmental pathway leading to the warning trait and to also share similar environments, due to niche conservatism [START_REF] Chazot | Mutualistic mimicry and filtering by altitude shape the structure of andean butterfly communities[END_REF][START_REF] Elias | Mutualistic interactions drive ecological niche convergence in a diverse butterfly community[END_REF][START_REF] Joshi | Evolutionary assembly of communities in butterfly mimicry rings[END_REF], likely to limit departure from ancestral trait values. Our results suggest that imperfect mimicry could not be promoted among closely-related species experiencing high levels of reproductive interference but limited differences in ancestral traits. Imperfect mimicry may rather be observed between phylogenetically-distant species, subject to more strikingly different historical constraints, where reproductive interference might be more limited. Distantly-related species indeed might have diverged in other traits, facilitating mate recognition through different cues.

For similar historical constraints, mimicry between defended species can then either be promoted or limited depending on predator discrimination. Low predator discrimination allows for the evolution of imperfect mimicry, since imperfect mimics are seen as similar by predators, allowing mutualistic relationship without implying strong cost of historical constraints. By contrast under high predator discrimination, mutualistic mimetic relationships necessitate a strong similarity between species, which is limited by historical constraints. Empirical studies based on vertebrates or on insects show that predators do not perceive difference between Batesian mimics and their models, or at least this difference does not entail a difference in behavior [START_REF] Dittrigh | Imperfect mimicry: a pigeon's perspective[END_REF][START_REF] Kikuchi | Predator cognition permits imperfect coral snake mimicry[END_REF][START_REF] Morris | Do crab spiders perceive Batesian mimicry in hoverflies?[END_REF]. Loose predator discrimination may therefore play a key role in the evolution of imperfect mimicry.

How important are historical constraints in the evolution of warning trait?

Our model predict that imperfect mimicry can arise through the interplay between different historical constraints, predation, and reproductive interference. However, estimating the actual level of historical constraints potentially shaping the evolution of warning traits is not straightforward. Genetic studies, reviewed in [START_REF] Joron | Heliconius wing patterns: an evo-devo model for understanding phenotypic diversity[END_REF], show that Heliconius species share the same 'toolkit' of genes, repeatedly recruited during both convergent and divergent evolutions of warning traits within and between species. The important lability in color patterns observed in this genus suggests a limited level of developmental constraints, facilitating the evolution of mimicry, even between species from different sub-clades within this genus [START_REF] Hines | Wing patterning gene redefines the mimetic history of heliconius butterflies[END_REF].

Such mimicry between distantly related species shows that selection due to predation can overcome historical constraints. By contrast, in butterflies from the tribe Ithomiini and in butterflies from the tropical forests of the Western Ghats, India, a strong phylogenetic signal on the warning trait is observed [START_REF] Chazot | Mutualistic mimicry and filtering by altitude shape the structure of andean butterfly communities[END_REF][START_REF] Elias | Mutualistic interactions drive ecological niche convergence in a diverse butterfly community[END_REF][START_REF] Joshi | Evolutionary assembly of communities in butterfly mimicry rings[END_REF], suggesting that historical constraints may limit the evolution of mimicry among distantly related species despite predation pressure.

Evolution of female preferences limiting the reproductive interference costs generated by mimicry

When considering reproductive interference, the relationship between female and predator discriminations is crucial to understand the evolution of warning traits. Surprisingly, when female and predator discriminations are low, higher predator than female discrimination promotes convergence of warning traits because selection due to predation and reproductive interference increase with predator and female discrimination respectively. By contrast, when female and predator discriminations are high, imperfect mimicry can evolve despite reproductive interference. When female discrimination is high, successful species recognition might occur without decreasing the protection brought by mimicry. Such situation arises when predators largely generalize, and therefore do not discriminate imperfect mimics. Some studies report similar female and predator discriminations [START_REF] Finkbeiner | Warning signals are seductive: Relative contributions of color and pattern to predator avoidance and mate attraction in heliconius butterflies[END_REF][START_REF] Mcclure | Does divergent selection predict the evolution of mate preference and reproductive isolation in the tropical butterfly genus melinaea (nymphalidae: Ithomiini)?[END_REF], suggesting that reproductive interference may act on mimetic species. On the other hand, differences in the discrimination of color patterns between prey and predators may exist in the wild. For instance, [START_REF] Llaurens | Cryptic differences in colour among müllerian mimics: how can the visual capacities of predators and prey shape the evolution of wing colours?[END_REF] showed that the variations in color pattern between co-mimetic species from the distantly related genera Heliconius and Melinaea might be better perceived by the Heliconius butterflies themselves but not by avian predators. The evolution of visual perception in females could also enhance species discrimination without impairing mimicry. The evolution of vision in females from the Heliconius butterflies indeed coincides with the evolution of the yellow pigments 3-OH-kinurenin displayed on their wings [START_REF] Bybee | Uv photoreceptors and uv-yellow wing pigments in heliconius butterflies allow a color signal to serve both mimicry and intraspecific communication[END_REF]. The evolution of high discrimination capacities in mimetic prey, as well as the evolution of mating cues undetected by predators could thus limit the cost of reproductive interference in mimetic prey.

In butterflies, mate choice indeed often relies on pheromones that may strongly differ among closely-related mimetic species [START_REF] Darragh | Male sex pheromone components in Heliconius butterflies released by the androconia affect female choice[END_REF][START_REF] González-Rojas | Chemical signals act as the main reproductive barrier between sister and mimetic Heliconius butterflies[END_REF]. Similarly, in non-mimetic species, chemical cues may reduce reproductive interference without entailing reproductive character displacement on a trait under natural selection. Females of the swordtails Xiphophorus pygmaeus prefer larger mate leading to reproductive interference with males of the Xiphophorus nigrensis species. However X. pygmaeus females avoid mating with heterospecific on the basis of chemical cues (Crapon de [START_REF] Crapon De Caprona | Conspecific mate recognition in swordtails, xiphophorus nigrensis and x. pygmaeus (poeciliidae): olfactory and visual cues[END_REF]. Micro-habitat differences among mimetic species may also allow reducing heterospecific encounters, while still benefiting mimicry by sharing the same predator community [START_REF] Estrada | Patterns of pollen feeding and habitat preference among heliconius species[END_REF]. For example the two sympatric ladybird species Harmonia axyridis and Harmonia yedoensis have similar body size and coloration [START_REF] Sasaji | Natural history of the ladybirds[END_REF] and experience reproductive interference [START_REF] Noriyuki | Asymmetric reproductive interference between specialist and generalist predatory ladybirds[END_REF].

These species nevertheless have different host specialization [START_REF] Noriyuki | Prey capture performance in hatchlings of two sibling harmonia ladybird species in relation to maternal investment through sibling cannibalism[END_REF], that may limit reproductive interference [START_REF] Noriyuki | Host selection in insects: reproductive interference shapes behavior of ovipositing females[END_REF]. Likewise, in three Morpho butterfly species displaying local convergence in wing patterns [START_REF] Llaurens | Convergence in sympatry: Evolution of blue-banded wing pattern in morpho butterflies[END_REF], temporal segregation in patrolling activity has been observed between species sharing similar color patterns (Le [START_REF] Roy | Convergent morphology and divergent phenology: unravelling the coexistence of mimetic morpho butterfly species[END_REF], which may strongly limit heterospecific rivalry.

The levels of reproductive interference among mimetic species might thus be modulated by the evolution of the converging traits themselves, as well as the evolution of other traits involved in species interactions.

Reproductive interference strongly impacts species with low relative density.

Our model shows that the effect of reproductive interference strongly depends on the relative abundances of interacting species, leading to surprising evolutionary outcomes. For example, in rare defended species, selection favoring mimicry towards a defended model species is expected to be strong. Nevertheless, our model shows that an elevated cost of reproductive interference prevents the evolution of mimicry in the rarest species, because females then encounter much more heterospecific than conspecific males.

Reproductive interference may particularly promote the emergence and persistence of a distinct warning trait in low-density populations of warning species coming into contact with a local mimicry ring that exhibits a different warning trait. Our model does not take into account the dynamics of population density, and therefore ignores the extinction risk of low-density populations. Such non-mimetic populations with low density might nevertheless persist in the wild, when the level of individual defense is sufficiently high.

Because undefended mimics have a negative impact on predator learning [START_REF] Lindström | Imperfect batesian mimicry-the effects of the frequency and the distastefulness of the model[END_REF], they are expected to be scarce compare to their models [START_REF] Kunte | Evolution of mimicry rings as a window into community dynamics[END_REF]. In line with this prediction, empirical studies report low density of undefended mimics compared to their defended models [START_REF] Long | A test of fundamental questions in mimicry theory using long-term datasets[END_REF][START_REF] Prusa | Umbrella of protection: spatial and temporal dynamics in a temperate butterfly Batesian mimicry system[END_REF]. Reproductive interference may act strongly on Batesian mimics because of their low density with respect to the model species. Our model thus suggests that Batesian mimicry among closely related species may be limited by strong reproductive interference acting on Batesian mimics due to phylogenetic proximity and unbalanced density. By contrast, Batesian mimicry may evolve between distantly-related species despite unbalanced density, because high phylogenetic distance reduces risk of reproductive interference. This is supported by the pattern of convergence observed in tropical forests of the Western Ghats in India: Müllerian mimicry is observed between closely related species whereas Batesian mimicry involves more distantly related species [START_REF] Joshi | Evolutionary assembly of communities in butterfly mimicry rings[END_REF].

Reproductive interference does not always promote divergence of reproductive character (here the warning trait) but can also provoke spatial segregation between species [START_REF] Gröning | Reproductive interference between animal species[END_REF]. The strong reproductive interference acting on scarce mimetic species may limit their coexistence with more abundant mimetic species displaying similar warning signals.

Reproductive interference may then restrict the spatial distribution of mimetic species with low abundance to the edges of the range occupied by more abundant co-mimetic species.

Our model also brings new insights to the ecological processes driving the direction of advergence in warning traits. In the absence of reproductive interference, mimicry is expected to evolve in the less defended species (e.g. low density populations and/or low level of individual defense) towards better-defended species living in sympatry [START_REF] Balogh | Müllerian mimicry: an examination of fisher's theory of gradual evolutionary change[END_REF][START_REF] Franks | The evolution of multicomponent mimicry[END_REF]. When considering the cost of reproductive interference, however, this general trend does not always hold. Our results show that warning traits in the most defended species can evolve toward the warning trait of the less abundant one.

Our model therefore highlights the interplay between mutualism and reproductive interference in sympatric species, which determines the strength and the direction of traits evolution involved in these ecological interactions.

Reproductive interference can explain the emergence of mimetic diversity

In our model, we consider the evolution of warning trait between two interacting species. In the wild however, natural communities involve a variable number of mimetic species, with mimicry ring size ranging from 2 to a dozens of species [START_REF] Kunte | Evolution of mimicry rings as a window into community dynamics[END_REF]. Assuming reproductive interference, species richness within a mimicry ring may influence warning trait evolution. We hypothesize that mimicry rings with high species richness are more likely to contain closely related species. The evolution of mimicry in a species may be limited if a closely related species is abundant in the ring because of the strong cost of reproductive interference generated. However, in a mimicry ring with high species richness, distantly-related species may not suffer from less reproductive interference but mutually benefit of sharing a similar warning trait. This increased advantage of mimicry between distantly-related species may counterbalance the more elevated costs due to reproductive interference with closely-related species, increasing the likelihood of having closely related species within large mimicry rings.

Our results shed light not only on the persistence of distinct warning traits within local communities of defended species in the wild, but also on the emergence of these distinct warning traits in the first place. Mimetic diversity is an apparent paradox but several hypotheses have been suggested to promote the persistence of different warning signals, such as the segregation of predators within microhabitats [START_REF] Beccaloni | Vertical stratification of ithomiine butterfly (Nymphalidae: Ithomiinae) mimicry complexes: the relationship between adult flight height and larval host-plant height[END_REF][START_REF] Devries | Associations of co-mimetic ithomiine butterflies on small spatial and temporal scales in a neotropical rainforest[END_REF][START_REF] Elias | Mutualistic interactions drive ecological niche convergence in a diverse butterfly community[END_REF][START_REF] Willmott | Maintaining mimicry diversity: optimal warning colour patterns differ among microhabitats in amazonian clearwing butterflies[END_REF]. The spread of distinct warning traits has frequently been shown to be promoted by demographic stochasticity, as in shifting balance models [START_REF] Mallet | Evolution of diversity in warning color and mimicry: Polymorphisms, shifting balance, and speciation[END_REF][START_REF] Sherratt | Spatial mosaic formation through frequency-dependent selection in müllerian mimicry complexes[END_REF] or in other models combining predator behaviors, such as neophobia, to stochastic effects [START_REF] Aubier | Diversity in müllerian mimicry: The optimal predator sampling strategy explains both local and regional polymorphism in prey[END_REF]. However, these models do not provide any selective mechanism explaining the emergence of warning signal with different levels of divergence, contributing to mimetic diversity. By contrast, reproductive interference selects for different levels of divergence in warning traits, and could be a major driver of the diversity of mimetic traits. Other mechanisms may generate gradual departure from the ancestral trait value and may also contribute to the diversity of mimetic traits: the evolution of aposematic signals in defended species away from those exhibited in Batesian mimics has been theoretically shown [START_REF] Franks | Warning signals evolve to disengage batesian mimics[END_REF], but empirical evidence of such effect of Batesian mimicry is still lacking. Artificial modification of the warning trait of mated females has also been demonstrated to reduce harassment by males in the butterfly H. erato, and would therefore allow them to lay more eggs, suggesting that evolution of slightly divergent trait could be promoted in females [START_REF] Merrill | Experimental manipulation of heliconius warning patterns reduces harassment of previously mated females[END_REF].

We hope our theoretical work will encourage experimental approaches investigating the impact of reproductive interference on mimicry. Such studies may shed lights on the actual role of reproductive interference on mimetic diversity.

Conclusion

Our analytical and numerical results show that reproductive interference and historical constraints can explain a wide range of levels of convergence, and even explain divergence of warning trait between sympatric species. Our results suggest that reproductive interference alone cannot explain imperfect mimicry, highlighting the role of historical constraints in the evolution of imperfect mimicry. Our study also highlights the importance of female and predator discriminations in the evolution of warning traits.

  exp [-b(tτ) 2 ] describes how much predators perceive the trait values t and τ as similar. The predator discrimination coefficient b thus quantifies how much predators discriminate different trait values.

Figure 1 :

 1 Figure 1: Illustration of four approximate patterns referred in this paper: (a) mimicry: the value of the trait in species 1 t 1 becomes very close to the mean value displayed in species 2 t 2 , (b)imperfect mimicry: the value of the trait in species 1 t 1 gets closer but stays distant from the mean value displayed in species 2 t 2 , (c) warning trait displacement: the value of the trait species 1 t 1 diverges away from the mean value displayed in species 2 t 2 , (d) ancestral warning trait: the value of the trait in species 1 t 1 stays very close to the ancestral trait value t a1 .

Figure 2 :

 2 Figure 2: Influence of the strength of reproductive interference c RI on the phenotypic distances between the two species t * 1t 2 when trait in species 2 is fixed (t 2 = t a2 ), using the analytical approximation (purple curve) or numerical simulations (green curves). The different values of ε′ allows to investigate the intermediate case between weak and strong female and predator

Figure 3 :

 3 Figure 3: Influence of female and predator discriminations (a 1 = a 2 = a and b) on the phenotypic distance between the two species |t * 1t * 2 |. The red solid line shows the case where the phenotypic distance between the two species is equal to the ancestral phenotypic distance

Figure 4 :

 4 Figure 4: Influence of the density and of the individual defense level in species 1 (n 1 and λ 1 ) on the traits displayed in both species and on the phenotypic distance between the two species t * 1t * 2 , for different strengths of reproductive interference (c RI ). (a)(b)(d)(e) Trait values greater than 2 (resp. lower than -1) are shown in red (resp. blue). The yellow solid line shows the case where both species have the same level of defense (λ 1 n 1 = λ 2 n 2 ). Below (resp. above) this line species 1 has a lower (resp. higher) level of defense than species 2. (c)(f) Phenotypic distances greater than 1.5 are shown in yellow. Yellow lines indicate equal levels of t * 1t * 2 . Different values of strengths of reproductive interference are assumed: (a), (b) and (c) c RI = 0, (d), (e) and (f) c RI = 0.005. We assume: G t 1 = G p 1 = G t 2 = G p 2 = 0.01, a 1 = a 2 = 1, d = 0.05,

Table 1 :

 1 Description of variables and parameters used in the model. The subscript i ∈ {1, 2} denotes the identity of the species.To identify the general impact of the strength of reproductive interference c RI on the phenotypic distance between the two species (|t * 1t * 2 |), we first look at the analytical resolution assuming that trait and preference are fixed in species 2 (t 2 = p 2 = t a2 ) and weak female and predator discriminations (a 1 = O(ε) and b = O(ε)) (see Supplement 5.2). This cover cases where species 2 is well defended and more abundant than species 1 (n 2 ≫ n 1 ) as in classical mimic/model interactions between species.

	Abbreviation			Description
	t i /p i		Mean trait/preference value in species i
	G t i /G p i		Genetic variance of trait t i /preference p i
	C t i p i	Genetic covariance between trait t i and preference p i
	β t i /β p i	Selection coefficient on trait t i /preference p i
	a i			Female discrimination in species i
	s i	Strength of stabilizing selection due to historical constraints on trait t i
	t ai			Ancestral trait in species i
	d			Basal predation rate
	b			Predator discrimination
	n i			Density of species i
	λ i			Individual defense level in species i
	λ i n i			Population defense level in species i
	c RI		Strength of reproductive interference
	Using Equation (11) we derive t	* 1 , t	* 2 , p * 1 and p * 2 the mean traits and preferences at equilibrium
	(see Supplements 5 & 6).

All variables and parameters used in the model are summed up in Table

1

. The effect of different parameters on the evolutionary outcome are presented in Supplements 5.4 and 6.3.

Results

Evolution of the warning trait in a mimetic species in sympatry with a model species

t

|) and to study the impact of the different evolutionary forces on the advergence between mimic and model.However, when (14) is not verified, the distances between mean trait and preference values in species 1 and t 2 become very large (not of order 1), and mimicry does not emerge (see

t *

| > |t a1t 2 |. (17)When assuming such an elevated cost of reproductive interference, imperfect mimicry is thus no

Acknowledgments

LM would like to thank Dorian Ni for feedbacks on the mathematical part of the study. LM and VL would like to thank Charline Pinna and the whole 'Evolution and Development of Phenotypic Variations' team for stimulating discussions on the evolution of warning traits. The authors would like to thank the ANR SUPERGENE (ANR-18-CE02-0019) for funding the PhD of LM, and the Emergence program from Paris city council for supporting the team of VL. This work was also supported by the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MNHN-F.X.

Data Availability

Codes are available online: github.com/Ludovic-Maisonneuve/limits-of-ev-conv (Zenedo: https://doi.org/10.5 Ludovic-Maisonneuve 2022).

Higher levels of female discrimination allow females to accurately distinguish between conspecific and heterospecific males even when they display similar traits. Accurate choice by females allows both species to harbor similar traits from the point of view of predators, without entailing heterospecific mating, relaxing divergent selection generated by reproductive interference. Surprisingly, when predator discrimination increases above a certain threshold, increased discrimination no longer promotes accurate mimicry (see sharp transition in Figure 3). When b is approximately greater than 5.5, mimicry is limited, even without reproductive interference (c RI = 0), because of historical constraints (see Figure S21). For intermediate predator discrimination (b ≈ 5), mimicry is limited when reproductive interference is strong and makes similarity too costly for females (a ≈ 1) (Figure 3).

When reproductive interference limits mimicry, it generally leads to warning trait displace-

reproductive interference promotes a large distance between female preference value in species 1 and the value of the trait displayed in species 2, therefore increasing phenotypic distance between the two species.

Reproductive interference can modify the model/mimic relationship.

The population defense levels in both species, i.e. λ i n i , i ∈ {1, 2}, are likely to impact the joint evolution of traits in both species. To investigate how the relative population defense levels of the two species affect the joint evolution of traits, we studied the phenotype at equilibrium in both species and also the phenotypic distance between the two species, for different values of the two components of the population defense level in species 1: the individual defense level (λ 1 ) and the density (n 1 ). Here we assumed that species 2 is already well protected (λ 2 = 0.1, n 2 = 10).

When assuming no reproductive interference (c RI = 0), the trait of the less defended species adverges towards the ancestral trait of the most defended species. In Figures 4(a

Conflict of interest disclosure

The authors of this preprint declare that they have no conflict of interest with the content of this article.