Pierre Bernhard

Frédéric M Hamelin

Simple signaling games of sexual selection (Grafen's revisited)

We investigate several versions of a simple game of sexual selection, to explore the role of secondary sexual characters (the "handicap paradox") with the tools of signaling theory. Our models admit closed form solutions. They are very much inspired by Grafen (1990)'s seminal companion papers. By merging and simplifying his two approaches, we identify a not so minor artifact in the seminal study. We propose an alternative model to start with Grafen's sexual selection theory, with several similarities with Getty (1998).

Introduction

Sexual selection was introduced by Charles Darwin as a mechanism complementary to natural selection in shaping evolution. In short, the number of offspring to any individual is a function of both of its survivorship, the source of natural selection, and its mating success, the source of sexual selection.

The investigation of sexual selection has been dominated by that of the role of secondary sexual characters, and by the "handicap paradox" -not to be mistaken with the "handicap principle" to be described hereafter. The paradox is that secondary sexual characters that seem to attract individuals of the other sex, often male characters attracting females, seem excessively developed, with a negative impact on the viability of the animals bearing them. Since females attracted by these males run the risk of having ill-adapted male progeny, such a preference should be counter-selected.

There has been a very large literature discussing this topic, proposing several possible mechanisms to explain the paradox. We refer to [START_REF] Kirkpatrick | The evolution of mating preferences and the paradox of the lek[END_REF][START_REF] Peter | A strategic taxonomy of biological communication[END_REF] for classifications of the proposed mechanisms, and to the review [START_REF] Kokko | Unifying and testing models of sexual selection[END_REF] and the bibliography therein. Over-simplifying, two mechanisms are in competition in the literature: the so called "Fisherian runaway" dynamic mechanism, see [START_REF] Ronald | The evolution of sexual preferences[END_REF][START_REF] Lande | Models of speciation by sexual selection on polygenic traits[END_REF], and Zahavi's static, or equilibrium, "handicap principle" [START_REF] Zahavi | Mate selection -a selection for the handicap[END_REF][START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF][START_REF] Grafen | Biological signals as handicaps[END_REF]].

Zahavi's intuition was made precise in a mathematical model by Grafen [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF][START_REF] Grafen | Biological signals as handicaps[END_REF]. In [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF], he develops a model in terms of population genetics. In [START_REF] Grafen | Biological signals as handicaps[END_REF], the handicap principle results from an analysis of the role of secondary sexual characters as signals, in the framework of signaling theory, a branch of game theory. This model is one of direct benefits in the classification of [START_REF] Kokko | Unifying and testing models of sexual selection[END_REF] (or direct selection D in the classification of [START_REF] Kirkpatrick | The evolution of mating preferences and the paradox of the lek[END_REF]). Yet, in [START_REF] Kokko | Unifying and testing models of sexual selection[END_REF], the handicap principle is quoted as belonging to the family of indirect benefits, more specifically the "good gene" mechanism 1 .

In this study, and consistent with, e.g. [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF], we will not need to decide whether the quality of males is genetic or environment dependent. The only thing that matters is that females have a benefit in reproductive efficiency to mate with good males, either because these are more fertile (a "direct" effect) leading to a larger progeny at the first generation, or because their own offspring will have a larger fitness (an indirect "good gene" effect: larger survivorship or better mating success -"sexy son"-), leading to a larger progeny at the second generation.

We shall develop an asymmetrical model where males signal to attract females. It is known that this is not always the case; the reverse exists, and even cases of mutual signaling are well documented, a most famous example being in the Great Crested Grebe Podiceps cristatus, which form stable pairs. But, following the main trend in the literature, we will assume that males keep foraging for mates, and that they have a potential reproduction rate much larger than what the number of females allow, so that one may consider that they can mate an arbitrary number of times. As a consequence, even if females are unequally fertile, there is no point for males to be choosy, and therefore no evolutionary pressure on females to develop signals of their quality. In contrast, females are assumed to mate only once per breeding season. Hence if males are of unequal qualities, there is a reproductive benefit for females in being choosy, and therefore evolutionary pressure on males to develop signals of their quality.

In this line of thought, our model may be viewed as a simple synthesis of Grafen's [START_REF] Grafen | Biological signals as handicaps[END_REF][START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF] to be revisited. As in [START_REF] Grafen | Biological signals as handicaps[END_REF], we will use reproductive efficiency as the payoff of the game; however, inspired by the companion paper [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF], we will use a mechanistic model, with explicit mating dynamics, for reproductive efficiency.

One major simplification compared to Grafen's model [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF] is that we do not allow females' fecundity to vary during the breeding season. A minor difference is that we deal with viability in a somewhat different way. While Grafen's example deals with a probability to survive the pre-breading season, which better fits semelparous species, we deal with an expected number of breeding seasons, which extends the model to iteroparous species as well ( [START_REF] Stephen | Older males signal more reliably[END_REF], [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF]). We also use from the start two simplifications of Grafen's model which he uses in the example that he solves numerically: a uniform distribution of males' qualities and a fecundity of females proportional to their mate's quality. As a benefit, we gain explicit solutions, through significantly simpler derivations.

We investigate first a model where these are essentially the only differences with Grafen's. Then we propose a second model, using ideas close to those of [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF], but again more explicit, to circumvent an undesirable feature in Grafen's model pertaining to the lower bound of admissible males' quality.

In Section 2, we give a quick account of the basics of signaling game theory, including the very simple derivation of what we call the handicap principle. In Section 3, we develop two variants of each of our two models of sexual selection, which all turn out to have explicit equilibrium strategies, simplifying the discussion of the model's conclusions.

Signaling

Definitions

Motivation and intuition

We consider the secondary sexual characters under investigation as "signals", i.e. means by which a male communicates to females his "quality", a "private" parameter that females cannot observe directly, in order to attract them. Hence their primary role is of carrying information.

The investigation of situations where two "players", here males and females, interact, and where one has to rely on the observed actions of the other one to recover some missing information about him, is the object of signaling game theory. We investigate equilibrium situations, which needs not be always the case in biological signaling. We assume that all the decision and signal variables range over a continuum, plus some technical assumptions to be given hereafter. The "private" parameter of player one, denoted θ in the next section, will be males' quality q in the sexual selection models.

In essence, the explanation of the handicap paradox via signaling theory, the so-called "handicap principle", simply states that, if reproductive success is max-imized by the selective pressure of evolution, and if it is a function of both life expectancy and mating success, and if the marginal effect of the signal size on mating success is positive, its marginal effect on survivorship has to be negative to balance the first one and get zero marginal value at equilibrium ( [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF]).

Another intuition, which is equally valid, is that if the signal were "cheap", a low-quality male could, for free, mimic a high quality one and so doing improve his pay-off, contradicting the equilibrium hypothesis.

In the next subsections, we give a short account of signaling theory, that we need to proceed.

A signaling game

A simple signaling game is as follows. There are two players, 1 and 2, choosing their decisions u 1 and u 2 in decision sets U 1 and U 2 respectively, most often sets of mixed strategies.

The payoffs of the players are functions J i (θ, u 1 , u 2 ), i = 1, 2, of the decisions and of a parameter θ belonging to some set Θ. But θ is a "private information" of player 1, i.e. player 1 is able to choose his control via a strategy ψ 1 as u 1 = ψ 1 (θ), while player 2 does not know θ.

The game proceeds as follows: player 1 plays first. His decision u 1 creates, possibly via a noisy -probabilistic-mechanism, a signal s observed by player 2. Then player 2 plays, and he is allowed to choose his decision u 2 as a function of the signal, say u 2 = ψ 2 (s).

The rules of the game, the sets Θ, U 1 , U 2 , the criteria J 1 and J 2 , and the mechanism that generates s as a function of θ and u 1 , leading to a probability law P 1 (s | θ, u 1 ), are common knowledge. (i.e. both players know them and know that the other one knows, . . . , ad infinitum.) But player 2 does not know θ, nor the realizations of the random variables involved in generating s.

Finally, we are compelled to adopt Harsanyi's device for games of incomplete information: both players share an a priori distribution law, say P 0 , upon θ.

The interesting feature of that kind of game is that in making his decision, player 1 should take into account the information about his private data that he is leaking to player 2 through his own decision.

Pure signaling game We shall consider a simplified form of signaling game, called pure signaling game, where, on the one hand, J 2 does not depend directly on u 1 , so that player two is only interested in the informational content of u 1 , which, on the other hand, he exactly observes, with no uncertainty.

Bayesian equilibrium

We now describe what we consider as an equilibrium of this game, called a bayesian equilibrium, the natural extension of the Nash equilibrium to signaling games.

First form Player 2 will form a conjecture as to how player 1 has formed his decision, of the form u 1 = χ(θ). Using that conjecture and the signal s, he will update his probability distribution over θ according to Bayes rule, producing a conditional law P 2 (θ | s):

P 2 (θ | s) = P 1 (s | θ, χ(θ))P 0 (θ) P 1 (s | η, χ(η))P 0 (dη) .
He then chooses u 2 = ψ 2 (s) to maximize his expected pay-off, i.e. such that

∀u 2 ∈ U 2 , E P 2 (θ|s) J 2 (θ, χ(θ), ψ 2 (s)) ≥ E P 2 (θ|s) J 2 (θ, χ(θ), u 2 ) .
Assume that player 1 has correctly "guessed" player 2's conjecture χ. He then knows completely how player 2 will construct his decision. i.e., the function ψ 2 . He may choose his own decision u 1 = ψ 1 (θ) to maximize his own expected payoff, i.e. such that,

∀u 1 ∈ U 1 , E P 1 (s|θ,ψ 1 (θ)) J 1 (θ, ψ 1 (θ), ψ 2 (s)) ≥ E P 1 (s|θ,u 1 ) J 1 (θ, u 1 , ψ 2 (s)) . (1)
Definition 1 The pair (ψ 1 , ψ 2 ) is a bayesian equilibrium if, in the above construction, ∀θ ∈ Θ , ψ 1 (θ) = χ(θ) .

(

) 2
In words: even knowing player 2's conjecture χ, player 1 cannot do better than using that same decision rule.

Remark 1 It is not necessary, in defining a bayesian equilibrium, to introduce a different name for player 2's conjecture χ and player 1's decision rule ψ 1 , since we want them to coincide. But doing so helps to stress that χ is part of player 2's strategy. As usual, a Nash-like equilibrium can be viewed as a fixed point: ψ 1 is a function ψ 1 = F(χ) and a bayesian equilibrium should satisfy F(ψ 1 ) = ψ 1 . At equilibrium, it holds that

P 2 (θ | s) = P 1 (s | θ, ψ 1 (θ))P 0 (θ) P 1 (s | η, ψ 1 (η))P 0 (dη) . (3)
Second form A simplification arises if J 2 is independent from u 1 . In that case, instead of forming a conjecture on player one's strategy, player 2 may use the signal s to form a conjecture, called a belief, on θ in the form of a probability law P 2 (θ | s). Then, he chooses his strategy ψ 2 (s) such that,

∀u 2 ∈ U 2 , E P 2 (θ|s) J 2 (θ, ψ 2 (s)) ≥ E P 2 (θ|s) J 2 (θ, u 2 ) .
If player 1 correctly infers this strategy, he may choose a strategy ψ 1 (θ) such that, as previously equations 1 holds.

Definition 2

The system ψ 1 , ψ 2 , P 2 is a bayesian equilibrium if equation 3 holds.

Necessary conditions 2.2.1 Types of bayesian equilibria

We do not attempt to write here first order necessary conditions that a general bayesian equilibrium should satisfy, because we shall only need a particular, and simpler, version of the second form above.

Definition 3 A bayesian equilibrium where ψ 1 is an invertible function, is called separating (or revealing). Otherwise, the equilibrium is called pooling.

Remark 2

1. If a separating equilibrium is played, after player 1 has played player 2 is exactly informed of the previously private information θ. The signal is said to be "honest".

2. Our models have two equilibria: a "mixed" equilibrium, separating in a range of the private parameter, pooling in its complement, but essentially behaving as a separating equilibrium, and a trivial pooling equilibrium.

Separating equilibrium of a pure signaling game

We consider a pure signaling game

J 1 : Θ × U 1 × U 2 → R, J 2 : Θ × U 2 → R, signal s = u 1 ∈ U 1 . Let ϕ : Θ → U 2
give the optimal control u 2 knowing θ:

∀u 2 ∈ U 2 , J 2 (θ, ϕ(θ)) ≥ J 2 (θ, u 2 ) .
We have implicitly assumed up to now that the set U 1 of possible first player's decisions is independent on θ, however, this will not be true in our first model. But this is critical only if an equilibrium strategy lies on the boundary of U 1 . We avoid that problem with the following assumptions:

Assumptions 1. Θ, U 1 and U 2 are subsets of R or R itself, 2 2. J 1 ∈ C 1 , J 2 ∈ C 2 ,
3. J 2 is locally strictly concave in u 2 in a neighborhood of its maximum and

∂ 2 J 2 ∂u 2 2 (θ, ϕ(θ)) < 0 . (4)
4. There exists a bayesian equilibrium with, for all θ, ψ 1 (θ) in the interior of

U 1 and ψ 2 • ψ 1 (θ) in the interior of U 2 . ( 3 )
Handicap principle Under these assumptions, the optimality of ψ 1 implies

dJ 1 (θ, u 1 , ψ 2 (u 1 )) du 1 = ∂J 1 ∂u 1 + ∂J 1 ∂u 2 ψ ′ 2 (u 1 ,u 2 )=(ψ 1 (θ),ψ 2 (ψ 1 (θ))) = 0. (5)
Hence, the direct effect of u 1 on J 1 and its indirect effect through player two's response are necessarily of opposite signs. Much of the literature fails to define precisely what is called "handicap principle". We use this phrase for the following proposition, which is a corollary of the above essentially trivial remark:

Proposition 1 If we assume that the aim of the signal is to obtain a favorable response from the other player, then, under assumptions 1 to 4 above, its direct effect ∂J 1 /∂u 1 has to be negative at equilibrium.

Computing the equilibrium strategies

According to the hypothesis that the optimal decisions are interior decisions, we have

∂J 2 ∂u 2 (θ, ϕ(θ)) = 0
and, by the implicit function theorem,

ϕ ′ (θ) = dϕ dθ = - ∂ 2 J 2 ∂u 2 ∂θ (θ, ϕ(θ)) ∂ 2 J 2 ∂u 2 2 (θ, ϕ(θ)) -1 . (6)
If the equilibrium is separating, the optimal ψ 2 is

ψ 2 (u 1 ) = ϕ(ψ -1 1 (u 1 )) ,
and hence, according to the chain rule

ψ ′ 2 (u 1 ) = ϕ ′ (ψ -1 1 (u 1 )) 1 ψ ′ 1 (θ)
.

Substituting this and equation ( 6) in ( 5), one obtains that for

u 1 = ψ 1 (θ), 0 = dJ 1 (θ, u 1 , ψ 2 (u 1 )) du 1 = ∂J 1 ∂u 1 (θ, u 1 , ψ 2 (u 1 )) - ∂J 1 ∂u 2 (θ, u 1 , ψ 2 (u 1 )) ∂ 2 J 2 ∂u 2 ∂θ (θ, ψ 2 (u 1 )) ∂ 2 J 2 ∂u 2 2 (θ, ψ 2 (u 1 )) 1 ψ ′ 1 (θ) , hence ψ ′ 1 (θ) = ∂J 1 (θ,ψ 1 (θ),ψ 2 (ψ 1 (θ))) ∂u 2 ∂J 1 (θ,ψ 1 (θ),ψ 2 (ψ 1 (θ))) ∂u 1 ∂ 2 J 2 ∂u 2 ∂θ (θ, ψ 2 (u 1 )) ∂ 2 J 2 ∂u 2 2 (θ, ψ 2 (u 1 )) . ( 7
)
The point is that, as a rule, the first order necessary conditions yield a differential equation on ψ 1 . There remains to choose an initial condition for it. As u 1 is costly, the best possible initial condition is the smallest possible one within the constraints of the problem at hand. Finally, notice that we know that, under our assumptions, both ∂J 1 /∂u 1 and ∂ 2 J 2 /∂u 2 2 have to be negative. Assuming that ∂J 1 /∂u 2 is positive (hence also ψ ′ 2 ), then, it follows from equation ( 7) that the equilibrium signal is indeed monotonously increasing, hence honest, as well as the receiver's response, if, at equilibrium

∂ 2 J 2 ∂u 2 ∂θ (θ, ψ 2 (θ)) > 0 .
This condition bears on the receiver's marginal sensitivities. It looks different from Getty [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF] equation ( 5), but only because Getty writes his condition in terms of the signaller's resulting fitness taking into account the receiver's response (P (a) in his notation). The receiver's criterion is implicit in his model, while it is explicit here.

3 Two models of sexual selection

The framework

Most previous studies have modelled fitness as the product of survivorship and mating success, where survivorship is a lifetime measure of the expected number of breeding attempts. ( [START_REF] Stephen | Older males signal more reliably[END_REF]). Let therefore the quality of an individual male denote his survivorship, more precisely the expected number of breeding seasons he will live through, in the absence of the negative impact of the secondary sexual characters used as a signal. The qualities of male individuals are distributed according to a probability law P 0 , that we shall quickly assume uniform. Quality will be denoted by q ∈ (q 0 , q 1 ]. The latter quantity can indeed be bounded above by q 1 = 1 for instance, which means that the species considered live through a single breeding season (annual or semelparous species) [START_REF] Grafen | Biological signals as handicaps[END_REF], or by a larger q 1 for iteroparous species enjoying as a rule several reproductive bouts. Grafen also considered that the expected number of breeding seasons can be bounded below by a strictly positive value q 0 > 0. It is obvious that from the female point of view, q cannot be zero. Yet, it could definitely be arbitrarily close to zero, implying q 0 = 0, and we will see that this parameter has a great influence on Grafen's model.

To show why q 0 has such an influence on Grafen's model, we will compare two closely related models. We call one the multiplicative signaling cost model, since in this model, the measure of the signal σ ∈ [0, 1] is the factor by which the survivorship L ♂ of a male is reduced: L ♂ = q(1 -σ). This model will be shown to be akin to Grafen's, as opposed to the additive signaling cost model, also alluded to in [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF] who attributes it to [START_REF] Nur | Phenotypic plasticity and the handicap principle[END_REF], in which the measure of the signal s > 0 is the quantity by which survivorship is reduced: L ♂ (q, s) = q -s. In this model, males have negative fitness as soon as they signal greater than q. These two models could be used to compare "index" (unfakeable) signaling with strategic (unconstrained by quality) signaling [START_REF] Peter | A strategic taxonomy of biological communication[END_REF]. Yet, we will focus on interior equilibria on the basis of local stability conditions, so this dichotomy will not be relevant in this study. As a last remark, we stress that in each model, contrary to [START_REF] Johnstone | Dishonesty and the handicap principle[END_REF] for instance, males vary by their quality only; males of same quality "pay" the same amount for the same signal.

In both cases, the expected number of successful matings per breeding season, or mating success rate, is denoted N m . It is a function of the signal, σ or s, and of how females behave in response to it. To evaluate that number, we assume that the breeding season is of length T each year; t is the time elapsed since the beginning of the breeding season. We however neglect the event of a male dying during that season. Male's fitness therefore is

F ♂ = L ♂ N m .
For each model, we shall consider two extreme cases, depending on whether the limiting factor for the males, beyond being accepted by females, is their (constant) potential reproduction rate (P.R.R.) µ, or the scarcity of females, measured by an (increasing) operating sex ratio (O.S.R.) ω(t).

Each female meets males in a Poisson process of intensity λ. She observes the signal exactly, and chooses a probability m = ϕ ♀ (σ) in the first model, m = ψ ♀ (s) in the second, to accept to mate. Her fitness will be taken to be the quality of the male she will mate with, its expectation Q 0 = F ♀ being the same for all females. After mating, she leaves the pool of females available for mating.

It will be convenient to use all along the following dimensionless parameters

k = 2 λT , τ = T -t T , ( 8
)
where λT is the mean number of males encounter by females in one breeding season.

Females' behavior

In this section, we use the same notation s for s or σ.

Females' strategy

According to the earlier introduction to signaling theory, we let χ(q) be the conjecture made by the females about the males' strategy, i.e. χ -1 (s), be her estimate of a male's quality given his signal s.

Let Q(t) be the expected quality of the male she will mate with for a female that has not yet mated at time t. In the forthcoming infinitesimal time interval dt, she will meet a male with a probability λ dt, whose quality q is distributed according to the probability law P 0 , and displaying an a priori unknown signal s, and she will accept to mate with him with a probability m, her choice, which may depend on t and s. We therefore get

Q(t) = E[λ dt mq + (1 -λ dt m)Q(t + dt)] .
In the limit as dt → 0, this yields

dQ dt + λE[m(q -Q(t))] = 0 , Q(T ) = 0 .
(In this equation, m is allowed to be a random variable measurable on the sigma algebra generated by q.) It follows from the theorem of comparison of the solutions of ordinary differential equations that to maximize Q(0), the females should maximize

E[m(q -Q)], i.e. choose m = 0 if q < Q(t) , 1 if q ≥ Q(t) .
However, they observe neither q nor Q. They will have to rely on q = χ -1 (s), and choose the threshold strategy

m = ψ ♀ (s; t) := 0 if s < θ(t) , 1 if s ≥ θ(t) . (9)
with

θ(t) = χ(Q(t)) , (10)
where Q(t) is calculated as the solution of the differential equation obtained on the basis that s = χ(q), i.e. that q is exactly recovered from s. Let, for any X ∈ R,

[X] + = 0 if X ≤ 0 , X if X > 0 .
The differential equation for Q(t) reads:

dQ dt + λE[q -Q(t)] + = 0 , Q(T ) = 0 ,
or, equivalently

dQ dt + λE[q -Q(t) | q ≥ Q(t)]P 0 ([Q(t), q 1 ]) = 0 , Q(T ) = 0 .
To carry the calculations further, we make the following assumption:

Assumption The probability law P 0 is uniform over [q 0 , q 1 ].

To go further, we need to discuss whether Q(t) is smaller or larger than q 0 .

1. If Q ≤ q 0 , then

E[q -Q] + = q 1 + q 0 2 -Q , hence dQ dt = -λ q 1 + q 0 2 -Q Q(T ) = 0 , and
Q(t) = q 0 + q 1 2 1 -e -λ(T -t)
from the time T 0 at which Q(T 0 ) = q 0 , i.e. Figure 1: Curves Q(t)/q 1 , to the left, for q 0 = 0, with various values of λT , to the right, for λT = 8 and different values of q 0 /q 1 . up to time T . We shall use the notation

e -λ(T -T 0 ) = q 1 -q 0 q 1 + q 0 , (11)
τ 0 := T -T 0 T = k 2 ln q 1 + q 0 q 1 -q 0 . ( 12
) 2. If Q ≥ q 0 , then E[q -Q] + = 1 q 1 -q 0 q 1 Q (q -Q) dq = (q 1 -Q) 2 2(q 1 -q 0 ) , hence dQ dt = -λ (q 1 -Q) 2 2(q 1 -q 0 ) , Q(T 0 ) = q 0 .
Integrating this last equation we summarize:

Q(t) =        q 1 k q 0 q 1 T + T 0 -t kT + T 0 -t if t ≤ T 0 , q 0 + q 1 2 1 -e -λ(T -t) if t ≥ T 0 . (13)
Notice that

∀t ∈ [0, T ] , Q(t) ≤ Q 0 := q 1 k q 0 q 1 T + T 0 kT + T 0 . (14)
Equations ( 9), ( 10), [START_REF] Kirkpatrick | The evolution of mating preferences and the paradox of the lek[END_REF], and (13) define the females' strategy in terms of their conjecture χ about the males' strategy. In figure 1, we have plotted curves of Q(t) against t/T .

Consequences for the males

A male displaying a signal s mates only when Q(t) ≤ χ -1 (s), i.e. when t ≥ t m with

τ m := T -t m T =    1 if χ -1 (s) ≥ Q 0 , τ 0 + k χ -1 (s) -q 0 q 1 -χ -1 (s) if χ -1 (s) ≤ Q 0 . (15)
It will be useful in the sequel to note that, for q < Q 0 ,

dτ m ds = τ ′ m = k 1 - q 0 q 1 q 1 χ ′ (χ -1 (s))(q 1 -χ -1 (s)) 2 . ( 16
)
Another consequence is that the number of females available for mating decreases during the breeding season. At time t, there are n ♀ (t) unmated females. Let N ♀ = En ♀ . During an infinitesimal time interval dt, an expected number N ♀ (t)λ dt of females meet a male, and a proportion (q 1 -Q(t))/(q 1 -q 0 ) of these males are accepted as mates, as long as Q < q 0 , i.e. for t < T 0 , and all are accepted once t ≥ T 0 .The females that have accepted a mate leave the pool of available females. Therefore we have for the expected number of available females

dN ♀ dt =    -N ♀ λ q 1 -Q(t) q 1 -q 0 = -N ♀ 2 kT + T 0 -t if t < T 0 , -λN ♀ if t ≥ T 0 .
and therefore

N ♀ (t) =          N ♀ (0) kT + T 0 -t kT + T 0 2 if t < T 0 , N ♀ (0) kT kT + T 0 2 q 1 -q 0 q 1 + q 0 e -λ(t-T ) if t ≥ T 0 . (17)
3.3 Males' strategy: multiplicative signaling cost

An abstract derivation

Notation We investigate our first model, i.e. one where the signal is σ ∈ [0, 1] and the survivorship

L ♂ = q(1 -σ) , σ ∈ [0, 1] . (18)
We let q ∈ [q 0 , q 1 ] with q 0 > 0.

The females' strategy is as above, and therefore t m (s) is given by equation ( 15) and N ♀ (t) by [START_REF] Vincent | Evolutionary Game Theory, Natural Selection and Darwinian Dynamics[END_REF]. We denote σ = ϕ ♂ (q) the males' strategy in this model.

We use the notation

F ♂ (q, σ, m(•)) = L ♂ (q, σ)N m (m(•)) , F ♂ (q, σ) := F ♂ (q, σ, ϕ ♀ (σ; •)) = L ♂ (q, σ)N m (ϕ ♀ (σ)) , (19) and ϕ ♀ (σ, t) = φ♀ (χ -1 (σ), t) , N m (q) := N m ( φ♀ (q, •)) , ( 20
) so that F ♂ (q, σ) = L ♂ (q, σ) N m (χ -1 (σ)) .
Deriving ϕ ♂ and F ♂ According to section 2, we derive ϕ ♂ (q) by differentiating F ♂ with respect to σ and equating to zero. Using (χ -1 ) ′ (σ) = 1/χ ′ (χ -1 (σ)), this yields, for q < Q 0 :

∂ F ♂ (q, σ) ∂σ = -q N m (χ -1 (σ)) + q(1 -σ) N ′ m (χ -1 (σ)) χ ′ (χ -1 (σ))
Replacing χ -1 (σ) by q, χ by ϕ ♂ and σ by ϕ ♂ (q), and equating to zero, we obtain

0 = -N m (q) + (1 -ϕ ♂ (q)) N ′ m (q) ϕ ′ ♂ (q) , or, equivalently ϕ ′ ♂ (q) 1 -ϕ ♂ (q) = N ′ m (q) N m (q) .
Using ϕ ♂ (0) = 0, it follows immediately that

(1 -ϕ ♂ (q)) N m (q) = N m (q 0 ) , F ♂ (q, ϕ ♂ (q)) = q N m (q 0 ) . (21)
When q ≥ Q 0 , there is no incentive for the males to signal more than for q = Q 0 , since anyway, with this signal τ m = 1. We let thus ϕ ♂

(q) = ϕ ♂ (Q 0 ), N m (q) = N m (Q 0 ), so that (1 -ϕ ♂ (q)) N m (q) = (1 -ϕ ♂ (Q 0 )) N m (Q 0 ) = N m (q 0 )
, and equation (21) still holds.

Equilibrium solutions of two variants

P.R.R. as the limiting factor

In this first simpler variant, we deviate somewhat from the stated hypothesis that males can mate an arbitrary number of times. We assume that the number of available females remains large and the meeting rate high, and that, once accepted by females, the limit on males' number of matings is their potential reproduction rate µ, limited, say, by the time it takes to produce sperm. Il this model, λ is the rate of females meeting a male ready for mating. In this case, we simply have N m (σ) = µT τ m (σ). Hence

1 -ϕ ♂ (q) = N m (q 0 ) N m (q) = τ 0 (q 1 -q 0 ) τ 0 (q 1 -q) + k(q -q 0 ) , or ϕ ♂ (q) = k(q -q 0 ) τ 0 (q 1 -q) + k(q -q 0 )
,

and F ♂ (q, ϕ ♂ (q)) = qµT τ 0 .
Remember that τ 0 is given by equation ( 12), and is zero if q 0 = 0. Therefore in that case, we get ϕ ♂ (q) = 1 and F ♂ = 0 for all q > 0, an absurd conclusion. We must therefore assume q 0 > 0. We shall come back to this point in the discussion.

Scarcity of the females as the limiting factor

In this variant, we come back to the hypothesis that the P.R.R. of males is large, and that the only limit on their number of matings, once accepted by females, is the increasing scarcity of mating opportunities while the breeding season goes on. We ignore explicit infighting between males to get access to females, which is a reasonable assumption as long as the initial operating sex ratio is low enough that males seldom meet each other.

The population under consideration is assumed to be in a steady state, with N ♂ males and N ♀ (0) females at the start of each breeding season. N ♂ being constant, so is the rate λ at which females meet males. Each useful meeting involves a male and a female. Hence, the process of males meeting an available female is Poisson with an intensity µ(t) such that N ♂ µ(t) = n ♀ (t)λ. Therefore, writing µ(0) = µ 0 ,

µ(t) = λn ♀ (t)/N ♂ = µ 0 n ♀ (t)/N ♀ (0) . ( 22
)
We can now estimate the expected mating rate using [START_REF] Vincent | Evolutionary Game Theory, Natural Selection and Darwinian Dynamics[END_REF] (and Fubini's theorem):

N m = E T tm µ(s) ds = µ 0 N ♀ (0) T tm N ♀ (s) ds = µ 0 N ♀ (0) T 0 tm N ♀ (s) ds + T T 0 N ♀ (s) ds = µ 0 k 3 T 3 (kT + T 0 ) 2 q 0 q 1 + q 0 + 1 3 1 + τ m -τ 0 k 3 - 1 3 .
We use this with equation ( 15) to get

N m (q) = K 3 3q 0 q 1 + q 0 + q 1 -q 0 q 1 -q 3 - 1
with

K := µ 0 T 3 k 3 (kT + T 0 ) 2 ,
and finally ϕ ♂ (q) = (q 1 -q 0 ) 3 -(q 1 -q) 3 (q 1 -q 0 ) 3 -q 1 -2q 0 q 1 +q 0 (q 1 -q) 3 , F ♂ (q, ϕ ♂ (q)) = K q 0 q 1 + q 0 q .

Note that this model again displays the feature that if q 0 = 0, then ϕ ♂ (q) = 1 and F ♂ = 0 for all q > 0, an absurd conclusion.

In figure 2, we have plotted the strategy ϕ ♂ , the relative survivorship L ♂ /q 1 and the overall fitness F ♂ /q 1 of males for λT = 8, and for different values of q 0 /q 1 in the case of a limiting availability of females. We have included q 0 /q 1 = .5 for clarity, but we consider this value beyond a realistic model. See the foregoing discussion.

Males' strategy: additive signaling cost

In this section, we investigate a model somewhat more different from Grafen's than the first one, in that we assume that, denoting s the signal,

L ♂ = q -s , s ∈ [0, q] . (23)
In this model, we will see that it is possible to take q 0 = 0, and hence T 0 = T , τ 0 = 0, and Q 0 = q 1 /(1 + k). We denote ϕ ♂ (s) the males' strategy, and we use the notation (19) mutatis mutandis.

Limiting P.R.R.

As for the first model, we first consider the case where µ is constant. The expected number of matings in any season is µ(T -t m ), and therefore the males' fitness is L ♂ /q 1 (bottom left) and overall fitness F ♂ /q 1 (bottom right) as a function of q/q 1 , and for λT = 8 and several values of q 0 /q 1 .

F ♂ (q, s) = µT (q -s)τ m (s) .
Clearly, for χ -1 (s) ≥ q 1 /(1 + k), F ♂ is strictly decreasing in s, so that it is useless for the males to use a signal larger than χ(q 1 /1 + k). For smaller s, we have, using the fact that (χ -1 ) ′ = (χ ′ ) -1 :

1 µT k ∂ F ♂ ∂s = - χ -1 (s) q 1 -χ -1 (s) + χ -1 (s) -s (q 1 -χ -1 (s))χ ′ (χ -1 (s))
Now, at equilibrium, s = ψ ♂ (q) = χ(q) so that also χ -1 (s) = q. Equating the above derivative to zero, we get

ψ ′ ♂ (q) = q 1 q -ψ ♂ (q) q(q 1 -q) , ψ ♂ (0) = 0 . (24)
We initialize this differential equation at ψ ♂ (0) = 0 in order to keep s as small as compatible with the equation. It turns out that equation (24) has a closed form solution Ψ(q): Ψ(q) = q 1 1 + q 1 -q q ln 1 -q q 1 if q = 0 ,

Ψ(0) = 0 . (25)
It takes a power expansion in the neighborhood of q = 0 to check that indeed, this Ψ is continuous at 0, and also Ψ ′ (0) = 1/2 and Ψ ′′ (0) = 1/3.

We therefore end up with definition (25) and

ψ ♂ (q) =      Ψ(q) if q ≤ q 1 1 + k , Ψ q 1 1 + k if q ≥ q 1 1 + k . ( 26
)
It remains to substitute this ψ ♂ to χ in equation [START_REF] Johnstone | Dishonesty and the handicap principle[END_REF] to have the complete bayesian equilibrium of the signaling game, which is pooling in the upper range q ≥ q 1 /(1 + k), and separating for smaller q.

Scarcity of females as limiting factor

We investigate now the full model with an essentially unlimited P.R.R. of the males. Their meeting rate of females is therefore given by (22). Using [START_REF] Vincent | Evolutionary Game Theory, Natural Selection and Darwinian Dynamics[END_REF] with T 0 = T , it follows that

Eµ(t) = µ 0 k + τ k + 1 2 . (27)
We use this in evaluating the expected number of matings of a male in one season as

N m = T tm Eµ(t) dt = µ 0 T τm 0 k + τ k + 1 2 dτ = µ 0 T (k + 1) 2 (k 2 τ m + kτ 2 m + 1 3 τ 3 m )
and,

F ♂ (q, s) = µ 0 T (k + 1) 2 (q -s)(k 2 τ m (s) + kτ m (s) 2 + 1 3 τ m (s) 3 ),
where τ m is still given by [START_REF] Stephen | Older males signal more reliably[END_REF]. We may again notice that for q > q 1 /(1 + k), it is useless for the males to use a signal greater than for q = q 1 /(1 + k). For lower q, we differentiate F ♂ with respect to s, and equate to zero, replacing χ -1 (s) by q and s by ψ ♂ (q), to get

ψ ′ ♂ (q) = q 3 1 q -ψ ♂ (q) q(q 1 -q)(q 2 1 -q 1 q + q 2 3 ) , ψ ♂ (0) = 0 . (28)
Moreover, this stationary point is indeed a maximum, since F ♂ (q, 0) = F ♂ (q, q) = 0, and F ♂ (q, •) is positive and differentiable for s ∈ (0, q). Thus, the stationary point being unique, it necessarily is a maximum. Unexpectedly, the differential equation (28) has a closed form, and even rational, solution. Since a direct check that formula (29) does satisfy the equation is not so easy, we provide a direct derivation in the appendix. This closed form solution of the differential equation ( 28) is Ψ(q) = q 1 27q 5 1 q + 18q 4 1 q 2 + 9q 3 1 q 3 + 3q 2 1 q 4 -q 6 2(27q 6 1

+ q 6 ) (29)
One may notice that, on the one hand, both ( 24) and (28) imply that ψ ′ ♂ (0) = 1/2, but the second one yields ψ ′′ ♂ (0) = 2/3, so that for small q, Ψ(q) is "above" Ψ(q). On the other hand, for s, q ∈ (0, q 1 ),

q 1 q -s q(q 1 -q) ≤ q 3 1 q -s q(q 1 -q)(q 2 1 -q 1 q + q 2 2 )
so that, by a standard comparison principle, for all q in [0, q 1 ], Ψ(q) ≤ Ψ(q). But since ψ ♂ (q) must remain less than q, it follows that necessarily, Ψ(q 1 ) = q 1 , which is indeed satisfied by (29). Finally, the equilibrium strategies of this model are given by equations ( 9), ( 13), [START_REF] Johnstone | Dishonesty and the handicap principle[END_REF] where we substitute ψ ♂ to χ, (29), and

ψ ♂ (q) =      Ψ(q) if q ≤ q 1 k + 1 , Ψ q 1 k + 1 if q ≥ q 1 k + 1 . (30)
In figure 3, we have plotted the strategy ψ ♂ /q 1 , the relative survivorship L ♂ /q 1 , and the overall fitness F ♂ /q 1 of males against q/q 1 , for λT = 8, for the case of a limiting P.R.R. (fixed µ) or a limiting availability of females (variable µ).

Discussion

General synthesis

The paradigm of bayesian equilibrium is a natural one to deal with the role of sexually selected characters, this independently of whether the underlying mechanism is one of direct benefit, as in [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF], or indirect benefit as allowed by our model. Indeed, what game theory teaches us is that, if these secondary sexual characters have for unique, or main, role in the evolutionary process, to transmit information about their bearer's quality to the individuals of the other sex, and if they are not physically constrained by that quality (see [START_REF] Peter | A strategic taxonomy of biological communication[END_REF]), then at equilibrium it must be that the direct effect of these characters is to lower the viability of their bearer, trading viability for greater sexual success. And the nature of a (separating) Bayesian equilibrium is that the possibility of profitable cheating has been defined out. Figure 3: Additive model: Males' strategy ψ ♂ /q 1 (top), survivorship L ♂ /q 1 (bottom left), and fitness F ♂ /q 1 (bottom right) as a function of q/q 1 , and for the two variants : limiting P.R.R. (fixed µ) or limiting availability of females (variable µ) for λT = 8.

Our models are based on reasonable simplifying assumptions. They allowed us to obtain closed form solutions for all four variants, a useful feature if one wants to discuss the effect of the parameters left: the mean number of males encounter by females in one breeding season, λT , and the minimum to maximum survivorship ratio q 0 /q 1 . The initial operating sex ratio λ/µ 0 is less significant, only playing as a multiplicative coefficient in males' fitness F ♂ .

As expected, they do yield a large signaling with a correlative large reduction in males' viability, this in the absence of any "decreasing marginal cost of signaling", deemed necessary by [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF], but in agreement with Getty [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF] [START_REF] Getty | Sexually selected signals are not similar to sports handicaps[END_REF]. As a matter of fact, in the additive signaling cost model, ∂ 2 F ♂ /(∂q∂s) = 0, and in the multiplicative signaling cost model, the marginal cost of signaling increases with the male's quality.We claim that the only requirements for what we call the "handicap paradox" to hold are those of our Proposition 1.

Lastly, our models exhibit a very low reproductive efficiency of males of low or average quality. This begs the question of maintenance of quality diversity in the population, a form of the lek paradox [START_REF] Kirkpatrick | The evolution of mating preferences and the paradox of the lek[END_REF].

Back to the main hypotheses

Beyond the hypothesis of the existence of a scalar "quality" for each male, reflecting its survivorship and translating into fitness for the females that mate with him, and of a scalar "signal" associated with quality, the simplicity of the model, and its closed form solution, hinge on 2 main hypotheses:

1. Linear relationship between survivorship and female fitness. Once such a correlation is hypothesized, a linear one is not less realistic than anything else, and obeys in some sort a principle of parsimony, or of simplicity.

2. Uniform distribution of males' quality. Although arbitrary, and difficult or impossible to check in any particular case, it does not seem to be exceedingly unrealistic. Indeed, we need a distribution with compact support, which normal or log-normal distributions cannot provide. This can again be seen as the application of a principle of parsimony.

Equilibrium selection as an open issue

The bayesian equilibrium outlined above is not unique. In all cases considered, another bayesian equilibrium pair of strategies is the trivial one where females accept all males, and males do not signal: ψ ♀ (t, s) = 1 for all t, s, and ψ ♂ (q) = 0 for all q. Indeed, if females do not expect males to signal, they ignore the signal if any, and choose χ(q) = 0. Then males' mating success is not influenced by their signal, which only has a detrimental impact on their fitness, so that they should keep it equal to 0, and therefore ψ ♂ (q) = χ(q). This raises the issue of whether the nontrivial and trivial Bayesian equilibria attract or repel evolutionary dynamics. This is discussed in the literature [START_REF] Eshel | Evolutionary and continuous stability[END_REF][START_REF] Eshel | On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution[END_REF], and is amenable to mathematical analysis in the framework of adaptive dynamics [START_REF] Dercole | Analysis of evolutionary processes: the adaptive dynamics approach[END_REF] or evolutionary games [START_REF] Vincent | Evolutionary Game Theory, Natural Selection and Darwinian Dynamics[END_REF].

Some qualitative findings

We have stressed that the females' strategy is essentially the same in all four cases, accepting males with an observed quality no less than Q(t). The literature on sexual selection has often introduced a variable measuring choosiness of females, discussing what cost it imparts on them. In our model with a time dependent threshold strategy as females' optimum strategy, an individual is more choosy than another one if her threshold curve lies "above" the other one's. We obtain a natural choice of threshold in terms of males' quality, since the curve Q(t) does not depend on the female's conjecture χ nor on the males' behavior. It only depends on the expected number λT of encounters with a male in a breeding season and on the minimum to maximum survivorship ratio q 0 /q 1 .

The larger the expected number λT of encounters per season the choosier the females are, since they have more opportunities, and correlatively, the larger best males' signals are. Correlatively, for a fixed quality, the male fitness F ♂ decreases as the frequency of meetings λ increases. The explanation is that when λ gets large, females become very choosy, and a male of a given quality is ignored for a longer time during each breeding season; an effect reinforced by the fact that late in the season, the number of available females drops. Perhaps an underappreciated, but rather straightforward, implication of search theory 4 . But of course, this fitness increases proportionally to the inverse initial operational sex ratio µ 0 /λ.

Concerning the ratio q 0 /q 1 , the larger it is the choosier the females are, since the average male's quality increases with this ratio allowing the females to hope for a better mate. But unexpectedly, in the multiplicative signaling cost model, males' signals decrease when the ratio q 0 /q 1 increases.

An artifact in Grafen's model, and a new model to support his theory

Grafen [START_REF] Grafen | Biological signals as handicaps[END_REF] similarly noticed that "as q 0 approaches zero, the general level of advertising increases indefinitely." He quickly proposed an interpretation based on the debatable argument that for a larger q 0 , a male of a given quality has less competitors of lower quality to distinguish himself from. However, there is a simpler and clearer explanation to what actually is an artifact. The point is that both in the multiplicative signaling cost model and in Grafen's, marginal signaling costs vanish as q approaches zero. Indeed, in our multiplicative model, the marginal cost is -∂L ♂ /∂σ = q, and in Grafen's, where L ♂ = q σ in our notations, it is -∂L ♂ /∂σ = -q σ log(q) (remember that q ≤ q 1 = 1 in Grafen's). Since in both models, lim q→0 -∂L ♂ /∂σ = 0 for any σ > 0, a male whose quality is almost zero has little penalty for signaling almost as much as possible, which leads other males to almost full signaling as well. This does not occur in the additive model, since in this case -∂L ♂ /∂s = 1, which implies that the marginal cost of signaling is never zero, whatever q and s. Hence the artifact in the multiplicative model.

In other words, in the multiplicative model akin to Grafen's, we cannot let q 0 tend to 0. As a matter of fact, in our formulas, if q 0 = 0, then ϕ ♂ (q) = 1 for all q > 0. Thus, not only is ϕ ♂ discontinuous at zero, but worse, the survivorship is zero for all males. This is of course an absurd result. Yet, it can be understood in the following way: perturb somewhat the model by assuming that L ♂ (q, σ) = q(1 -σ) + ε for some small positive ε, but keep the constraint σ ≤ 1. And assume q 0 = 0. Then a bayesian equilibrium is ϕ ♂ (0) arbitrary in [0, 1] , ϕ ♂ (q) = 1 ∀q > 0 .

As figure 2 shows, this is almost the equilibrium found for very small q 0 . Yet, it was shown above that it depends very specifically on the fact that the multiplicative form hides zero marginal signaling costs when quality tends to zero. Since the additive model has, to our knowledge, no such hidden bias, we believe it makes a better job as a minimal signaling model to start with.

It is nevertheless interesting to give a thought to the multiplicative model game equilibrium for q 0 = 0. Should this signal be termed honest [START_REF] Számadó | The cost of honesty and the fallacy of the handicap principle[END_REF]? Hardly so in the sense that all males mimic the signal of the best quality males, but yes in the sense that it does not deceive the females. As a matter of fact, in the framework of bayesian equilibrium, "honesty" is merely a consequence of the existence of a unique equilibrium signaling strategy that it is advantageous for the males to use, given the female's conjecture; in essence, existence of a profitable cheating behavior has been defined out by definition 1 and equation [START_REF] Eshel | Evolutionary and continuous stability[END_REF]. In the additive model, the strategy is revealing in the range of qualities below a critical level, and pooling, for high quality males. Yet being unique, it is honest even for these high quality males, revealing that their quality is above the critical level, and it is a sufficient information for the females.

Conclusion

Any model involves a trade-off between realism on the one hand and mathematical simplicity and tractability on the other hand. We settled somewhat more on the second side of this dilemma than Grafen [START_REF] Grafen | Sexual selection unhandicapped by the fisher process[END_REF], [START_REF] Grafen | Biological signals as handicaps[END_REF], allowing us to obtain closed form solutions for all four variants investigated. Yet, our models display some interesting features of the bayesian equilibria of signaling games of sexual selection. Of course, we claim no quantitative results, as so many features of all these models are over-simplifications of the real world: not only many phenomena, such as male infighting, have been ignored, but more profoundly, a scalar "quality", moreover taken as a proxy for the benefit a female has to mate with this male, may be an oversimplification of the biological processes at work, and a scalar signal is surely so most often.

At least, comparing our two models let us propose an explanation of the curious fact that previous models forbade a zero minimum quality, and we offer a simple model in which this is not the case. from which formula (29) follows. (As a cross check, we performed a numerical integration with a Runge Kutta scheme of order 4 and a step size of 0.01, starting from r = 0.05 and ψ(0.05) estimated with a tenth order power expansion. The result agreed with the closed form formula up to the fifth decimal place at r = 1.)

5 Figure 2 :

 52 Figure2: Multiplicative model: Males' strategy ϕ ♂ (top), relative survivorship L ♂ /q 1 (bottom left) and overall fitness F ♂ /q 1 (bottom right) as a function of q/q 1 , and for λT = 8 and several values of q 0 /q 1 .

However, the explanation of the handicap principle in[START_REF] Kokko | Unifying and testing models of sexual selection[END_REF] contains a mistake concerning the transitivity of positive correlations.

In our model, the females' decision is a time function.

Note that this assumption is violated in the case of "index signals"[START_REF] Peter | A strategic taxonomy of biological communication[END_REF] to be discussed, as well as in our trivial equilibrium.

We thank an anonymous reviewer for pointing out the relationship with search theory.

A Solving for Ψ

We are interested here in the differential equation (28). For simplicity, we re-write it in terms of the dimensionless r = q/q 1 and ψ ♂ (q) = q 1 ψ(r). The equation now writes

Write ψ as a power expansion in r, i.e.

and identify terms of like power in r. This yields

and for n > 3,

We consider now ψ(r) + 1/2, (i.e. we add a coefficient ψ 0 = 1/2), we compute the 17 first coefficients ψ n and write them in the following table, with six columns labelled by the powers of r as a function of the line number ℓ starting at 0:

This suggests the following general rule:


Pierre Bernhard

Evolutionary dynamics of the handicap principle: an example

For a model developed in a companion paper, we investigate how adaptive dynamics might have led to the Bayesian equilibrium found. In an appendix, we also explain from a dynamic viewpoint a lack of equilibrium in a closely related model also invesitgated in the companion paper.

Introduction

Ever since Darwin, biologists interested in evolutionary biology have been puzzled by the "handicap paradox", i.e. the fact that secondary sexual characters, most often in males, that clearly attract females, be so exceedingly developed as to be a threat to the male's viability. To (over-)simplify, two mechanisms have been proposed to explain that state of affairs. One is due to Fisher [START_REF] Ronald | The evolution of sexual preferences[END_REF], made into a mathematical model by Lande [START_REF] Lande | Models of speciation by sexual selection on polygenic traits[END_REF]. It is known as the "Fisher runaway". The other one, indeed much simpler, was originally proposed by Zahavi [START_REF] Zahavi | Mate selection -a selection for the handicap[END_REF], and was made into a mathematical model by Grafen [START_REF] Grafen | Sexual selection unhandicapped by the Fisher process[END_REF][START_REF] Grafen | Biological signals as handicaps[END_REF]. It is known as the "handicap principle".

A notable difference is that while Fisher's runaway is a dynamical model, exhibiting a divergence as time goes to infinity, Zahavi's handicap principle describes an equilibrium, and does not say how the dynamics of evolution may have reached that equilibrium. Grafen's article [START_REF] Grafen | Sexual selection unhandicapped by the Fisher process[END_REF] does contain a dynamic argument, but only in the neighborhood of the equilibrium to show its stability. (Moreover, Siller [START_REF] Siller | A note on errors in Grafen's strategic handicap models[END_REF] has invalidated part of the dynamic argument of [START_REF] Grafen | Sexual selection unhandicapped by the Fisher process[END_REF].)

In the article [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF], we emphasize the exact nature of the equilibrium of the handicap paradox: a bayesian equilibrium of signaling theory, where males of different "qualities" signal their qualities via the secondary sexual character of interest to identical females to induce them into accepting to mate with them. In this equilibrium the performance indices are measures of the resulting fitness of the players, or a reasonable proxy of it: the expected quality of her mate for a female, and his expected number of matings for a male. Then we investigate two variants of each of two models of sexual selection directly inspired by Grafen's model [START_REF] Grafen | Sexual selection unhandicapped by the Fisher process[END_REF], and in a lesser extent by Getty [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF]. All four variants yield closed form bayesian equilibria. The models themselves are partly dynamical, because the behaviour of males and females depend on the time of the year, but it is an equilibrium that we compute. And because a bayesian equilibrium is a rather involved concept, it is not a priori clear how evolution may have reached it.

Our aim in this article is to provide one plausible explanation of how the dynamics of adaptation may have reached the signaling bayesian equilibrium. This is the more interesting that there actually are two bayesian equilibria in our models, one where no signaling occurs and one with a large signaling reminiscent of the observed extravagant ornaments of males of certain species.

We shall focus on the model that we deem the most interesting of all four: the "additive signaling cost" model with zero minimal male quality, and scarcity of the females as the limiting factor for the males. Concerning the "multiplicative signaling cost model" of [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF], we shall only give in the appendix a dynamical explanation of an undesirable feature also found in [START_REF] Grafen | Sexual selection unhandicapped by the Fisher process[END_REF].

The model 2.1 General description

We recall here our "additive" model and the equilibria we found. For a discussion of the underlying hypotheses and the validity of the models, and also of the implications of our solutions, we refer the reader to [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF].

There is a short breeding season of length T each year, and we call t ∈ [0, T ] the time elapsed within the breeding season.

There is a fixed number N ♂ of males. The fitness of each is taken to be proportional to the number of matings he achieves over his life time. All have a potential reproduction rate large enough that they are only limited in their number of matings each season by the availability of females and the willingness of the females they meet to mate with them.

Each male has a quality q statistically uniformly distributed over [0, q 1 ] ⊂ R + . 1Each displays a signal s ∈ [0, q], "chosen" according to a strategy s = ψ ♂ (q). Except for their quality, all males are identical and share the same strategy. The fecund life expectancy L ♂ of a male linearly increases with q, but decreases with the signal. (This last fact is the manifestation of the handicap principle. It is not hypothesized a priori, but we show in [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF] under which hypotheses it is a necessary feature of the bayesian equilibrium.) More specifically, we assume that L ♂ = q-s.

There is a fixed number N ♀ (0) of females at the begining of each breeding season. All females are identical. They mate only once during any breeding season. We take as a proxy of their fitness the expectation of the quality of the male they will eventually mate with. Hence they have a benefit in choosing a high quality male as a mate. This in turn puts a selective pressure on males to signal their quality q. Females cannot observe the quality of the males they meet, but only their signals. They accept a given male with a probability m = ψ ♀ (s, t), depending on the male's signal and the time during the breeding season.

Once a female has mated, she leaves the pool of available females, causing the number n ♀ of available females and its expectation En ♀ = N ♀ to decrease, and thus the operational sex ratio ω = N ♂ /n ♀ to increase, during the breeding season.

Females meet fecund males in a Poisson process with a fixed intensity λ, while males meet females in a Poisson process with intensity µ. Since any such meeting involves one male and one female, one necessarily has N ♂ µ = n ♀ λ, hence µ decreases according to µ(t) = λ/ω(t).

We recapitulate the notation in Table 1.

The bayesian equilibrium: concept and notation

We are looking for a stable strategy pair (s = ψ ⋆ ♂ (q), m = ψ ⋆ ♀ (s, t)). What "stable" means is the object of signaling theory. Its concept of bayesian equilibrium simplifies here, for this "pure signaling game", (the signal is exactly observed by the females and does not directly impact their fitness) in the following way. (We shall use super stars to denote equilibrium strategies.)

The uninformed player, here the females, forms a conjecture χ about the males' strategy, meaning that they conjecture, some s = χ(q). If at equilibrium this conjecture is strictly monotonic (here increasing), the equilibrium is said to be separating. In that case, the females can conjecture q = χ -1 (s), and choose a strategy ψ ♀ (q, t) and play

m = ψ ♀ (s, t) = ψ ♀ (χ -1 (s), t). A bayesian equilibrium, is one where ψ ⋆ ♂ is optimal against (χ, ψ ♀ ), ψ ♀ is optimal against ψ ⋆ ♂ ,
k = 2/(λT ). τ τ = (T -t)/T . τ m τ m = (T -t m )/T .
Table 1: The main notation used in the model

Definition 1 A separating bayesian equilibrium is a pair (ψ ⋆ ♂ , ψ ⋆ ♀ ) such that ψ ⋆ ♀ (s, t) = ψ ♀ ((ψ ⋆ ♂ ) -1 (s), t) , (1)
where the pair

(ψ ⋆ ♂ , ψ ♀ ) is a Nash equilibrium of the complete information game.
Of course, this is a construction of thought. Females are not assumed to actually "conjecture" anything, and equation ( 1) should be seen as a useful representation of the equilibrium strategy.

In our model, the bayesian equilibrium is a slight deviation from the above definition, in that ψ ⋆ ♂ is not strictly increasing, but yet, it provides all the needed information on q for the females to determine their optimal behaviour.

Females' behaviour

It turns out that there is a separating bayesian equilibrium, meaning that at equilibrium, females exactly infer males' quality from their signal. They adopt a threshold policy where a male is accepted if his quality is larger than the expected quality Q ⋆ (t) of their future mate if they have not yet mated at that time.

We show in [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF] that Q ⋆ (t) obeys the differential equation

Q + λE max{q -Q, 0} = 0 , Q(T ) = 0 .
the solution of which is given by

Q ⋆ (t) = q 1 T -t (k + 1)T -t . (2)
The a priori expected quality of her mate for any female at equilibrium is thus:

Q ⋆ 0 = Q ⋆ (0) = q 1 k + 1 . (3)
The females conjecture as to the males' strategy is s = χ(q), with χ(0) = 0. Then the threshold strategy translates into

θ ⋆ (t) = χ(Q ⋆ (t)) and m = ψ ⋆ ♀ (s, t) := 0 if s < θ ⋆ (t) , 1 if s ≥ θ ⋆ (t) . (4)
2.4 Males' strategy

Mating opportunities

It follows that a male with a signal s mates no earlier than a time t m given by

τ m := T -t m T =    1 if χ -1 (s) ≥ Q ⋆ 0 , k χ -1 (s) q 1 -χ -1 (s) if χ -1 (s) ≤ Q ⋆ 0 . (5)
We assume that the potential reproduction rate of all males is large enough, so that the limiting factor in their number of matings is the scarcity of available and willing females. Females dynamics as outlined above imply that the expected number of available females at any time during the breeding season is (see [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF])

N ♀ (t) = N ♀ (0) (k + 1)T -t (k + 1)T
The expected rate of males meeting available females is

Eµ(t) = λN ♀ (t)/N ♂ = µ 0 N ♀ (t)/N ♀ (0)
. The expected number of matings per breeding season for a male is the integral of Eµ from t m to T , i.e.

N m (s) = K 3 q 1 q 1 -χ -1 (s) 3 -1 , (7)
with

K = µ 0 T k 3 /(k + 1) 2 .

Males' strategies

The males fitness function is given by

F ♂ = F ♂ (q, s) = K 3 (q -s) q 1 q 1 -χ -1 (s) 3 -1 . (8)
Equations ( 5) and [START_REF] Grafen | Biological signals as handicaps[END_REF] show that for χ -1 (s) > Q 0 , F ♂ is strictly decreasing in s, so that there is no incentive for such a male to advertise with a signal larger than for χ -1 (s) = Q 0 . For smaller qualities, we differentiate F ♂ (q, s) with respect to s and equate to zero. At the bayesian equilibrium, χ = ψ ♂ , and this yields a differential equation for ψ ♂ , specifically

ψ ′ ♂ (q) = q 3 1 q -ψ ♂ (q)
q(q 1 -q)(q 2 1 -q 1 q + q 2 3 )

, ψ ♂ (0) = 0 . (9)
We show in [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF] that the differential equation ( 9) has a closed form solution Ψ(•):

Ψ(q) = q 1 27q 5 1 q + 18q 4 1 q 2 + 9q 3 1 q 3 + 3q 2 1 q 4 -q 6 2(27q 6 1

+ q 6 ) (10)
leading to the equilibrium strategy

ψ ⋆ ♂ (q) =      Ψ(q) if q ≤ q 1 k + 1 , Ψ q 1 k + 1 if q ≥ q 1 k + 1 . ( 11
)
3 Evolutionary dynamics

Fundamental hypothesis

We formulate here an hypothesis which seems reasonable, and greatly simplifies the investigation of the evolutionary dynamics, in effect uncoupling the males and females dynamics. It is based upon the remark that the females' trait of interest is a behavioural trait, which may evolve essentially as an ecological character, while the males' trait is a physical trait, evolving as a genetic character.

Hypothesis 1

The dynamics of evolution of the females' behaviour is much faster than that of the males' signaling strategy.

As a consequence, using Kolmogorov's theorem, if the females' behaviour dynamics turn out to be stable, we may investigate them with a constant males' strategy, and the males' strategy dynamics may be investigated with χ = ψ ♂ , which will prove to be the limit χ.

Females' behavioural "fast" dynamics

The fitness of a female does not depend, in our model, on the behaviour of her conspecifics, so that essentially any evolutionary dynamics are simple gradient dynamics. Assuming it is essentially an ecological process, we call u the ecological time.

General strategy

We assume that the females adopt a strategy ψ ♀ (s, t). It may have been constructed as some ψ♀ (χ -1 (s), t) with some conjecture χ. Now, before reaching the bayesian equilibrium, if ever, χ is different from ψ ♂ and ψ♀ is different from the equilibrium strategy ψ ♀ . Recall, though, that now ψ ♂ is assumed fixed. We may set ψ ♀ (q, t) = ψ ♀ (ψ ♂ (q), t) . This is the strategy that a female happens to play.

In both cases, we use the expected quality Q(t) of her future mate for a female which has not yet mated at time t. Let a ∈ {0, 1} be the random boolean variable describing the choice of a female meeting a male, a = 1 for "accept" to mate. With P(a = 1) = E(a) = m. One has :

Q(t) = λdtE(aq) + (1 -λdtEa)Q(t + dt) , Q(T ) = 0 .
Taking the limit as dt → 0, we obtain

dQ dt = -λE[a(q -Q(t))] , Q(T ) = 0 . (12)
We consider two cases, depending on whether females use a mixed strategy, with m ∈ [0, 1], or whether they use a threshold strategy, with m ∈ {0, 1}, more specifically m = 1 if s ≥ θ(t) for some time dependent threshold θ(t).

Mixed strategies

In this case, each female adopts a mixed strategy, i.e. she randomizes a, choosing its probability m = ψ ♀ (q, t) of being 1, which is also its expected value, and with a -m independent from q, as is customary for a mixed strategy. It follows that E[(a -m)q] = E(a -m)Eq = 0. Therefore

E(aq) = E(mq) = q 1 0 ψ ♀ (q, t) qdq q 1 .
Now, Q obeys the differential equation

dQ dt = -λE[ ψ ♀ (q, t)(q -Q)] . (13)
Let DQ(t) be the Fréchet derivative of Q(t) with respect to ψ ♀ . We evaluate the variation δQ(t) = DQ(t), δ ψ ♀ for a given variation δ ψ ♀ . It satisfies:

dδQ dt = λE ψ ♀ (q, t)δQ -λ q 1 0 δ ψ ♀ (q, t)(q -Q) dq q 1 ,
The first variation of the females' fitness is therefore obtained as a function of

h(v, u) = exp λ v u E ψ ♀ (q, t) dt as δQ(0) = λ T 0 q 1 0 h(0, t)δ ψ ♀ (q, t)(q -Q(t)) dq q 1 dt , (14)
exhibiting the gradient at q as the function t → h(0, t)(q -Q(t)).

Let M be the positive mutation intensity factor (see [START_REF] Dercole | Analysis of evolutionary processes: the adaptive dynamics approach[END_REF]). Finally, the adaptive dynamics are just

d ψ ♀ (q, t) du =          [M h(0, t)(q -Q(t))] -if ψ ♀ (q, t) = 1 , M h(0, t)(q -Q(t)) if ψ ♀ (q, t) ∈ (0, 1) , [M h(0, t)(q -Q(t))] + if ψ ♀ (q, t) = 0 .
They lead to

d dt dQ(t) du = λE[ ψ ♀ (q, t)] dQ(t) du -λh(0, t)E[M (q -Q(t)) 2 ] if d ψ ♀ /du = 0, with dQ du (T ) = 0 .
Now, the semi-group h(•, •) is positive. Hence dQ(t)/du > 0, pointing to a constantly increasing Q(t). But Q(t) is bounded above by Q ⋆ (t). Therefore Q(t) converges, uniformly according to Dini's theorem, to some Q(t), and ψ ♀ (q, t) converges to

ψ ♀ (q, t) = 0 if q < Q(t) , 1 if q ≥ Q(t) . ( 15
)
Because of the continuity of the solution of an ordinary differential equation with respect to its right-hand side, it follows that Q(t) = Q ⋆ (t), and the limiting strategy (15) coincides with the equilibrium strategy

ψ ⋆ ♀ (s, t) = 0 if s < ψ ♂ (Q ⋆ (t)) , 1 if s ≥ ψ ♂ (Q ⋆ (t)) .

Threshold strategy

We now consider the case where the females' strategy evolves from the start as a threshold strategy. Let it be

ψ ♀ (s, t) = 0 if s < θ(t) , 1 if s ≥ θ(t) .
We set q(t) = ψ -1 ♂ (θ(t)), so that the same strategy can be expressed as

ψ ♀ (q, t) = 0 if q < q(t) , 1 if q ≥ q(t) .
In equation [START_REF] Siller | A note on errors in Grafen's strategic handicap models[END_REF], we now have a = 1 if q ≥ q, and E(aq) = EaE(q | a = 1), i.e. E(aq) = P(q ≥ q)E(q | q ≥ q). The expected mate's quality now obeys

dQ dt = -λdt q 1 - q q 1 q 1 + q 2 -Q , Q(T ) = 0 .
We compute its gradient with respect to q(•). Let δQ(t) = DQ(t), δ q . It obeys

dδQ dt = λ q 1 - q q 1 δQ - λq 1 q 1 (Q -q)δ q , δQ(T ) = 0 . Let g(v, u) = exp λ v u q 1 -q(t) q 1 dt .
We obtain

δQ(0) = λ T 0 g(0, t)(Q(t) -q(t)) dt .
Thus, the gradient is of the sign of Q -q. We conclude as above that the adaptive dynamics dq(t) du = M g(0, t)(Q(t) -q(t)) make Q(t) converge monotonically, hence uniformly, toward Q ⋆ (t) while the threshold q converges toward q(t) = Q ⋆ (t) for all t, and that this generates the same equilibrium strategy as previously. We have therefore proved the following:

Theorem 1 Whether they use mixed strategies or threshold strategies, the females' behaviour converges toward the signaling equilibrium threshold strategy.

Therefore, in the limit, females behave as if they were making the right conjecture about the males' strategy.

3.3 Males' strategic "slow" dynamics

Adaptive dynamics

It follows from the hypothesis 1 and from the previous subsection that we may investigate the males' strategy dynamics as if χ = ψ ♂ and the females use the threshold q = Q ⋆ . Let v denote evolutionary time.

For lack of a better choice, we shall make use of the adaptive dynamics approach (see [START_REF] Dercole | Analysis of evolutionary processes: the adaptive dynamics approach[END_REF]), although it has been completely justified [START_REF] Hans) Metz | How should we define fitness for general ecological scenarios[END_REF][START_REF] Dieckmann | The dynamical theory of coevolution, a derivation from stochastic ecological processes[END_REF][START_REF] Dieckmann | Can adaptive dynamics invade ?[END_REF] in the case of clonal reproduction, while we investigate sexual behaviour. One possible hypothesis to better fit the theoretical framework of adaptive dynamics would be to assume that the trait considered is inherited from the father only (say on the Y chromosome of mammals, but this does not account for birds). But we do not wish to insist on this justification, simply taking adaptive dynamics for a plausible description of evolution. In so doing, we shall also assume that the mutation intensity is independent of the quality.

Let ψ ♂ be the resident strategy among males. At equilibrium, thus, χ = ψ ♂ . Assume a mutation produces a small deviation away from that strategy in some males. The fitness of the mutants depends on the resident strategy through the females' behaviour according to their "conjecture" χ = ψ ♂ . If this fitness is larger than the resident's, this strategy will invade and, with the males' population dynamics, replace the resident strategy. Then the females' behaviour will adapt again. Accordingly, the dynamics of the trait ψ ♂ (•) will be obtained by taking the gradient Lemma 1 Under the hypotheses of the theorem, the solution of equation ( 16) cannot converge to ψ ♂ (q) = 0.

Proof We write the equations of the characteristic curves of equation ( 16), using the classic notation ψ ′ ♂ = p. They read

dq dv = KM q 3 1 q -ψ ♂ p 2 (q 1 -q) 4 , ( 17
)
dψ ♂ dv = KM 1 (q 1 -q) 3 2q 3 1 q -ψ ♂ p(q 1 -q) -q 2 1 q + q 1 q 2 - q 3 3 , (18)
dp dv = KM q 3 1 (q 1 -q) 4 4 q -ψ ♂ p(q 1 -q) + 1 p -2 . ( 19
)
Note that q(v) = 0, ψ ♂ (q(v)) = 0 is a characteristic curve. Then, we claim:

Proposition 2 The solution of ( 16) satisfies for all q ∈ [0,

Q ⋆ 0 ], ψ ♂ (q, v) < q.
Proof It follows from equations ( 17) and ( 18) that (we denote with a dot differentiation with respect to v)

(q -ψ ♂ ) • = KM q 3 1 (q 1 -q) 3 1 -2p p 2 (q 1 -q) (q -ψ ♂ ) + q q 1 - q 2 q 2 1 + q 3 3q 3 1 ,
with q(0) -ψ ♂ (q(0), 0) > 0. Because q/q 1 < 1, the last three terms in the right hand side above add to a positive value. Let a(v) = KM q 3 1 (1 -2p)/(p 2 (q 1 -q) 4 ). By the theorem of comparison of the solutions of ordinary differential equations, we have q -ψ ♂ (q, v) > y(v) where y(v) is the solution of the differential equation ẏ = a(v)y , y(0) = q(0) -ψ ♂ (q(0), 0) > 0 , as long as q < Q ⋆ 0 < q 1 . Therefore y(v) remains positive for all relevant v, proving the proposition.

Proposition 3

The characteristic curves fill a neighborhood of the v axis in the (v, q) plane.

Proof To use the local surjectivity theorem, check that the differential

δq(v) = dq(v) dq(0) δq(0) + dq(v) dψ ♂ (0) δψ ♂ (0, 0)
remains positive for all v, provided that δq(0) -δψ ♂ (0, 0) > 0. This last inequality and the previous proposition guarantee that the differentials satisfy δq(v)δψ ♂ (0, v) > 0, driving the variational equation

δ q(v) = 1 p 2 (v) (δq(v) -δψ ♂ (0, v)) , δq(0) > 0 ,
to positive values. We now prove the lemma. Indeed, the characteristic equation for p along the characteristic q(v) = 0, ψ ♂ (0, v) = 0 is

ṗ = KM q 1 1 p -2 . ( 20
)
Hence, as long as p < 1/2, it is increasing. Hence for all v, p(v) > p 0 . By continuity, there is a neighborhood of the v axis of the (v, q) plane where p(v) > p 0 /2. Hence in that neighborhood ψ ♂ (q, v) ≥ p 0 q/2, forbidding a convergence of ψ ♂ (q, v) toward zero. This proves the lemma, and together with proposition 1, the theorem.

Further evidence

We have not conclusively proved that for all q ∈ [0, Q ⋆ 0 ], the solution ψ ♂ (q, v) of equation ( 16) converges toward Ψ(q), solution of ∂ψ ♂ (q, v)/∂v = 0, i.e. toward Ψ(q). Yet we offer two other indications.

On the characteristic curve q(v) = 0, ψ ♂ (0, v) = 0, p(v) is governed by equation ( 20), from which it follows that p(v) → 1/2 = dΨ/dq(0). Moreover, we were able to prove mathematically that the second derivative (∂ 2 ψ ♂ /∂q 2 )(0) converges to 2/3, the second derivative of ψ ⋆ ♂ . More convincingly, we used the characteristic equations (thus getting around a severe numerical instability of the finite difference method) with a Runge Kutta method of order 4 to integrate equation ( 16) from ψ ♂ (q, 0) = 0.1q, obtaining numerical approximations of ψ ♂ (q, v) for various values of v. Using q 1 = 1, and arbitrarily, KM = 10 -3 , the results up to v = 1000 appear in Table 2 together with ψ ⋆ ♂ . For v = 1500, the difference with ψ ⋆ ♂ is below 5 × 10 -5 . The precision of the approximation of ψ ⋆ ♂ is the more striking that the characteristic curves themselves do not stabilize as v → ∞.

Figure 1 shows the graphs of ψ ♂ (q, v) for various values of v. The graph for v = 1000 is indistinguishable from that of ψ ⋆ ♂ .

q v = 0 v = 5 v = 10 v = 100 v = 1000
ψ ⋆ ♂ 0, 1 0, 01 0, 0145 0, 0175 0, 0370 0, 0533 0, 0535 0, 2 0, 02 0, 0322 0, 0395 0, 0833 0, 1144 0, 1148 0, 3 0, 03 0, 0550 0, 0684 0, 1420 0, 1846 0, 1849 0, 4 0, 04 0, 0859 0, 1079 0, 2167 0, 2650 0, 2653 0, 5 0, 05 0, 1303 0, 1644 0, 3113 0, 3570 0, 3571 0, 6 0, 06 0, 1987 0, 2498 0, 4274 0, 4614 0, 4615 0, 7 0, 07 0, 3140 0, 3855 0, 5610 0, 5791 0, 5791 0, 8 0, 08 0, 5188 0, 5975 0, 7038 0, 7097 0, 7097

Table 2: Numerical values of ψ ♂ (q, v) obtained with the method of characteristics, and a fourth order Runge Kutta integration scheme and a fixed step size of 1.

Conclusion

Our analysis is entirely dependent on our hypothesis of separation of time scales between the dynamics of behavioural adaptation of females and that of physical evolution of males. Under that hypothesis, the natural evolutionary dynamics of the females' behaviour converge toward the signalling equilibrium behaviour under two hypotheses on the nature of this behaviour during adaptation : mixed strategy or threshold strategy. That is, in the limit, females behave as if they had conjectured the right strategy of males.

Concerning the males, we feel that we have shown that indeed, for the model of sexual selection emphasized here, adaptive dynamics do converge toward the signaling equilibrium we have analyzed in our previous article [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF]. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 We wish to allow for arbitrarily large, and if necessary infinite, slope for ϕ ♂ . One way of achieving this will be to use its inverse function q = R(σ). Recalling that r ≤ 1, we have dF ♂ dσ > K(1 -σ) R ′ (σ) -R(σ) 1 -σ .

The coefficient (1 -σ) is always positive. Moreover, the function R is increasing. It follows that

ς 0 R ′ (σ) - R(σ) 1 -σ dσ > R(ς)[1 + ln(1 -ς)],
which is positive for ς < 1 -e -1 . Hence, in every right neighborhood of σ = 0, there are σ values, and hence also in any right neighborhood of q = 0 values of q, for which dF ♂ /dσ is strictly positive. These individuals would therefore have an incentive to deviate from ϕ ♂ , preventing the existence of an equilibrium.

Figure 1 :

 1 Figure1: graph of ψ ♂ (q, v) as computed by the method of characteristics. The graph for ψ ⋆ ♂ is indistinguishable from that for v = 1000.

We could, without loss of generality, normalize q1 = 1, taking the life expectancy of a best male that would not signal as the unit of time. For a non-unit q1, the fitness F ♀ and F ♂ , the females'

threshold θ and the males' strategies ψ ♂ would just be multiplied by q1. We do not do that to keep better track of the dimension of the variables involved.

The fact that x and x ′ in equation (2) of [4] belong to a Banach space makes it very difficult to give a meaning to M (x ′ , x) and to the integral: the symbol dx ′ cannot stand for the Lebesgue measure which does not exist in a Banach space.

For both models, we also investigated the case where the limiting factor for the males is their potential reproduction rate.

of the males' fitness with respect to the signal, assuming the females' conjecture χ(•) fixed at the resident value of ψ ♂ (•).

Since this trait is function valued, this will yield a partial differential equation, as emphasized in [4]. As compared to the treatment there, beyond the mathematical difficulties 2 raised by the definition of their σ 2 (q ′ , q), in such a general framework as ours we have no ground to postulate a specific cross-correlation between different values of q. Therefore, we neglect any such effect and choose to ignore it, or, equivalently, set σ 2 (q ′ , q) = δ(q -q ′ ). (Note that this is essentially what the authors do in the examples provided in the article.)

Mathematical analysis

According to the above analysis, we differentiate F ♂ (q, s, ψ ⋆ ♀ (s)) with respect to s assuming χ(•) fixed at χ = ψ ♂ . Call v the evolutionary time, and denote ψ ′ ♂ = ∂ψ ♂ (q, v)/∂q. Assume a constant mutation factor M . We obtain

The problem now is to investigate the asymptotic behaviour of the solution of this partial differential equation (PDE) as v goes to infinity, for any positive, supposedly small, initial condition ψ ♂ (q, 0), and for q ∈ [0, Q 0 ].

We were not able to prove conclusively its convergence toward the equilibrium strategy. However it is clear that it does in view of the following development.

We claim:

Theorem 2 In equation ( 16), let ψ ♂ (0, 0) = 0, p 0 ∈ (0, 1/2) and, ∀q ∈ [0,

) converges in that range as v → ∞, it is toward Ψ, given by equation [START_REF] Lande | Models of speciation by sexual selection on polygenic traits[END_REF].

Proof Note first the following:

Proof Indeed, for any limit, the right-hand side of (16) must be either zero, meaning that ψ ♂ (•, ∞) = Ψ(•), or indeterminate, which happens only on the trivial equilibrium solution ψ ♂ (q) = 0.

A Appendix: Multiplicative model

In our article [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF], we investigated another model 3 , closer to Grafen's, that we called "multiplicative" (as opposed to the "additive" model investigated here) where we called σ a dimensionless signal, and assume L ♂ = q(1 -σ). We concluded that that model is less satisfactory on two counts. On the one hand, it yields an excessively large equilibrium signal, and thus an exceedingly small residual survivorship at equilibrium, and, on the other hand, in that model it is not possible to let q range over (0, q 1 ] as we did here. One has to assume a positive minimum q 0 . As q 0 approaches zero, the equilibrium male signaling strategy goes to 1 for all positive q, an absurd conclusion. Yet, as low quality males may die before reaching adulthood, there is no reason why the adult life expectancy could not be arbitrarily small.

Beyond the tentative explanation given in [START_REF] Bernhard | Simple signaling games of sexual selection: Grafen's revisited[END_REF] of why this undesirable feature arises, we provide here a dynamic viewpoint:

Theorem 3 In the multiplicative model with q 0 = 0, no equilibrium is possible with any males' strategy σ = ϕ ♂ (q) with a strictly increasing ϕ ♂ (•).

Proof We may, without loss of generality, assume that ϕ ♂ (0) = 0, since any other choice could lead to a translated signaling strategy conveying exactly the same information but "cheaper" in terms of loss of viability.

Females' behaviour and formula [START_REF] Getty | Handicap signalling: when fecundity and viability do not add up[END_REF] (substituting σ to s) remain unchanged from the additive model, but now F ♂ = q(1 -σ)N m (σ). We differentiate it with respect to σ assuming χ(•) fixed and equal to ϕ ♂ (•). We write everything in terms of r = q/q 1 . We obtain

) .