2.3.1 Directional and quadratic selection gradients.

We now assume weak mutation effects and write the mutant’s allelic value as a small perturbation of the resident’s allelic value, where is a trait-specific random mutation effect and is a small parameter (assuming that mutations are such that always lies within the domain of , for any trait l in any class j). The basic reproductive number of a mutant allele can then be Taylor expanded around as

(3a)

where we used the fact that in a resident population at demographic equilibrium, , and is a remainder of order . Here, is the effect of the mutant in class j; is a vector of directional selection coefficients, with element l given by

(3b)

is a matrix of quadratic selection coefficients with -element

(3c)

where we used the facts that implicitly depends on and that . Finally, ‘·’ denotes the dot product, and ‘’ denotes transposition in eq (3a).

The quantity (eq 3b) is the effect of a marginal change in the mutant allelic value on the mutant’s reproductive number. It can be decomposed as the product of two components (right-hand side of eq 3b): the marginal effect of on , and the effect of a change in allelic value on trait expression. Because we assume additive gene action, this second term is a constant independent of trait values for a given class j: it gives the relative genetic contribution of a mutant allele to the individual’s trait value (e.g., one-half under diploidy or one under haploidy). If , selection favours an increase in that trait; if , it favours a decrease (see Table 2 for a summary of the types of selection considered throughout).

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Table 2. Summary of selection types.

https://doi.org/10.1371/journal.pcbi.1013591.t002

Meanwhile, the quantity (eq 3c) is proportional to the multiplicative effects of joint changes in and on the mutant’s reproductive number, beyond their individual effects. When or , it quantifies how a change in the first trait modulates the effect of a change in the second. In other words, describes correlational selection on traits and : positive values indicate that increases faster with one trait when the other is larger – meaning that quadratic selective effects favour similar changes in the two traits – whereas negative values indicate that increases faster with one trait when the other is smaller, favouring opposite changes. For and , a negative value of indicates that decelerates with increasing deviation from the resident trait along a single trait axis , while a positive value of indicates that accelerates along that axis – meaning that quadratic selection favours trait values close or away from the current resident value, respectively.

2.3.2 Mutation limited evolutionary dynamics.

Directional and quadratic selection coefficients inform gradual trait evolution when mutations are rare and have weak effects on the traits. Specifically, when mutations have small effects (i.e., ), the ‘invasion implies substitution’ principle for class-structured populations [33] states that a mutant will increase (respectively, decrease) in frequency and eventually replace the resident (be lost) whenever is positive (respectively, negative). This implies the existence of a mutation-limited regime in which evolution proceeds gradually, as a sequence of trait substitutions along the direction favoured by selection. This sequence halts only when the population reaches a phenotypic point v where

(4)

for all mutations in a close neighbourhood of v. When the trait space is bounded and v is at the boundary, new mutations can only move traits inward or along the boundary. Condition (4) can then be satisfied if directional selection opposes all admissible mutations (i.e., or equivalently if selection favours trait values outside of the authorized range, such that corresponding elements of and always have opposing sign). In what follows, we focus on cases where condition (4) is met in the interior of trait space.

When mutation effects on trait values are not exactly equal across traits – that is when the distribution of mutation effect is not confined to a lower-dimensional subspace of trait space (e.g., is sampled from a multi-normal distribution over with full-rank covariance matrix) – there always exists some mutation that affects one trait without changing the others. Therefore, on the long term and unless mutation effects are fully correlated across traits, traits can be thought of as being unconstrained genetically. In this case, condition (4) is satisfied in the interior of trait space if, for some trait value vector ,

(5)

Any satisfying eq (5) will be called a singular trait value and constitutes a candidate evolutionary equilibrium. But will evolution converge to such a singular trait?

A sufficient condition for to act as a local attractor of the evolutionary dynamics – i.e., for the population to evolve toward via a trait substitution sequence – is that the Jacobian matrix is negative definite ([34, p. 189]). This matrix is a block matrix whose block is a matrix with element given by

(6)

The matrix is negative definite if the leading eigenvalue of its symmetric part, , is negative. In that case, is said to be strongly convergence stable (sensu [34]). The term ‘strongly’ reflects that convergence occurs from all nearby trait values, regardless of the mutational covariance structure. In contrast, when is not negative semi-definite, there is some mutational covariance structure for which is a repellor of the evolutionary dynamics ([34, p. 198]).

By eq (3a), if the population has reached a singular trait value , then a sufficient condition for local uninvadability, i.e., for any nearby mutant to be selected against and thus for selection to be stabilising at , is that

(7)

Eq (7) holds if and only if the block matrix , whose block is , is negative definite (i.e., for all ). Since is a symmetric real matrix, it is negative definite if and only if all its eigenvalues are negative.

When has at least one positive eigenvalue (i.e., is not negative semi-definite), there exists a set of mutation directions – specifically those aligned with the leading eigenvector of – that are favoured on both sides of the singular strategy . In other words, selection is disruptive at : it favours divergence away from the resident trait value. A sufficient condition for to not be negative semi-definite is that any diagonal block is not negative semi-definite (since diagonal blocks are symmetric matrices, [35, Theorem 2.2, p. 1062]). In turn, a diagonal block is not negative semi-definite if any of its diagonal elements is positive, i.e., if for some trait . At a singular strategy, the quantity captures whether selection on trait is locally stabilising (if negative) or disruptive (if positive) when that trait evolves in isolation from all others. More generally, when is not negative semi-definite and is strongly convergence stable, selection may lead to evolutionary branching, whereby two alleles with slightly different trait values near are both maintained and subsequently diverge as they accumulate further mutations (the exact conditions for this to happen when multiple traits co-evolve remains an open question, [36]).

Eqs (3a)–(7) show that under the assumption of rare mutations with small phenotypic effects, key features of the gradual long-term evolution of traits can be predicted from the selection gradients and the Jacobian and Hessian matrices. Eqs (5)–(7) assume that mutation effects on trait values are not exactly equal across traits, thereby excluding cases where genetic constraints force traits to be expressed identically across classes. We will later visit such genetically constrained trait expression, which can in fact be analysed using the same results as above, with only minimal adjustments (see appendix A.6 in S1 Text for details). In the following, we give explicit expressions for the directional and quadratic selection coefficients.

3.1.1 Age-specific coefficient of directional selection.

We show in appendix A.2 in S1 Text that the coefficient of directional selection on trait can be written as

(8)

where

(9)

is an age a specific coefficient of directional selection. This coefficient involves three quantities. (i) The leading right-eigenvector of the resident next-generation matrix , such that gives the frequency of alleles carried by class-j individuals among resident newborns. (ii) The leading left-eigenvector of , such that is the reproductive value of an allele found in a resident newborn of class i (i.e., the expected number of copies this allele will produce over the lifetime of its carrier). We refer to and as allelic class frequencies and reproductive values at birth, respectively. Finally, (iii) the marginal effect of variation in trait expression on the so-called Hamiltonian , which we detail below in eq (10). For now simply note that this derivative measures how a change in trait affects the production of class-i offspring by a class-j individual: directly, through a change in fecundity at age a; and indirectly through changes in survival and development at that same age (i.e., influencing future reproduction).

With these definitions in mind, in eq (9) can be read from right to left as follows. First, gives the probability that a mutant allele occurs in a newborn of class j, so that a variation in trait will be expressed in such an individual and affect its life history across all ages a. Second, quantifies how such age-specific variation affects the production of class-i offspring, accounting for both current and future contributions, i.e., at age a and later. Third, converts offspring into transmitted mutant allele copies, and finally weights each such descendant copy by its reproductive value in class i. Summing over all offspring classes i gives the total fitness consequences of variation in trait via its effects on fecundity, survival and development at age a. Directional selection on trait is finally obtained by integrating these age-specific coefficients over all ages (eq 8).

3.1.2 The Hamiltonian: partitioning age-specific fitness effects.

Quantifying the fitness effects at a specific age relies on the Hamiltonian:

(10)

where and are so-called costate variables that are given by the solution to the system of ODEs:

(11)

The costates and respectively measure how marginal changes in survival and in internal state m at age a affect the expected remaining reproductive output of a class-j individual via offspring of class i. In particular is a class-oriented equivalent of Fisher’s definition of the reproductive value of an individual of age a (see Box 1). The boundary conditions in eq (11) are obtained as follows. At birth, and provided fecundity is finite, we have ; namely, the resident expected lifetime class-oriented fecundity (e.g., in a haploid population without class structure owing to demographic consistency; see appendix A.2.3 in S1 Text). Biologically, this reflects that death at birth would eliminate the entire reproductive potential of the individual. By contrast, the development costates vanish as , under the assumption that traits do not affect internal states at birth but do affect their finite terminal values (see appendix A.2.3 in S1 Text). Intuitively, this is because at very old ages the expected remaining reproductive output approaches zero, such that marginal changes in state values no longer influence fitness.

Box 1. Costates and reproductive values in age- and class-structured populations.

Consider the remaining number of offspring of class i that a mutant individual of class j and age a is expected to produce until its death,

(I-A)

such that (see eq 1). Using results from [9] (their appendix B.1), we obtain that the co-states and satisfy

(I-B)

Costates can therefore be interpreted as the change in the expected remaining class-oriented reproductive output of an individual of class j and age a, resulting from a change in its current survivorship or internal states values. The costate associated to survivorship, in particular, can be expressed as

(I-C)

We see from eq (I-C) that is the expected remaining class-oriented reproductive output of a resident individual, conditional on this individual having survived to age a. Thus, is the class-oriented equivalent of Fisher’s reproductive value [37–39]. Note that, since according to eq (I-C) we have , we obtain from the definition of the left-eigenvector of (eq. A-3) that

(I-D)

This connects costates to the elements of the leading left-eigenvector of the next-generation matrix (i.e., allelic reproductive values at birth). While costates are age-specific measures of future reproductive output at the individual level, reproductive values in are lifetime measures at the allelic level that take into account the value of each type of descendants (i.e., in eq I-D).

The Hamiltonian eq (10) quantifies how a class-j individual, at age a, contributes to its lifetime production of class-i offspring through three distinct components: (1) current offspring production through fecundity ; (2) future reproductive output depending on mortality rate at that age, weighted by ; and (3) future reproductive output depending on development rate at that age, weighted by . This decomposition reflects the fact that at any given age, an individual can influence its eventual production of class-i offspring not only through its immediate fecundity, but also through its current survival and developmental rates, which determine its future reproduction. The Hamiltonian can thus be interpreted as an age-specific measure of reproductive success—one that, when integrated over age (as in eq 8), allows us to quantify the total effect of a trait change on an individual’s lifetime reproductive success.

An important practical point is that in the coefficient of directional selection (eq 9) the derivative of the Hamiltonian is evaluated at resident trait values. As a result, when computing the Hamiltonian and solving the ODEs for the costates (eq 11), we only need to consider resident survivorship and state dynamics. Concretely, it is sufficient to solve

(12)

rather than the mutant system in eq (2). This can greatly simplify the analysis as we will illustrate later in our example.

3.1.3 Life history trade-offs in structured population.

The age-specific coefficient of directional selection (eq 9) is expressed in terms of the effect of a trait change on the Hamiltonian, which in turn separates the relevant fitness components at each age. To gain further insights, one can substitute eq (10) into eq (9), giving

(13)

Eq (13) highlights the fundamental trade-off between current and future reproduction, while accounting for age, class structure, ontogeny and ploidy. This is made explicit by showing that directional selection boils down to a weighted effect of a trait change on the three core life-history components: fecundity, mortality, and development. The weights , and capture the relative importance of different parent and offspring classes, weights age-specific effects by the probability of surviving to age a, and the costates and measure how a trait change at age a influences future reproduction through survival and development. Taken together, these weights quantify how selection balances all the potential first-order effects of trait expression. They make it possible to distinguish three types of trade-offs: (i) within-age trade-offs, between the contributions to fecundity, survival or development at a given age (the term within parentheses in eq 13); (ii) among-class trade-offs, reflecting the relative value of producing offspring of different classes (arising from the summation over classes in eq 13); and (iii) among-age trade-offs, arising when eq (13) is integrated over ages in eq (8), and which depend on how survivorship declines with age and on how the costates and vary with age—either decreasing or increasing depending on developmental effects and life history.

3.1.4 Connections with previous expressions for directional selection.

Eq (13) connects to several results from the literature. First, in the absence of ontogeny, it reduces to the standard selection gradient in class-structured populations [40–43]. Second, in the absence of class structure, it reduces to the selection gradient on constant traits (controls) affecting life history [9,11,12], extended to to arbitrary ploidy. Third, the term within parenthesis in eq (13) is conceptually similar to expressions for the fitness effects of age-specific trait expression under complete plasticity (usually derived in the absence of class structure, e.g., [1–3,8,38,44–46], but see [47]).

Finally, when mutations act only on mortality at age a, eq (13) reduces to Hamilton’s first indicator of the strength of selection ([13], eq. 9, p. 17; see also [48]), with selection being proportional to i.e., to the expected remaining reproductive output (eq. I-C); while for mutations acting only on fecundity at age a, eq (13) reduces to Hamilton’s second indicator, i.e., survivorship to age a, . Eq (13) shows that selection generally acts not only through mortality and fecundity, but through development as well, taking into account the specificities of those processes within each class. A relevant implication of including development, in particular, is that it can significantly alter among-age trade-offs between life-history components. Indeed, developmental effects can be strongly non-monotonic with respect to age (e.g., during metamorphosis), and as a result the costates associated with internal states do not necessarily decline with age. Consequently, the age-specific strength of selection also need not decrease monotonically with age [49,50], contrary to a common interpretation of Hamilton’s seminal results [13]. We illustrate this in our example model (Section 4).

3.2.1 Age-specific direct trait effects.

The first element in the right-hand side of eq (15) with is given by

(16a)

Eq (16a) describes the direct joint effects of traits and on the Hamiltonian (eq 10), summed over all offspring classes m with appropriate weights. Biologically, the derivative in eq (16a) quantifies whether simultaneous changes in the two traits, expressed in the same individual of class j, increase or decrease its production of class-m offspring. As with directional selection (eq 13), these fitness effects can manifest through fecundity at age a or through survival and development at that same age, which influence future reproduction.

Eq (16a) can thus be viewed as the direct extension of the classical measure of correlational selection (i.e., described in the absence of any population structure, [51]). In that setting, correlational selection is due to multiplicative effects of traits on reproduction only. By contrast, correlational selection here can also arise because traits have multiplicative effects on survival rates (i.e., on ) or on development rates (i.e., on ) at age a. For instance, if two traits interact to determine the growth rate – say one trait sets foraging effort, which determines resource acquisition, and the other governs how much of these resources are allocated to growth – and if larger size later increases reproduction, eq (16a) indicates that selection will favour a positive association between these two traits.

These different paths, which become apparent by substituting eq (10) into eq (16a) (though we refrain from doing so explicitly for succinctness), show that population structure and ontogeny create additional opportunities for correlational selection, and thus for the emergence of polymorphism via disruptive selection. As we will now see, there are in fact many other paths for correlational selection generated by ontogeny and class structure.

3.2.2 Age-specific indirect effects.

The next element in the right-hand side of eq (15), which for is given by

(16b)

describes the effect of joint change in all pairs of internal states at age a ( and ), that are induced by changes in traits and . Put differently, this term captures how simultaneous changes in two traits alter state variables at age a in ways that synergistically influence the different components of life history—fecundity, survival, or development—where fecundity affects reproduction at that age, while survival and development determine future reproduction (as per eq 10). As an example where eq (16b) may be relevant, consider a scenario where one trait regulates allocation to body growth while another controls allocation to sexual maturation. These respectively determine the internal state variables ‘size’ and ‘maturity’ at a given age. When particular combinations of these internal states yield higher fitness than intermediate ones (e.g., small but early-maturing individuals, or large but late-maturing ones, as observed in Atlantic salmon, [18]), eq (16b) will be positive, indicating that selection favours an association between the two traits that induce these internal state combinations.

Correlational selection may also be due to the third element in the right-hand side eq (15), which for reads

(16c)

and which describes the interaction effects between traits and internal states, i.e., it captures situations where the state-mediated influence of one trait on fitness contributions at age a depends on another trait. More specifically, eq (16c) shows that correlational selection associates two traits whenever some internal state varies with the value of one trait, while the effect of that internal state on the Hamiltonian depends on the other trait.

For example, suppose the first trait determines the growth and thus morphology of some sexual traits (e.g., genital morphology) and the second determines the preference for mates with a compatible morphology. In this case, correlational selection arises because the benefit of expressing a given reproductive morphology is conditional on simultaneously expressing a preference for mates with the matching morphology. Conversely, the benefit of the preference trait depends on having the compatible morphology. Eq (16c) captures this interdependence. Similar mechanisms apply whenever behaviours or other traits have state-dependent effects, such as risk-taking behaviours that depend on body condition, or condition-dependent costs of sexual traits as in handicap models.

Another interaction with traits that can contribute to correlational selection is given by the fourth element in the right-hand side of eq (15), which for is

(16d)

This term describes the interaction effects between traits and survivorship to age a. In biological terms, correlational selection arises because of this term whenever one trait modifies survivorship to age a, while another affects fecundity or mortality at that age (but not growth, since growth rates do not depend on survival, see eq 10).

As an illustration, consider a trait that improves survival to later ages, and another trait that enhances reproductive success specifically at those later ages (e.g., investment in traits expressed after maturity, such as secondary sexual displays). In this case, the benefit of the survival trait is conditional on the reproductive trait, and, conversely, the benefit of the reproductive trait depends on surviving long enough to express it. Such an interaction is captured by eq (16d), with correlational selection favouring an association between longevity and late-life reproductive investment, and disruptive selection potentially arising if intermediate strategies perform poorly compared to the extremes.

Interaction effects that are entirely mediated by life history can also generate correlational selection. This occurs when trait changes influence both internal states at a certain age and survivorship to that same age, as long as the Hamiltonian depends multiplicatively on these two components. This case is captured by the fifth element in the right-hand side of eq (15) that for reads

(16e)

Eq (16e) shows that correlational selection arises whenever an internal state at age a affects the fecundity or mortality rates (since both rates are multiplied by survivorship in eq 10), provided traits influence both survivorship and the internal state.

For example, one trait may enhance survival through investment in immune function while another governs allocation to sexual development. The benefit of improved immunity is conditional on surviving long enough to express the sexual state, while the benefit of sexual development depends on immunity ensuring survival. Eq (16e) would therefore be positive in this case, indicating that correlational selection favours a positive association between these two traits through their joint effects on survivorship and development.

The final element in the right-hand side of eq (15) is

(16f)

which consists of the product between the effect of one trait on the Hamiltonian for a given parental class and the effect of the other trait on the probability that mutant alleles are found in newborns of that parental class (i.e., on mutant allelic frequencies at birth ). In essence, this means that correlational selection will favour an association between two traits when one increases the likelihood of being in a certain class, while the other increases reproductive success in that class. For example, one trait may affect the development of forms that are particularly suited to reproduction as a female but not as a male, while another biases the sex ratio at birth so that a larger fraction of offspring are female. In this case, eq (16f) would be positive, indicating that correlational selection favours an association between these two traits. This is because such an association leads genotypes that produce better female forms to be more often expressed in females, while those producing better male forms are more often expressed in males.

In contrast to the previous components (eqs 16a–16e), this term contributes to correlational selection even when (see eq 15). The reason is that a change in a trait expressed in one class i can modify the allelic class frequencies at birth in another class with . For example, if sex ratio is under paternal control, then a male-expressed trait that biases the proportion of daughters directly affects the allelic frequency of the female class at birth.

Taken together, the different parts of eq (15) show that correlational selection can arise not only through direct interactions among traits (eq 16a), but also indirectly via their effects on internal states, survival, or class representation at birth (eqs 16b–16f), thereby encompassing all pathways by which trait associations influence fitness in class-structured populations with ontogeny.

3.2.3 Trait effects on internal states, survivorship and class structure.

The indirect pathways through which correlational selection acts (eqs 16b–16f) ultimately depend on how traits affect internal states, survivorship, and class structure. Here we summarise how these effects, or perturbations, can be computed (see appendix A.4 in S1 Text for details).

First, we show in appendix A.4.1 in S1 Text that the effect of a trait l on internal states at age a in class j is obtained as

(17)

where is the so-called fundamental matrix collecting the cumulative effects of states on their own rates of change and satisfying

(18)

where I is the identity matrix. Usefully, in the special case where rates of change in states are constant with respect to time (i.e., ), the fundamental matrix has an explicit exponential form that can be plugged directly into eq (17) (appendix A.4.1 in S1 Text).

The effect of trait l on survivorship to age a in class j is given by

(19)

that is, as the discounted integral of all past (direct and indirect) effects of trait expression on mortality.

Finally, appendix A.4.2 in S1 Text shows that derivatives of mutant allelic class frequencies at birth with respect to mutant traits at can be obtained as

(20)

where is a matrix of ones of dimension . See appendix A.4.2 in S1 Text for the derivatives of mutant allele frequencies when .

Together, the above results provide all the ingredients needed to compute the indirect pathways of correlational selection in class-structured populations with ontogeny. We will return to how to combine these ingredients more precisely in section 3.3, where we summarise our results and highlight potential computational challenges.

3.2.4 Connections with previous expressions for quadratic selection.

Eqs (14)–(16) connect to previous results on quadratic selection in class-structured populations found in the literature. First, in the absence of ontogeny, age-, and class-structure, eq (14) reduces to the standard multivariate quantitative genetics estimates of quadratic selection, where correlational selection is given by the cross derivative of individual fitness with respect to individual trait values [51–53]. Second, in the absence of ontogeny but with class structure (so that Hamiltonians reduce to fecundity rates), eq (14) reduces to the sum of eqs (16a) and (16f), consistent with the results of [54] for a single trait (see their eq 34 without relatedness). Finally, in the absence of class-structure and for a single trait, eqs (14)–(16) reduce to eq (3.13) of [12] (which can be recovered by substituting the Hamiltonian into our expressions). The model therefore extends these results not only to class-structured populations but also to arbitrary ploidy and any number of evolving traits. Moreover, whereas [54, eq. 35] provided a system of equations for the perturbation of class frequencies in terms of resident frequencies and derivatives of fitness components (that cannot always be solved by direct methods), and whereas [12] left the effects of traits on internal states and survivorship implicit, here we resolved these expressions fully (see section 3.2.3). In particular, eq (20) shows that the perturbation of class frequencies can be expressed directly in terms of resident frequencies and derivatives of fitness components. This then allows for a complete characterisation of directional, correlational, and disruptive selection in class-structured populations with and without ontogeny, opening the way for the analysis of adaptation under many biologically relevant scenarios.

4.1.1 Population and traits.

We consider a diploid, dioecious, age-structured population with two classes, females and males, denoted by subscripts and respectively (for readability, we use these instead of using 1 and 2 to denote classes here). Sex is determined at birth with a fixed primary sex ratio c (the proportion of males in newborns, see Table 3 for symbols). Class proportions at birth are thus constants in this model. This leads to several simplifications that are detailed in Box 2. In the notation introduced there, we have and . Age-specific vital rates (survival, development, and offspring production) are allowed to differ between the sexes and to depend on sex-specific traits as we detail below.

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Table 3. List of symbols used in section 4.

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We study the co-evolution of female and male traits and that regulate the allocation of resources into the growth of a sex-specific morphological, physiological, or behavioural module, represented by the internal states and (and as a baseline we present the case without sex or development, where in appendix B.2 in S1 Text). Biologically, the most straightforward way to think about this module is as overall body size or body mass, where female fecundity scales with size and male compatibility with females is mediated by size matching. For ease of presentation, this is the interpretation we will adopt throughout. However, the model also applies to specialised reproductive modules such as genital morphology, gamete recognition characters (e.g., fertilisation proteins), or sexually selected phenotypes (e.g., eyestalks, ritualised dances) when female choice correlates with her internal states.

4.1.2 Growth and survival.

We assume that the rate of change in the internal state of an individual of sex at age a does not depend on its environment (i.e., no density dependence, no environmental effect in resource availability), and write

(22)

where is the rate at which size (or mass) is lost due to physiological constraints (e.g., respiration), and is the trait value of a mutant individual, corresponding to its sex-specific basal growth rate, which can be thought of as the baseline energetic investment into growth. Since there is a single trait per sex, we simplify notation by writing , , and .

There is a trade-off between allocation to growth and survival such that the sex-specific mortality rate of a mutant individual of sex j is

(23)

where is a constant extrinsic mortality rate (e.g., due to sex-specific risks) and modulates the cost of growth on survival. The rate of change of survivorship in a focal mutant of sex j and age a is thus

(24)

(from eq 2).

Solving eqs (22) and (24) at resident allelic values v, the size and survivorship of a resident individual of sex j and age a are

(25)

where and . We show in appendix B.1 in S1 Text that the corresponding equilibrium probability density of individuals of size and sex j in the resident population is

(26)

which is defined for values of between zero and the maximal resident size (obtained from eq 25 in the limit of and entailing that ). Accordingly, the shape of the size density distribution in each sex qualitatively depends on the balance between and the mortality rate . When , the density increases with size , indicating that many individuals survive to reach large sizes. When , the density decreases with size, indicating that most individuals die before reaching their maximum size. This is illustrated in Fig 1, and plays an important role in evolutionary dynamics.

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Fig 1. Equilibrium distribution of size in the resident population.

Each line represents the equilibrium probability density of individuals of size and sex j among residents. With increasing investment into growth, the maximum size increases (compare black, brown and orange lines), but so does the resident mortality rate , such that the proportion of smaller-sized individuals also increases. Under a stronger cost of growth on survival, the switch from a distribution with a majority of larger individuals to a population with mostly smaller individuals occurs at smaller values of (compare plain and dotted lines). Fixed parameter values: and .

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4.1.3 Female and male fecundity.

The remaining model components we need to specify are the female and male fecundity rates, denoted and . We assume that a female produces eggs at a rate and that each egg is fertilised (i.e., no mate or sperm limitation). Male fecundity then reflects the outcome of competition among males for the fertilisation of the eggs.

For females, we assume that fecundity increases linearly with size and decreases with total population density, for example due to competition for resources. Specifically, we assume:

(27)

where controls (inversely) the strength of density dependence (large values of K mean low density-dependent competition), and is the equilibrium total density of individuals in the resident population (sum of male and female densities). We show in appendix B.1 in S1 Text that this is given by

(28)

where is the expected lifetime production of female offspring by a single resident female in the absence of density-dependence.

For males, we assume that fecundity — that is, the fertilisation of eggs — depends on matching with female size (e.g., due to mechanical constraints during mating or female preferences for size-similar partners) and on competition with other males. Specifically, we write the fecundity rate of a focal mutant male of size , i.e., the total number of offspring he is expected to sire across all females in the population, as

(29)

where we recall that is the maximum size attainable by residents of sex j, and where

(30)

is the resident secondary sex-ratio (i.e., the proportion of males in the resident population; see appendix B.1.1 in S1 Text for derivation). Accordingly, in eq (29) is the total number of females of size , and is the rate at which eggs are produced by these females collectively (i.e., the total fecundity rate of females of size ). For each of these eggs, we assume that the probability that it is fertilised by a focal mutant male of size is given by

(31)

where the numerator gives the competitive ability of the focal male to fertilise a female of size and the denominator normalises over all resident males. We assume that competitive ability declines with the difference between male and female size according to

(32)

where is a parameter controlling the strength of size matching: when , all males are equally competitive regardless of their size (i.e., ); as increases, fertilisation becomes increasingly size-assortative.

Equation (32) is equivalent to the compatibility function ψ used in earlier models of sexual selection (e.g., [56,57]), where trait matching determines mating success and can favour polymorphism. However, whereas other models assume fixed trait values, here we apply this mating function to phenotypes that change with development over individuals’ lifetimes. From a mechanistic point of view, eq (32) applies indiscriminately to systems where female directly express preference for similar-sized males (e.g., as in the convict cichlid Archocentrus nigrofasciatus; [58]), where they co-occur more often with similar-sized males (e.g., as in the striped goby cichlid Eretmodus cyanostictus; [59]), or where mating success depends on size-similarity due to morphological constraints (e.g., as in the Jerusalem cricket Stenopelmatus sp.; [60]). Finally, note that eq (29) links male and female fecundity such that it satisfies Fisher’s condition (i.e., any newborn has one parent of each sex; [61,62]; see appendix A.5.2 in S1 Text).

4.2.1 Hamiltonians and costates.

Since class proportions at birth are constant in this model, we only need class–specific Hamiltonians and costates (see Box 2). Substituting the model ingredients (eqs 22, 23, 27, and 29) into eq. (II-G) of Box 2 gives

(33)

where the sex-specific costates satisfy the system of ODEs:

(34a)

(34b)

(34c)

(34d)

These have boundary conditions and since in a resident sexual population at demographic equilibrium, each individual has an expected lifetime reproductive output of two offspring (one daughter and one son on average). We solve these ODEs numerically.

4.2.2 Directional selection: balancing future effects of growth across ages.

Age-specific coefficients of directional selection are obtained by substituting eq (33) into eq. (II-E) of Box 2. In addition, since we are considering a diploid population with constant primary sex ratio, the class frequencies and reproductive values are , and respectively (see Box 2 and appendix A.5.2 in S1 Text for derivation). Using these quantities, we have

(35a)

(35b)

Eq (35) shows that age-specific selection at age a on the basal growth rate balances two antagonistic effects on future reproduction. On the one hand, future fitness effects mediated by growth tend to favour larger especially at a young age (captured by in eq (35), see Fig 2A). On the other hand, future fitness effects mediated by survival favour smaller (captured by the term in eq 35). The reason is that higher growth raises mortality and hence lowers the expected remaining reproductive output, here, which itself declines with age (Fig 2B). In our model, declines more steeply than with age a, so the balance between these two antagonistic effects changes with age: selection favours faster growth at younger ages but slower growth later in life (Fig 2C).

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Fig 2. Age-specific directional selection arises from opposing effects of growth on reproductive value and survival.

A. Age-specific benefit of increased size, captured by the costate , declines with age. This reflects the diminishing marginal returns of size on future reproduction as individuals age, especially when size loss is high ( large). B. Expected remaining reproductive output at age a, given by , also declines with age. However, it is more sensitive to in females than in males (lighter curves). This is because higher reduces the maximum attainable size and thus lifetime fecundity in females. These opposing age-specific effects (A vs. B) jointly generate the pattern of directional selection in panel C: early selection favours growth due to future size benefits, while late selection disfavours it due to survival costs. C. Age-specific directional selection on growth (from eqs 35 and 38) at the convergence stable strategies ( for independent traits; for a shared trait). Each curve integrates to zero (owing to eqs 5 and 8), but its shape reveals when selection favours increased or decreased allocation to growth. Selection tends to favour faster growth at early ages but reduced allocation later in life. Parameter values: , , , , and . Parameter has value in panel C.

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Selection eventually favours a compromise between these antagonistic effects. This is formalised by unique convergence stable trait values in females and males that satisfy

(36)

for sex j (obtained by plugging eq 35 into eq 8). Note that eq (36) is implicit as both costates and internal states depend on . It nevertheless makes clear that the equilibrium growth rate increases when the cumulative benefits of growth (i.e., ) outweigh the cumulative benefits of survival (i.e., ).

A numerical analysis of eq (36) together with eq (34) allows us to investigate how key parameters influence and thus life histories and demography at evolutionary equilibrium. Unsurprisingly, greater growth costs select for slower growth in both females (Fig 3A) and males (S1A Fig), leading to longer lives with delayed reproduction. By contrast, higher extrinsic mortality reduces the expected remaining reproductive output and therefore selects for faster growth in sex j (Fig 3B for females, S1B Fig for males), leading to “live-fast–die-young” types of life histories [63]. In females, faster growth partly compensates for the reduction in average size (and hence fecundity) caused by extrinsic mortality, such that population extinction () occurs only at relatively high values of (Fig 3B, dark gray area).

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Fig 3. Female mortality and growth parameters influence sex-specific evolution and extinction risk.

Each panel shows convergence stable growth rates (solid lines) and corresponding average sizes (dashed lines) for females (red) and males (blue), as a function of the parameter indicated on the x-axis (computed from eq 36 for trait values and B-99 for average sizes). Trait values under genetic constraint (i.e., identical growth rate in males and females, ) and the resulting female size are shown in green (from eq 39). A. Higher cost of growth to survival in females () selects for slower growth and smaller average size. Males evolve smaller size in response, to match females owing to size-matching sexual selection. B. Higher female extrinsic mortality () favours faster growth. Above a threshold, female fecundity becomes insufficient to maintain recruitment and the population goes extinct (; dark grey area). This extinction threshold is lower when genetic correlations constrain male and female growth to evolve identically (light grey area), because sexual antagonism prevents females from evolving growth rates high enough to offset increased mortality (green). Note that convergence stable growth rates and average sizes are reported beyond the extinction thresholds (transparent lines) for graphical purposes, since the model provides result even when . C. A higher rate of size loss () prevents individuals from reaching larger sizes. This favours faster growth in females, but slower growth in males. See main text for explanation. Parameter values (unless varied on x-axis): , , , , , and .

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The parameter , which determines the rate at which size is lost (eq 22), has opposite consequences in the two sexes. In females, greater selects for faster growth (larger ), while in males it selects for slower growth (smaller ; Fig 3C). The reason is that when is large, it prevents males and females from reaching higher sizes. This is reflected in the size distributions being shifted towards smaller values when is increased (compare the histograms of Fig 4A and 4C). In females, this prevents individuals from reaching high fecundity, lowering their remaining reproductive output at any given age (Fig 2B), and thus favouring growth over survival. For males, meanwhile, because greater reduces average female size and male success depends on matching female size (eq 32), this favours slower growth as long as .

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Fig 4. Evolutionary branching of male growth generates male life-history polymorphism.

Each panel shows results from individual-based simulations (appendix B.3 in S1 Text). The top part shows the expressed trait values in females (red) and males (blue): randomly sampled allelic values (dots; 10 copies every 10 time units), population mean (solid lines), and analytically predicted convergence stable trait values (dashed lines). Reported values give: the male-specific component (eq 14); its size-mediated component (see eqs 37 and A-37, with for diploidy); and the leading eigenvalue of . As expected, when this eigenvalue is negative, the population remains monomorphic (panels A–C). When it is positive, evolutionary branching occurs in males (panel D, split in blue lines). The bottom part of each panel shows male and female size distributions at equilibrium as histograms (averaged between time steps and ), along with the analytically predicted distribution at (solid lines, eq 26). The Kullback–Leibler divergence between male and female distributions is also reported. A–B: Male and female distributions are similar in shape as both sexes have the same demographic parameters (see left-hand side of panels for parameter values). However, females grow larger on average due to direct selection on size whereas males only track female size via selection for compatibility. C–D: Sex-specific demography generates asymmetries in size distributions with a higher frequency of small females. When sexual selection is strong enough, this leads to evolutionary branching (panel D) with one morph specialising on mating with smaller females. The smaller allele is seldom found at a homozygous state and the smaller large size mode is composed of heterozygotes. Other parameters: , , .

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In fact, the strength of size-based competition plays a decisive role in male evolution. When , size brings no reproductive benefit and males are selected to invest nothing in growth (). When is large, size matching is crucial for male fitness and selection favours growth rates that allow males to match female size. Simulations confirm this: has no effect on female evolution (red curves, left vs. right columns in Fig 4), but evolves to be larger when is large (left curves, left vs. right columns in Fig 4), thereby reducing the mismatch between female and male size distributions (compare red and blue histograms in Fig 4).

4.2.3 Quadratic selection: sexual selection based on trait matching favours life-history polymorphism in males.

Once the population has converged to the equilibrium characterized by eq (36), does it remain monomorphic under stabilising selection or can it become polymorphic due to disruptive selection? Numerical evaluations of eq (14) show that the Hessian matrix can be positive-definite, and thus selection can be disruptive, but only when

(37)

is positive and large, i.e., when positive multiplicative effects of size on male fitness at age a induced by changes in growth rate are large (eq 16b, as all other components of eq 15 were found to be too small to drive disruptive selection). Such multiplicative effects via size suggest that disruptive selection should favour males with either fast or slow growth such that small and large males of the same age coexist. Individual-based simulations confirm this (appendix B.3 in S1 Text for procedure). These show evolutionary branching leading to the coexistence of two diverged alleles coding for different growth rates, such that three types of males occur: two homozygotes with opposite growth strategies and an intermediate heterozygote (Fig 4D, blue dots). Some males thus grow substantially slower, remaining smaller for most of their lives but being longer-lived. At the population level this produces a multimodal male size distribution and a closer match with the female size distribution (compare panels C and D of Fig 4).

A closer inspection of eq (37) reveals the drivers of polymorphism. Disruptive selection on male growth is favoured when the number of competitors is large (i.e., when c is high or male survivorship is high), when size matching is strong (large ), and when individuals can attain large sizes (small ). In addition, the integral in eq (37) shows that a necessary condition is that the fecundity-weighted average of is positive, which occurs when male and female size distributions diverge. In particular, large positive values of , and thus disruptive selection, occur when males are predominantly large (; see Fig 1) whereas females are predominantly small (). In this case, at the convergence stable trait value most males are close to the average female size, while only a few match the smaller sizes more common in females (see Fig 4C). Under high , this leads to intense competition among larger males and thus to a fitness advantage for rare smaller males. Polymorphism therefore evolves because males specialize: some grow rapidly and die young, specializing into fertilising larger and more fecund females, while others grow slowly and live longer, and mate mostly with smaller females. In our simulations, such disruptive selection can be so strong that it overpowers directional selection before the population has completely converged to the singular strategy, leading to early branching (see Fig 4D). Under parameter values that favour a very wide distribution of female sizes, strong disruptive selection can also lead to the coexistence of more than two male morphs (see S2 Fig).

Evolutionary branching in our model hinges upon an advantage for males to specialize in different female sizes. Our results therefore connect to resource-competition and niche-partitioning models, where polymorphism evolves as individuals specialise to escape competition [e.g., 64–66], except that here the “resource” consists of potential mating partners. They also build on compatibility-based sexual selection models [56,57], which treat individual states as fixed traits determined at birth. In those models, male diversification arises in response to female genetic variation in compatibility traits generated by sexual conflict. By contrast, in our model females remain genetically monomorphic: variation in female states arises because individuals reproduce continuously while changing size through ontogeny. As a result, the population maintains a standing distribution of female sizes, which creates multiple mating niches; competition for these niches makes male fitness frequency dependent and can favour trait diversification. Another feature necessarily absent from these previous studies is that trait diversification here has knock-on effects on demography, altering the age and state structure of the population. In our model, the trait under sexual selection in males (investment into growth or sexual structures) also affects mortality via a survival–growth trade-off (eq. 23). Consequently, when selection favours alternative male growth strategies, evolutionary branching in that trait necessarily induces divergence in survival and lifespan. The resulting polymorphism therefore spans both mating strategy and life history, feeding back on population age structure and demography with the coexistence of fast-growing, short-lived males and slow-growing, long-lived males.

Alternative male mating morphs with markedly different life history traits are widespread (e.g., sneaker versus dominant males in fish, fighter versus sneaker males in beetles, and multiple morphs in isopods and ruffs). These polymorphisms are generally explained not by compatibility per se, but more often by sexual conflict (e.g., coercion versus resistance) or by condition-dependent development, where high- and low-condition males follow different trajectories [67,68]. While our results here apply mostly to the evolution of genetically determined alternative reproductive strategies [69], our model could readily be adapted to incorporate developmental plasticity and thus help disentangle the relative contributions of sexual selection, sexual conflict, and condition dependence in the evolution of sex-specific development. More generally, it provides a way to better understand the emergence and coexistence of alternative life-history strategies, together with their demographic causes and consequences.

4.2.4 Extensions.

We extended the model to explore two cases. First, we investigated the sensitivity of our results to a change in ploidy by considering a haplodiploid system, where males are haploid and females are diploid. This boils down to adjusting the weights in the sums that define selection, using , , , , (see Box 2 and appendix A.5.2 in S1 Text), and (i.e., assuming complete dosage compensation). Because vital rates are defined as functions of individual phenotype and genetic effects are additive, haplodiploidy does not affect the singular trait values (S3 Fig). Male haploidy does however affect quadratic selection by fully exposing mutations in the male trait. By increasing , haploidy amplifies the male-specific component of the Hessian matrix (see eq. 14). As a result, disruptive selection is stronger, especially under low male mortality and high sexual selection (S3D Fig). Evolutionary branching then produces the coexistence of two allelic types where the rarer, smaller type encodes larger male trait values than in the diploid case (compare S3D Fig and Fig 4D).

Second, we considered the effect of genetic constraints across sexes by assuming that males and females share the same growth rate (i.e., see eq 21). In a diploid population with additive gene action (i.e., ), the age-specific directional selection coefficient on the shared trait is the sum of the female and male gradients shown in eq (35):

(38)

resulting in sexual antagonism (i.e., when selection favours different trait values in males and females showing genetic correlations, [70,71]; Fig 2C, green line). Solving the integral of eq (38) for zero, we obtain a unique convergence-stable singular trait

(39)

Eq. (39) captures the lifetime trade-off between growth (numerator) and survival (denominator), averaged across sexes. In spite of this averaging, the equilibrium growth rate does not fall between the female and male optima found in the absence of genetic constraints (i.e., between and given by eq. 36) but instead evolves below both (Fig 3). This results from an evolutionary feedback: selection on males favours being smaller than females (owing to the costs of growth), genetic constraints then entail a reduction in female size, which in turn selects males to be even smaller, and so on. As a consequence of this sexually antagonistic feedback, females cannot evolve sufficiently large growth rates in response to high extrinsic mortality, making populations more prone to extinction than in the absence of genetic constraints (compare the green and red lines in Fig 3B).

Once the population has converged to , whether or not polymorphism evolves depends on the age-specific quadratic selection coefficient which is the sum of the female and male components (given by eq. 15; note that the mixed term is always zero in our model, see eq. A-86). By computing numerically, we find that genetic constraints across sexes prevent disruptive selection through two mechanisms. First, stabilizing selection on females is stronger, i.e., is more negative at all ages, than when male and female traits evolve independently (S4 Fig, red lines). Sexual antagonism thus prevents disruptive selection from materialising in males because selection in females counteracts it. Second, because the equilibrium growth rate (eq 39) is lower owing to sexual antagonism, the variance in female size in the population is not large enough to generate disruptive selection on males, and for rare smaller males to gain an advantage (S5 Fig). This effect is especially pronounced under strong sexual selection (see S5D and S4 Figs).