# The selection force weakens with age because ageing evolves and not vice versa

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195 Detailed exposition and derivations are in the Supplementary Methods.

### Demography, selection, and genetic effects

We consider a demographically stable population of constant, essentially infinite size where mj is average fecundity, i.e., number of offspring, at individual age j and pj is average survival, i.e., fraction surviving, from age j to age j + 1. Reproduction starts at age α and continues indefinitely. We define

• ageing in fecundity as mj > mj+1 with j = α, α + 1, …

• negative ageing in fecundity as mj < mj+1 with j = α, α + 1, …

• ageing in survival as pj > pj+1 with j = α − 1, α, …

• negative ageing in survival as pj < pj+1 with j = α − 1, α, …

The mean fitness in the population is

$$bar{w}=mathop{sum }limits_{i=1}^{infty }{m}_{i}{p}_{0}{p}_{1}ldots {p}_{i-1},$$

(1)

which can be seen as the reproductive value at birth in the demographically stationary state15,51,52. Mean fitness in the neutral population is 1. In this setting, the classic theory8 computes the selection force on age-specific fecundity and survival as proportional to the gradients

$$frac{partial bar{w}}{partial {m}_{j}}={p}_{0}{p}_{1}ldots {p}_{j-1}quad {{{{{{{rm{and}}}}}}}}$$

(2a)

$$frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}}=mathop{sum }limits_{i=j+1}^{infty }{m}_{i}{p}_{0}{p}_{1}ldots {p}_{i-1},$$

(2b)

respectively, which are always positive, as fecundity and survival are direct fitness components. Supposing fecundity does not cease and survival is never perfect (<1) at any age,

$$frac{partial bar{w}}{partial {m}_{j+1}} , < , frac{partial bar{w}}{partial {m}_{j}}quad {{{{{{{rm{and}}}}}}}},$$

(3a)

$$frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j+1}} , < , frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}},$$

(3b)

which are the classic theory result8 of the steady decline of these gradients with age. Equation 2 presupposes that genetic effects are additive on fecundity, whence the identity function implicitly operating on mj on the left-hand side of eq. 2a, and proportional on survival, whence the natural logarithm operating on pj on the left-hand side of eq. 2b. To make this explicit, we introduce the functions fM and fP, which act on age-specific fecundity and on age-specific survival, respectively. We only assume these functions be strictly increasing, to preserve positive selection on fM(mj) and fP(pj), and twice differentiable. We use them to add appropriate weights to the classic selection gradients and get the general gradients,

$$frac{partial bar{w}}{partial {f}_{{{{{{{{rm{M}}}}}}}}}({m}_{j})}=frac{1}{f^ {prime}_{{{{{rm{M}}}}}} ({m}_{j})}frac{partial bar{w}}{partial {m}_{j}}quad {{{{{{{rm{and}}}}}}}},$$

(4a)

$$frac{partial bar{w}}{partial {f}_{{{{{{{{rm{P}}}}}}}}}({p}_{j})}=frac{1}{{p}_{j}f^{prime}_{{{{{rm{P}}}}}} ({p}_{j})}frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}},$$

(4b)

which, as expected13, show no obvious age pattern. As an example of usage of these gradients, we recover the classic theory of additive effects on fecundity and proportional effects on survival when fM(mj) = mj and ({f}_{{{{{{{{rm{P}}}}}}}}}({p}_{j})={{{{{{mathrm{ln}}}}}}},{p}_{j}). We can reverse the classic assumptions, as in Fig. 1, by setting ({f}_{{{{{{{{rm{M}}}}}}}}}({m}_{j})={{{{{{mathrm{ln}}}}}}},{m}_{j}) and fP(pj) = pj. But the minimal assumptions about the f functions allow virtually any form of genetic effects, and not only additive and proportional, on fecundity and survival. For example, since (-{{{{{{mathrm{ln}}}}}}},{p}_{j}) is average mortality between ages j and j + 1, when ({f}_{{{{{{{{rm{P}}}}}}}}}({p}_{j})=-{{{{{{mathrm{ln}}}}}}},(-{{{{{{mathrm{ln}}}}}}},{p}_{j})) we have proportional decrements in mortality. We emphasize that the f are not fitness functions. They only capture how age-specific genetic variation acts on fecundity and survival.

### Model

To build our dynamical model, we use the breeder’s equation from quantitative genetics20,

$${bar{z}}_{j}(t+1)={bar{z}}_{j}(t)+{g}_{j}frac{partial bar{w}}{partial {bar{z}}_{j}}(t)+mathop{sum}limits_{ine j}{g}_{ji}frac{partial bar{w}}{partial {bar{z}}_{i}}(t),$$

(5)

which describes the time-discrete evolution of a mean trait (({bar{z}}_{j})) in a very large population under the action of selection when other traits (({bar{z}}_{i}) with i ≠ j) are also under selection. In this equation, the change in a focal trait over one-time step is the sum of two terms. The first term is the product between the selection gradient (partial bar{w}/partial {bar{z}}_{j}) and the additive genetic variance gj for that trait. This term captures the role of the selection force alone on the trait, provided there is some variability for it (gj > 0). The second term, the sum in eq. (5), captures trade-offs between the focal trait and all other traits concomitantly subject to selection with gji the genetic covariance between traits i and j. In our usage of the breeder’s equation, we set covariances to zero so that trade-offs are absent. The additive genetic variance, the fuel of selection, is assumed very small, constant over time and equal for all traits, gj = δ with 0 < δ 1. Thus, all traits share and retain, the same potential to evolve and selection is weak. These assumptions are also implicit in the classical work by Hamilton8. We study fecundity evolution and survival evolution separately.

When we let fecundity evolve, survival at each age is kept a positive constant at all times but it may vary between ages. We set ({bar{z}}_{j}={f}_{{{{{{{{rm{M}}}}}}}}}({m}_{j})) and use the selection gradient in eq. 4a inside the breeder’s equation. We then track change over time at the level of each mj via

$${m}_{j}(t+1)={{{Omega }}}_{{{{{{{{rm{M}}}}}}}}}(t){f}_{{{{{{{{rm{M}}}}}}}}}^{-1}left({f}_{{{{{{{{rm{M}}}}}}}}}({m}_{j}(t))+delta frac{1}{T(t)f^{prime}_{{{{{rm{M}}}}}} ({m}_{j}(t))}frac{partial bar{w}}{partial {m}_{j}}(t)right),$$

(6)

where ({f}_{{{{{{{{rm{M}}}}}}}}}^{-1}) is the inverse function of fM and the quantity T is the average generation time, which is required to get the change per time step from the change (delta frac{1}{f^{prime}_{{{{{rm{M}}}}}} ({m}_{j})}frac{partial bar{w}}{partial {m}_{j}}) per generation. The factor ΩM(t) describes density dependence. We assume a constant population size regulated by ecological factors, which is achieved by scaling fecundity at all ages equally to ensure that (bar{w}(t)=bar{w}(t+1)=1), as suggested before53.

When we look at how survival evolves, we set ({bar{z}}_{j}={f}_{{{{{{{{rm{P}}}}}}}}}({p}_{j})) and use the selection gradient in eq. 4b inside the breeder’s equation. Fecundities are set to positive constant parameters independent of one another. We then track change over time at the level of each pj via

$${p}_{j}(t+1)={{{Omega }}}_{{{{{{{{rm{P}}}}}}}}}(t){f}_{{{{{{{{rm{P}}}}}}}}}^{-1}left({f}_{{{{{{{{rm{P}}}}}}}}}({p}_{j}(t))+delta frac{1}{T(t){p}_{j}(t)f^{prime}_{{{{{rm{P}}}}}} ({p}_{j}(t))}frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}}(t)right).$$

(7)

Considerations analogous to eq. (6) apply.

As we presuppose potentially infinite ages, eqs. (6) and (7) give each rise to a separate dynamical system in the space of non-negative sequences of real numbers. The metric we use for this space is

$${d}_{i}(({a}_{n}),({b}_{n}))=mathop{sum }limits_{j=i}^{infty }frac{1}{{2}^{j}}frac{| {a}_{j}-{b}_{j}| }{1+| {a}_{j}-{b}_{j}| },$$

(8)

with (an) and (bn) sequences and starting index i depending on the specific tail sequence of interest. The distance between (an) and a subset B of the space is ({d}_{i}(({a}_{n}),B)={inf }_{({b}_{n})in B}{d}_{i}(({a}_{n}),({b}_{n}))).

### Analysis of fecundity

We change coordinates for the system in eq. (6) to yj = mj+1/mj. With a first-order Taylor expansion around δ = 0, the transformed dynamics of fecundity are,

$${y}_{j}(t+1)-{y}_{j}(t)={M}_{j}(t)left[left(dfrac{partial bar{w}}{partial {m}_{j+1}}(t)Bigg/dfrac{partial bar{w}}{partial {m}_{j}}(t)right)-dfrac{{m}_{j+1}(t){f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{j+1}(t))}^{2}}{{m}_{j}(t)f^{prime}_{{{{{rm{M}}}}}} {({m}_{j}(t))}^{2}}right],$$

(9)

where the factor Mj remains positive at all times. Setting to zero the left-hand side of eq. (9) to get equilibria (denoted by asterisks), we find that, for ages j = α, α + 1, …,

$$frac{{p}_{0}{p}_{1}ldots {p}_{j}}{f^{prime}_{{{{{rm{M}}}}}} ({m}_{j+1}^{* })}=frac{{p}_{0}{p}_{1}ldots {p}_{j-1}}{f^{prime}_{{{{{rm{M}}}}}} ({m}_{j}^{* })}sqrt{{p}_{j}frac{{m}_{j+1}^{* }}{{m}_{j}^{* }}}.$$

(10)

From eqs. 2a and 4a we see in eq. (10) that the equilibrium selection gradient on fecundity declines with age when ageing in fecundity is an equilibrium. Independently of our choice of fM, there is no realistic equilibrium of the form ({m}_{j}^{* }={m}_{j+1}^{* }) with j = α, α + 1, … , as this would lead to pj = 1 for j = α, α + 1, …  implying the total absence of mortality during adulthood. Depending on the sign of (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i}), equilibrium fecundity, if it exists, shows ageing or negative ageing (Fig. 2). When (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i} , > , 0), ageing in fecundity is an equilibrium (({y}_{j}^{* } , < , 1) with j = α, α + 1, …). The distance between the evolving fecundity schedule (yn) = yα, yα+1, … and the set ({A}_{,{ < },}^{m}) of life histories with ageing in fecundity is

$${d}_{alpha }(({y}_{n}),{A}_{,{ < },}^{m})=mathop{sum }limits_{i=alpha }^{infty }frac{{{Theta }}({y}_{i}-1)}{{2}^{i}}frac{| {y}_{i}-1| }{1+| {y}_{i}-1| },$$

(11)

where Θ is the unit step function. This distance can be shown to decrease under eq. (9) when (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i} , > , 0). In particular, when ({f}_{{{{{{{{rm{M}}}}}}}}}({m}_{i})={m}_{i}^{q}) with q > 1/2,

$${V}^{m}=mathop{sum }limits_{i=alpha }^{infty }frac{1}{{2}^{i}}frac{{Big({p}_{i}-{y}_{i}^{2q-1}Big)}^{2}}{1+{Big({p}_{i}-{y}_{i}^{2q-1}Big)}^{2}}$$

(12)

is a Lyapunov function and when ({f}_{{{{{{{{rm{M}}}}}}}}}({m}_{i})={a}_{1}exp (c{m}_{i})/c+{a}_{2}) with a1 > 0 and c > 0, any equilibrium can be shown linearly stable. When (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i} , < , 0), negative ageing in fecundity may be an equilibrium (({m}_{j+1}^{* } , > , {m}_{j}^{* }) with j = α, α + 1, …). Therefore, ({y}_{j}^{* } , > , 1) with j = α, α + 1, … . The distance between (yn) and the set of ({A}_{ ,{ > },}^{m}) of life histories with negative ageing in fecundity is

$${d}_{alpha }(({y}_{n}),{A}_{ ,{ > },}^{m})=mathop{sum }limits_{i=alpha }^{infty }frac{{{Theta }}(-({y}_{i}-1))}{{2}^{i}}frac{| {y}_{i}-1| }{1+| {y}_{i}-1| }.$$

(13)

This distance can be shown to increase with time under eq. (9) when (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i} , < , 0) and (yn) is not in ({A}_{ ,{ > },}^{m}). Moreover, keeping mα at equilibrium value and looking at the system in eq. (6) after a first-order Taylor expansion about δ = 0, it can be shown that the distance

$${d}_{alpha +1}({({m}_{n}(t))}_{alpha +1},{({m}_{n}^{* })}_{alpha +1})=mathop{sum }limits_{j=alpha +1}^{infty }frac{1}{{2}^{j}}frac{| {m}_{j}(t)-{m}_{j}^{* }| }{1+| {m}_{j}(t)-{m}_{j}^{* }| }$$

(14)

between the subsequence ({({m}_{n})}_{alpha +1}={m}_{alpha +1},{m}_{alpha +2},ldots ,) and its equilibrium, if this exists, increases with time when (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i} , < , 0). Irrespective of the sign of (partial ({m}_{i}{f}_{{{{{{{{rm{M}}}}}}}}}^{prime}{({m}_{i})}^{2})/partial {m}_{i}), it can be shown from eq. (10) that at most a single equilibrium with either ageing or negative ageing is possible—otherwise a contradiction with demographic stationarity is obtained.

### Analysis of survival

We change coordinates for the system in eq. (7) to xj = pj+1/pj. With a first-order Taylor expansion around δ = 0, the transformed dynamics of survival are

$${x}_{j}(t+1)-{x}_{j}(t)={P}_{j}(t)left[left(dfrac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j+1}}(t)Bigg/dfrac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}}(t)right)-dfrac{{p}_{j+1}^{2}(t){f}_{{{{{{{{rm{P}}}}}}}}}^{prime}{({p}_{j+1}(t))}^{2}}{{p}_{j}^{2}(t)f^{prime}_{{{{{rm{P}}}}}} {({p}_{j}(t))}^{2}}right],$$

(15)

where the factor Pj remains positive at all times. Setting the left-hand side of this expression to zero to calculate equilibria, we find that, for ages j = α − 1, α, …,

$$frac{1}{{p}_{j}^{* }f^{prime}_{{{{{rm{P}}}}}} ({p}_{j}^{* })}{left({left.frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}}right|}_{({p}_{n}) = ({p}_{n}^{* })}right)}^{frac{1}{2}}=frac{1}{{p}_{j+1}^{* }f^{prime}_{{{{{rm{P}}}}}} ({p}_{j+1}^{* })}{left({left.frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j+1}}right|}_{({p}_{n}) = ({p}_{n}^{* })}right)}^{frac{1}{2}},$$

(16)

which, by eqs. 2b and 4b, implies that the equilibrium selection gradient always declines with reproductive age. Independently of our choice of fP, there is no equilibrium of the form ({p}_{j}^{* }={p}_{j+1}^{* } , > , 0) with j = α − 1, α, …, as substituting this into eq. (16) and using eq. 2b we would get ({m}_{j}{p}_{0}^{* }{p}_{1}^{* }ldots {p}_{j-1}^{* }=0) for j = α, α + 1, …. The same problem derives from the assumption that fP is the natural logarithm. Since this is what the classic theory assumes, this theory fails to have an equilibrium in our model. Depending on the sign of (partial ({p}_{i}{f}_{{{{{{{{rm{P}}}}}}}}}^{prime}({p}_{i}))/partial {p}_{i}), equilibrium survival, if it exists, shows ageing or negative ageing (Fig. 3). When (partial ({p}_{i}{f}_{{{{{{{{rm{P}}}}}}}}}^{prime}({p}_{i}))/partial {p}_{i} , > , 0), only ageing in survival may be an equilibrium (({x}_{j}^{* } , < , 1) with j = α − 1, α, …). The distance between the evolving survival schedule ({({x}_{n})}_{alpha -1}={x}_{alpha -1},{x}_{alpha },ldots ,) into reproductive ages and the set ({A}_{,{ < },}^{p}) of life histories with ageing in survival is

$${d}_{alpha -1}({({x}_{n})}_{alpha -1},{A}_{,{ < },}^{p})=mathop{sum }limits_{i=alpha -1}^{infty }frac{{{Theta }}({x}_{i}-1)}{{2}^{i}}frac{| {x}_{i}-1| }{1+| {x}_{i}-1| }.$$

(17)

It can be shown that this distance tends to zero with time under eq. (15) when (partial ({p}_{i}{f}_{{{{{{{{rm{P}}}}}}}}}^{prime}({p}_{i}))/partial {p}_{i} , > , 0). When (partial ({p}_{i}{f}_{{{{{{{{rm{P}}}}}}}}}^{prime}({p}_{i}))/partial {p}_{i} , < , 0), negative ageing in survival may be an equilibrium (({x}_{j}^{* } , > , 1) with j = α − 1, α, …). The distance between the set ({A}_{ ,{ > },}^{p}) of life histories with negative ageing in survival and an evolving survival schedule outside of this set is

$${d}_{alpha -1}({({x}_{n})}_{alpha -1},{A}_{ ,{ > },}^{p})=mathop{sum }limits_{i=alpha -1}^{infty }frac{{{Theta }}(-({x}_{i}-1))}{{2}^{i}}frac{| {x}_{i}-1| }{1+| {x}_{i}-1| }.$$

(18)

This distance can be shown to increase with time under eq. (15) when (partial ({p}_{i}{f}_{{{{{{{{rm{P}}}}}}}}}^{prime}({p}_{i}))/partial {p}_{i} , < , 0).

### Reporting summary

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