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 m_{j} is average fecundity, i.e., number of offspring, at individual age j and p_{j} 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 m_{j} > m_{j+1} with j = α, α + 1, …

negative ageing in fecundity as m_{j} < m_{j+1} with j = α, α + 1, …

ageing in survival as p_{j} > p_{j+1} with j = α − 1, α, …

negative ageing in survival as p_{j} < p_{j+1} with j = α − 1, α, …
The mean fitness in the population is
$$bar{w}=mathop{sum }limits_{i=1}^{infty }{m}_{i}{p}_{0}{p}_{1}ldots {p}_{i1},$$
(1)
which can be seen as the reproductive value at birth in the demographically stationary state^{15,51,52}. Mean fitness in the neutral population is 1. In this setting, the classic theory^{8} computes the selection force on agespecific fecundity and survival as proportional to the gradients
$$frac{partial bar{w}}{partial {m}_{j}}={p}_{0}{p}_{1}ldots {p}_{j1}quad {{{{{{{rm{and}}}}}}}}$$
(2a)
$$frac{partial bar{w}}{partial {{{{{{mathrm{ln}}}}}}},{p}_{j}}=mathop{sum }limits_{i=j+1}^{infty }{m}_{i}{p}_{0}{p}_{1}ldots {p}_{i1},$$
(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 result^{8} 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 m_{j} on the lefthand side of eq. 2a, and proportional on survival, whence the natural logarithm operating on p_{j} on the lefthand side of eq. 2b. To make this explicit, we introduce the functions f_{M} and f_{P}, which act on agespecific fecundity and on agespecific survival, respectively. We only assume these functions be strictly increasing, to preserve positive selection on f_{M}(m_{j}) and f_{P}(p_{j}), 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 expected^{13}, 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 f_{M}(m_{j}) = m_{j} 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 f_{P}(p_{j}) = p_{j}. 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 agespecific genetic variation acts on fecundity and survival.
Model
To build our dynamical model, we use the breeder’s equation from quantitative genetics^{20},
$${bar{z}}_{j}(t+1)={bar{z}}_{j}(t)+{g}_{j}frac{partial bar{w}}{partial {bar{z}}_{j}}(t)+mathop{sum}limits_{ine j}{g}_{ji}frac{partial bar{w}}{partial {bar{z}}_{i}}(t),$$
(5)
which describes the timediscrete 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 onetime 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 g_{j} for that trait. This term captures the role of the selection force alone on the trait, provided there is some variability for it (g_{j} > 0). The second term, the sum in eq. (5), captures tradeoffs between the focal trait and all other traits concomitantly subject to selection with g_{ji} the genetic covariance between traits i and j. In our usage of the breeder’s equation, we set covariances to zero so that tradeoffs are absent. The additive genetic variance, the fuel of selection, is assumed very small, constant over time and equal for all traits, g_{j} = 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 Hamilton^{8}. 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 m_{j} 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 f_{M} 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 Ω_{M}(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 before^{53}.
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 p_{j} 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 nonnegative 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 (a_{n}) and (b_{n}) sequences and starting index i depending on the specific tail sequence of interest. The distance between (a_{n}) 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 y_{j} = m_{j+1}/m_{j}. With a firstorder 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 M_{j} remains positive at all times. Setting to zero the lefthand side of eq. (9) to get equilibria (denoted by asterisks), we find that, for ages j = α, α + 1, …,
$$frac{{p}_{0}{p}_{1}ldots {p}_{j}}{f^{prime}_{{{{{rm{M}}}}}} ({m}_{j+1}^{* })}=frac{{p}_{0}{p}_{1}ldots {p}_{j1}}{f^{prime}_{{{{{rm{M}}}}}} ({m}_{j}^{* })}sqrt{{p}_{j}frac{{m}_{j+1}^{* }}{{m}_{j}^{* }}}.$$
(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 f_{M}, there is no realistic equilibrium of the form ({m}_{j}^{* }={m}_{j+1}^{* }) with j = α, α + 1, … , as this would lead to p_{j} = 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 (y_{n}) = y_{α}, y_{α+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,
$${V}^{m}=mathop{sum }limits_{i=alpha }^{infty }frac{1}{{2}^{i}}frac{{Big({p}_{i}{y}_{i}^{2q1}Big)}^{2}}{1+{Big({p}_{i}{y}_{i}^{2q1}Big)}^{2}}$$
(12)
is a Lyapunov function and when ({f}_{{{{{{{{rm{M}}}}}}}}}({m}_{i})={a}_{1}exp (c{m}_{i})/c+{a}_{2}) with a_{1} > 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 (y_{n}) 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 (y_{n}) is not in ({A}_{ ,{ > },}^{m}). Moreover, keeping m_{α} at equilibrium value and looking at the system in eq. (6) after a firstorder 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 x_{j} = p_{j+1}/p_{j}. With a firstorder 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 P_{j} remains positive at all times. Setting the lefthand 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 f_{P}, 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}_{j1}^{* }=0) for j = α, α + 1, …. The same problem derives from the assumption that f_{P} 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).
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