Idea Transcript
J. Theoret. Biol. (1966) 12, 12-45
The Moulding of Senescence by Natural Selection W. D. HAMILTON Imperial
College Field Station, Silwood Park, Sunninghill, Berks., England (Received 16 October 1965)
The consequences to fitness of several types of small age-specific effects on mortality are formulated mathematically. An effect of given form always has a larger consequence, or at least one as large, when it occurs earlier. By reference to a model in which mortality is constant it is shown that this implication cannot be avoided by any conceivable organism. A basis for the theory that senescence is an inevitable outcome of evolution is thus established. The simple theory cannot explain specially high infant mortalities. Fisher’s “reproductive value”, the form of which gave rise to an erroneous opinion on this point, is shown to be not directly relevant to the situation. Infant mortality may evolve when the early death of one infant makes more likely the creation or survival of a close relative. Similarly, post-reproductive life-spans may evolve when the old animal still benefits its younger relatives. The model shows that higher fertility will be a primary factor leading to the evolution of higher rates of senescence unless the resulting extra mortality is confined to the immature period. Some more general analytical notes on the consequences of modifications to the reproductive schedule are given. Applications to species with populations in continual fluctuation are briefly discussed. Such species apart, it is argued that general stationarity of population can be assumed, in which case the measurement of consequences to fitness in terms of consequences to numerical expectation of offspring is justified. All the age-functions discussed are illustrated by graphs derived from the life-table of the Taiwanese about 1906, and the method of computation is shown. 1. Introduction Consider four hypothetical genes in man. Suppose all are limited in their expression to the female sex and also age-limited in the following way: each gives complete immunity against some lethal disease but only for one particular year of life. Suppose the first gives immunity for the first year, the second 12
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for the fifteenth, the third for the thirtieth, and the fourth for the forty-fifth. What are the relative selective advantages of these genes? If for further simplicity parental care is ignored and it is assumed that the menopause always comes before age 45, it is at once obvious that the fourth gene is null, whereas all the others do confer some advantage. It is also fairly obvious that the third gives less than the second. But how much less? Does the second give a maximum because it occurs at the age of puberty? Does the first give less than the second? The importance of questions of this kind for an evolutionary theory of senescence has been realized for some time. Most of the answers that will be given in this paper agree with the theory of Williams (1957). Although perhaps not obvious, they are so simple that it is surprising to find almost no indication that they had been realized earlier. Several writers have in effect answered the last two questions in the affirmative, which is for the one inexact and for the other wrong. Even Williams’ discussion failed to clear up completely the previous confusion of thought on the subject. Thus he regarded his theory as consistent with the views of Medawar (1952,1955). But Medawar in his 1952 lecture combined the development of a model which did lead him to the outlines of what we believe to be the correct theory with tentative adherence to a logically inconsistent opinion about the forces operating in the immature period. This latter seems to have been taken over uncritically from Fisher (1930, p. 29) who had written that he thought it “probably not without significance . . . that the death rate in man takes a course generally inverse to the curve of reproductive value”. As may be seen from the diagrams given in this paper a human curve of reproductive value (Fig. 3) rises to a maximum shortly after the attainment of reproductive maturity, while the curve of force of mortality (Fig. 2(a)) has a minimum at or slightly before it. Fisher argued that with an earlier age at marriage such as is very probable for our remote human and semi-human ancestry, the peak of reproductive value would have been earlier. Indeed it does seem quite likely that under primitive ancestral conditions the two turning points would have closely approximated the age of puberty. Hence apparently came Medawar’s idea that with the onset of reproduction there is reversal of selection for age-of-onset modifiers such that for deleterious effects before this age selection tends to make them occur earlier instead of postponing them. We hope to make it clear that the correspondence to which Fisher draws attention in the above statement is really largely trivial and that in the context to which they were restricting themselves the idea which he tacitly and Medawar explicitly assumed is without foundation. It must be admitted, however, considering the peculiar form of pre-adult mortality in man, the
14
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hypothesis was attractive, and we shall show reason to think that with a bio-economic basis something like it could be revived (section 9(~)). This paper outlines a fairly general approach to the problem of assessing how the age at which a gene acts affects its influence on fitness. Every effect on fitness must be capable of analysis into components manifesting in the schedules of mortality and fertility separately. Thus as regards the individual’s own fitness the problem has two sides. The argument will be developed in terms of hypothetical effects on mortality and most of the discussion will also be confined to this side. The other has fundamental interest but it will merely be shown at the end (section 9(c)) that effects on fertility are tractable, mathematically at least, by the same mathematical approach: it will be seen at the same time that the simplest implications are here rather obvious while more detailed ones are biologically doubtful. 2. Measures of Fitness and Mortality For an organism with “non-overlapping” generations the obvious measure of Darwinian fitness is the expectation of offspring as measured at birth. If the organism practises parental care “birth” should be considered to occur, for the purposes of this definition, at the age at which the offspring becomes independent. Although difficulties may arise, due to the participation of mates of different types in the parental care, the continuance of care right up to the period when the offspring themselves reproduce, and so on, the concept is essentially straightforward. The logarithm of the expectation is a parallel measure of fitness which may have advantages for some purposes. For an organism which reproduces repeatedly the concept of fitness is not so easily defined. The expectation of offspring suffers from the objection that early births are worth more than late in an increasing population, and vice versa in a decreasing one, and that there is no single measure of generation time which will serve as the unit for progress under natural selection. If the fitness differentials are all small and the population almost stationary these objections are of little importance, but in general the measure of fitness known as the Malthusian parameter (Fisher, 1930) has the advantage that it does take into account the relative values of early and late offspring by giving them appropriate weightings in the expectation. If I, is the fraction of a birth cohort of the type which survives to age x and f, the age specific fertility rate at age x, the Malthusian parameter m is defined to be the one real positive root for m of the equation
m s0 e-mx I,..,
dx = 1.
(1)
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Clearly m is a logarithmic measure of fitness, the correspondingmultiplicative measure being I = I”. To make these measures correspond to the “per generation” measures of the “non-overlapping” organisms we have to pick a generation time T and use mT and LT as the logarithmic and multiplicative measures, respectively. But a single unchanging T applicable to all types in an evolving population cannot be found; the present case is in this respect more complex than the non-overlapping one. It is best to be content with L or m based on some convenient unit of time. This would still have ditliculties for exact mendeliandemographic models but is good enough for the limited aims of the present study. The remaining preliminary is to decide on the measure in which the changes in mortality are to be specified. Given statistics of deaths by age and census population by age the life-table may be constructed in the actuarial manner. At an early stage qx is tabulated, estimating the proportion of people who have reached their xth birthday who will die before their (x+ 1)th birthday. This leads to the corresponding proportion of survivors px = 1 -qI and then to L
=
PO.Pl.PZ.
(2)
* *Px-1.
There are, of course, other possible procedures in life-table construction but this is the one preferred when the data is as stated and it is also the one most convenient for reference in the present discussion. qx is an average death rate for a period during which ability to resist death is not necessarily constant. We now define an instantaneous death rate, representing in negative the concept of ability to resist death, by a limiting process. Attention is transferred from qx, which applies to the unit age interval following age X, to a.& which is defined to apply to the part of this interval extending from x to x+6x. We then have l-
dxtix -=-=
6x
dxPx
lx-lx+dx
6x
1,6X
1 . -@s
= i.
f--
6X
dx+o
1 .--xcdl 1, dX
km
(3)
The instantaneous death rate pX is usually known as the “force of mortality”. The cologarithm of &..,., the natural logarithm taken with changed sign, is also relevant to this limit, for
colog dxpx 6x
log
=
I,-log
Ix+dx
6x
-6
=
log
1,
6x dx-ro1 dl,
=-y&=px.
d log I, dx
(4)
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And for estimating pX++ from the ordinary life-table colog px = log 1, -log I,+ 1 (5) is preferable to qx in being free from the particular slight bias apparent in
3. The Effect of a Brief Mortality
Change In practice to obtain an estimate of the parameter m defined in equation (1) an analogous equation is used: cc
(7) in which F, is the mean number of same-sex births in the age-interval x-3 to x++ to a parent living through it. Exactly, supposing the Z, and F, are exact, equation (7) gives the Malthusian parameter as it would be for a parthenogenetic organism if the births in each age interval (x--) to x++) were actually concentrated on the birthday. This might very nearly happen if breeding was strictly regulated by the seasons, but in other cases the inaccuracy will not be great so long as the age intervals are not too long. In the case of the human life-schedules, for example, so far from it mattering that reproduction is hardly seasonal at all we may take a quinquennial age-interval as giving a result quite as accurate as the data, the restriction to one-sex reproduction and the other artificialities, will justify. We require to differentiate 1 in the above equation with respect to colog pa where a is the age at which the effect is supposed to occur. From equations (4) and (5) it is evident that this is analogous to differentiating equation (1) w.r.t. ,uXSx and in value the derivative will correspond to the effect of a constant change of force of mortality between ages x and x+ 1. Clearly a finite instantaneous effect on ,uXcan have no effect on ~lt and in the continuous treatment we can only arrive at the fitness differential due to a very localized effect by finding the effect of 6~~ 6x = AX and then letting 6x --+0 while AX is held constant. Discussion in terms of the discrete treatment is here simpler on the whole: it not only steps off from the potentially available forms of schedule, but should be easier to follow. We shall therefore concentrate on this treatment and merely write the analogous expressions for the more ideal continuous one as we go along. We now proceed by differentiating 3, in each term of the sum represented in equation (7) w.r.t. pp. From the whole, noting that pa is only a factor in the terms for which x < a we arrive at
(x+l)ZxFx+L g l.-xZxFx=O. Paa+t
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Therefore
Since
it follows that
This is the derivative continuous treatment is
we require. The corresponding
formula
for the
m s dm dL\, = - :
I-” 1,f, dx
I XL-“1,
(9)
f, dx
The standard iterative calculation fo:solving equation (7) (e.g. Coale, 1957) to obtain the Malthusian parameter can be made to yield both the numerator and denominator expressions in the above formula. The value of the former for each successive value of a is given by cumulating the column which sets out A-xlxFx from the bottom; the total will equal unity provided I has been accurately found. The actual procedure is illustrated in Table 1, columns 7 and 8. The denominator expression is also familiar in demographic mathematics and may be denoted by W, with W used for the ideal quantity as it appears in equation (9). This is one of the parameters which can be considered as measuring the length of a generation, being the mean age of mothers at childbirth for all births occurring in the stable population. 4. The Effect of a Prolonged Mortality Change Rather as Z, can be regarded as compounded of the chances of surviving each successive year of life up to age x (equation (2)), so px itself can be regarded as compounded of the chances of evading the various independent factors which threaten life during the xth year. Among those of such factors which persist from year to year a few remain constant or nearly so, but most must tend to change with the intrinsic changes of habits and physiology that T.8.
2
I8
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accompany ageing. Despite such actual inconstancies, the illumination of which is the object of our study, we will proceed to discover the effect on m of multiplying every pX for which x > a by a constant factor slightly greater than unity. This is equivalent to adding a constant to log pX (which is necessarily negative), and to subtracting a constant from p,, throughout the same age range. In other words, the effect we are considering can be thought of as the total or partial elimination, from a certain age onwards, of a certain constant risk of death. Every Z, for which x d a is unchanged by this effect. For x > a, every I, now contains a factor h?-‘. Hence with k as defined
Writing the differential d log k as d logp,... m) so as to be more explicit we obtain corresponding to equation (8) by similar reasoning 03
In the continuous treatment the corresponding m dm 4k.,,
=-
formula is
sa (x-u)A-xE,fx * i XL-“l,f, 0
dx
01) dx
The denominator expressions are W and W as before. The value of the numerator in equation (10) is very easily obtained from the standard calculation: we simply cumulate the previous column which gave
again from the bottom, as illustrated in column 8 of Table 1. Like the denominator the numerator now has the form of a moment but differs in referring to only a part of the total unit of area under the function A-“I,F, and in being the moment about the abscissa a of the partial area instead of about the origin. When a = 0 the derivative is numerically unity. We then have which means that a mutant giving a life-long
mortality
reduction
of O-01
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TABLE 1
Age-functionsof mortality andfertilityfor Chinesewomenin Taiwan about 1906 1 Age
2
3
4 Fertility
Survivorship
Gross
Net
Barclay----- ___-___ “... .. ...” ---.----.--” Interp. Tuan
__._Interp. ..“.... 2x3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Totals
5 6 Weights Weighted for exp. net growth reprod.
7
8
9 Stable d:%b.
4x5 ____“II.......“““‘.-~---~-~
__._.__“““‘_.___. C7t
B6t
EY. 2+5
10 Basic reprod. value
-.._._ 2.. 7~9
1~00000 560479 1~00000l.OOoOO 1.00000 l-42914 0.7300 0.95852 l-oooo0 0.69972 0.6498 0.91876 460479 0*6100 0.88065 l*oooQo 0.54790 1.82515 0.5945 O-84412 3 m479 10lOOO O-47029 2.12635 0.5810 0*80910 0.5680 0.03246 0.77554 O-02518 2+X)479 2.37715 0.5520 0.25600 0.14122O-74337 0.97482 O-41008 0.5353 029987 0.71253 0.21366 1.62997 0.5151 0.80235 0.41325 0.68298 0.76116 0.35177 2.16380 0.4948 0.42876 O-65465 0.28068 0.86881 0.48048 0.29687 1.61849 0.473 1 0.82955 0.39246 O-62749 0.4514 O-35621 060146 0.21426 0.38833 1.07707 04287 0.75690 O-32452 0.57651 0.26623 0.24718 0.4061 O-29088 O-55260 0.16074 0.12210 0.3838 0.64001 0.24564 O-52968 0.10549 0.20329 0+.1891 0.01661 0.3615 0.17568 O-50770 0.08919 0,01630 0.16565 0,09840 0.3404 0.27713O-094330.48664 0.3193 0.03428 046646 0.01599 omO31 0@0232 0.2973 0.02500 0.00743 0.44711 omO31 0.13293 O-2753 O-00073 0.42856 0.00031 0.10347 0.2519 0.41078 0.2285 0.39374 0.2052 0.37741 0.07744 0.1819 0.36176 0.1581 0.34675 0.05484 0.1344 O-33237 0.03562 0.1118 O-31858 0.0892 0.30536 0.02040 O-0697 o-29270 0.0502 O-28056 0.0367 0.26892 0.00987 0.0232 O-25776 0.0167 0.24707 om414 0*0103 0.23682 0.00152 o-0067 O-22700 om31 0.21758 1+lOOOO l-61887 ---. -.- .---- __________ _--___.____ _____.______.____ __.____ .___ _______ ______ _______________.____.____._.__.__...-..._______________
l*OOOO
3.58694
I.61885
Except in column 4 all figures are in units of 5 years.
560479 italicized
are basic
data
or values
known
a priori.
Ages
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would confer a selective advantage of about 0.01. This fact has no particular novelty: it emerges equally well in the treatment of stable population theory in which the rate of increase or Malthusian parameter is conceived in terms of the imbalance of births and deaths. If b is the birth rate per head and li the death rate per head, for a population in its stable age distribution, the rate of natural increase is given by (Lotka, 1924, p. ill), r = b-d whence the stated result. Of course, it must be remembered that the change in mortality by affecting the rate of increase alters the stable age distribution with consequent effects on b and d: hence such statements are only approximate. If the mortality effect terminates at age b we would have m
dx+(b-a) [Pi& dx i‘a (x-u),l-xl,fx dm ----= (12) W d/q,. . .b) This gives equation (9) as b *co and tends to zero as b + a. By defining A,, . . . bj analogous to A,, we can find the substitution
dp,,.
.b)
=
-!b-a
d&a..
.b)*
Hence, if equation (12) is correct, we have b
j (x-u)~-xl,fx dx 1 n-Xl,fX dx dm ~ ------E dA (a.. .b) (b-a)W + Lwin which the left-hand term vanishes as b --f a, leaving equation (11). 5. The Effects of Age-of-Onset
By integrating
(13)
Modifiers
by parts we find that 04)
Thus adopting the notation
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we have Sob> = -Nsda)) = mszw. This fact, that the age-functions appearing in equations (9) and (11) are successive indefinite integrals of a function, is implicit in the cumulation method by which, as we have indicated, the corresponding discrete age functions can be obtained. With a corresponding notation for these functions g,(a) = n-aLzF,,
(18) (19)
gz(u) = f (x - a)n-x l,F,, lZ+1 we see that each gl(u) will give an estimate of the corresponding g&) value. The cumulation indicated in (19) will give a column of values which are, reading from the top, g,($),g,(l&), . . . , whereas the cumulation in (20) gives &(o),g,(l), * * * * This is conveniently shown in the tabulation by setting the numbers a half-row up as has been done in Table 1. It is easily seen that g*(O) = W, the estimate of IV. As derivatives with respect to age, g,(u) and gl(a) can evidently be regarded as measuring the effect of fitness of changes in age of action and age of onset, respectively, of mortality changes of the two kinds we have discussed. The occurrence of age of onset modifying genes is a plausible genetical hypothesis. For animals which grow continuously and reproduce repeatedly they would seem at least as plausible as the genes whose effects they are supposed to modify. In effect, of course, a gene which brings forward the age of onset of, say, a specific immunity is the same as a gene which directly confers an equal immunity for the limited period in question. The occurrence of age of action modifiers seems somewhat less plausible. This is partly because it is less easy to envisage the effects which they are supposed to modify. From the relation of go(u) to the reproductive schedule, it is evident that even if suitable subject effects did occur, only those manifesting within the fertile period would be subject to this kind of evolutionary improvement. 6. The Indirect Relevance of Reproductive Value The quantities co m
22
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are the expectations of births occurring after age a to persons chosen at age 0 and at age a, respectively. The quantities mb w, = gl(a) = J A-” l,., dx (21) cl and m
v, -
vo
=-
1”
A-*l,f,
dx
(22)
1, I
are clearly similar except that in them the births have been weighted in a particular way. As mentioned before, the weights are those necessary to correct for the different values of early and late births in a non-stationary population from the point of view of contribution to the population of the distant future. Roughly, each is inversely proportional to the size of the population into which the birth occurs relative to the size at the respective reference ages 0 and a, although exactly this statement is only true if the population is in its stable age distribution and therefore growing exponentially. All our other results are likewise reducible to simpler statements about the effects of mortality changes on expectations of offspring, as can be seen when the weights are removed from expressions as they stand. However, although it may be justifiable actually to ignore the weights in a wide range of cases, we retain generality for the present discussion. Thus ,v, can be considered to measure more exactly what Williams (1957) meant by “reproductive probability”. Unfortunately, it seems impossible to have a phrase which combines this brevity with greater precision, but “expected reproduction beyond age a” is at least more explicit. It should be clear from the analysis in preceding sections that u,, which is Fisher’s reproductive value, is not directly relevant to the rate of selection of a genotype whose special effect manifests at age a. Through various causes a proportion of people, (1 -I,), fail to survive to age a and in them the genotype is never expressed. Thus although v, indicates the magnitude of the effect of a set age-localized genotypic effect on mortality uhen this is aliowed to express itself, in rating the consequent natural selection of the genotype this effect on fitness must be considered diluted owing to a kind of “poor penetrance” of the genotype: IV, takes account of this dilution, whereas v, is so constructed as to ignore it. It may be argued that reproductive value does have at least an indirect relevance to senescence through its relevance to parental care. Acts of parental care necessarily involve living individuals, and it should now be
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evident that the ratio of the reproductive values is just as important as the coefficient of relationship in determining ideally adaptive social behaviour: the coefficient gives the chance that the offspring carries a replica of a behaviour-causing gene of the parent (Hamilton, 1964a,b), while the ratio gives the relative conditional expectation of its reproduction. The inclusive fitness of an individual is maximized by its continually acting in ways that cause increases in its inclusive reproductive value. It is implied that the selfsacrificing tendencies of parental care should follow a course inverse to that of reproductive value. But against this, it must be remembered that our main thesis concerns the necessary failure of ideal adaptation, and the reason why this should tend to increase with age. As Williams emphasized, ideally with non-social species natural selection improves viability for all ages for which there is prospect of any future reproduction. And if it is plausible to postulate for the working of his theory of senescence the existence and promotion of genes which reduce viability in late life and raise it in early life, then surely the natural selection of genes which are not physically pleiotropic, and cause unchanging patterns of parental care, is even more plausible. If so, the fact that the intensity of parental devotion does not increase very noticeably with age is not surprising. 7. The Model of a Non-senescing Organism The absurdity of the idea that reproductive value outlines the forces of selection tending to prevent senescence may be shown by reference to the concept of a non-senescing organism. Such an organism is supposed to have a mortality which does not change with age. As regards reproduction it can be assumed that so far from showing any senescent decline, fertility actually increases exponentially as the organism gets older. No real organism could increase its fertility indefinitely, of course (although some probably do keep up a gradual increase for a long time, for example some fish), but this assumption prejudices the case against the evolution of senescence as strongly as possible. To give an idea of how such reproductive expansion could be supposed to occur, a volvox-like organism may be imagined with all its cells undergoing synchronous division. After two divisions each cell has given rise to a tetrad; suppose one cell of each tetrad is expelled as a spore which starts a new colony, while the other three separate to take up as far as possible equipotential positions on the sphere ready for the next round of growth. Such an organism both grows and expands its fertility according to a geometric progression (in this case the growth ratio is three per two cell-generations). Its population dynamics ought strictly to be treated by means of series (as used in Cole,
24
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1954). But approximating fertility by a continuous exponential function would be generally very nearly correct and leads to simpler-looking mathematics. Suppose, therefore, the following schedules of survivorship and fertility : Age-range
Survivorship
xa
Not defined 1(1e-Jl(r-a)
Fertility
where c is the logarithmic growth rate of fertility, f, the fertility at the beginning of reproduction, 1, the survivorship up to age CI,and p the constant force of mortality that supervenes at least from age CI onwards. For the present discussion the course of mortality from age 0 to the age c(need not be defined. From equation (1) the fundamental equation for this case is co I e-mfa I, ,(c-P)(x-a) dx = 1 a or e -““zA
1 e(C-P-m)z& = 1.
It is clear that it is possible to choose m sufficiently large to make the integral converge, and therefore that a value exists which is the solution of the equation. Thus integrating and evaluating at the limits: e -““l,f, = m+p-c. (23) By some numerical method this may be made to yield the real root for m to any desired approximation. Using equations (21) and (22) we may now write down for this case formulae for the “expected reproduction beyond age a” and for reproductive value at age a, valid in each case for the age range a > CI. w, = e-ma1,f, j: e-(m+P-cC-a)dx _ e-““?zfa I- (mfp-c)(o-a) m+p-c =e -(m+p-c)(cJ-a) from (23) =e -e-“‘orJ,f.(o-a) > VI7 -= VO
em(a-a)lafa me-t~+e-ejCx-a)d X. 4 s (I
(24)
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Substituting for I,,, evaluating the integral as before and performing cancellations : V&l f, c(a-a) -c-e m+p-c vo =e c(a-cr)+ma,from (23). It is at once seen that w,, has the form of a descending exponential while v, has the form of an ascending one provided c is positive. The expansion of v, in this case simply reflects the fact that as we examine organisms from later and later age groups we find them not only still “as good as new” in viability but at the same time displaying more and more enlarged reproductivecapacity, so that the expectation of offspring to a still living organism is continually increasing. On the other hand, w, must always have a descending form if it exists at all for otherwise the integral of the fundamental equation (1) would not converge, and the stable population theory would be inapplicable. From the point of view of biology the range of cases where it is possible to find a root m to satisfy the equation is quite wide en0ugh.t One of the most extreme may be illustrated from the above model: mortality is zero at all ages and c is p0sitive.S We have e -ma 2,f, = m-c. It is easily seen that there is still a real positive root for m, and that w, retains its descending form. The circumstances of this organism have to be imagined to be such that although individuals increase their fecundity exponentially, and continue to do so indefinitely, the population has still met with no checks to increase whatsoever, so that all its members are immortal. It is striking to find that even under these utopian conditions selection is still so orientated that, given genetical variation, phenomena of senescence will tend to creep in. The form of w,, which is the same as the gl(u) discussed earlier, shows unequivocally that any mutation causing an improvement in early fecundity at the expense of an equal detriment later will give a raised Malthusian parameter, so that the mutant form will gradually come to numerical preponderance in the population; and if we allow any incipient incidence of mortality we likewise see that selection will favour resistance to it at early ages to a certain extent at the expense of greater vulnerability at later ages. t Very hypothetical organisms are easily defined whose characteristics take them outside the range of the mathematics of the present discussion. An example would be one whose fertility increased indefinitely like eza. $ It will be observed that while increases of c always reduce senescence, the same is not always true for increases in m. If the increase in m is caused by increases in 1. orf, or by decrease in a, senescence is actually enhanced.
26
W.
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Thus it may be stated that for organisms that reproduce repeatedly, senescence is to be expected as an ineaitable consequence ?f the working oj natural selection. There is no need to emphasize that senescence, or rather a tendency to complete exhaustion by the reproductive effort, and consequent death, is implied a fortiori for all organisms with reproductive schedules more limited than the one we have been discussing. In the last example we were concerned with an extreme and impossible case. Returning to a condition of the model which is biologically possible, indeed the only one which is permanently possible, consider the case III = 0. Equation (24) becomes wII = e-‘“fm’“-a’ Hence tendency to senesce should be weakest if matters are arranged such that l,f, is as small as possible: immature survival should be low and starting fertility low. In other words, immature mortality should be high and reproduction should not begin suddenly. From (23) l,f, = p-c ; therefore the growth rate of fertility would be unimportant if it were balanced wholly by a general increase in adult mortality, but if the rise in c causes any rise in immature mortality as well, as is very likely, it will tend to reduce the rapidity of senescence. The preceding equations support some of the points made by Williams about apparent connections between the phenomena of survival and reproduction and the rapidity of senescence. In saying this we are applying the present model outside its strict field, to organisms that already do show senescence. But arguments such as were used by Williams seem to show that conclusions of similar general nature would be obtained if the present model could be reworked with p replaced by some increasing function of age. Thus it seems justified to use the model to emphasize certain necessary connections which Williams says little about. If c = 0 we have Williams’ point that high rates of adult mortality should lead to high rates of senescence. The illustration which he gives in this connection, some evidence that birds and bats have low rates of senescence compared to flightless birds and mammals is certainly striking. It is likely, however, that high levels of adult mortality are usually more a consequence of high fertilities than of degree of adaptation in adult life. Bats are very much less fecund than rodents. Among species that do not have a special infant mortality (e.g., as implied in section 9(B), dispersive animals which lay their eggs separately) it should be found that the highest rates of senescence accompany the highest fertilities.
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SELECTION
27
Regarding Williams’ third point we have already shown that an increasing schedule of fertility does not necessarily retard senescence since it can be wholly balanced by increase of p (in the model at least); but if it is even partly balanced by increase of immature mortality his point holds, and this is very likely to be the case. It does seem likely at least with the molluscs, crustacea and poikilothermic vertebrates which he cites; we shall shortly refer to indications of high “infant mortalities” in the first and last. It may seem rather contrary to the previous paragraph that these long-lived animals are so fecund. But it may be that the state of gradually increasing fertility could only be established if the progeny are relatively small at birth and therefore numerous. 8. Notes on the Computed Examples As a further preliminary to discussing the fit of our apriori concepts to the reality, graphs illustrating the various functions for certain real cases are given. The notes of this section are necessary to explain the choice of data for these graphs and the computational procedures, and also to give some cautions as to generality. They may be skipped by the reader only interested to follow the main theme of the paper. Man is the species for which much the best data is available. The physiological phenomena of the human life span must have become established very nearly in their present-day form long before the advent of any civilization, and during a period when cultural advance was on the whole slow. The statistical manifestations of these phenomena, however, both as regards senescence and expressed fertility, are being changed very rapidly by modern cultural advances, as is well known. Therefore in attempting to reconstruct the forces of selection that were operative in the stem of hominid evolution it is appropriate to start from a present-day population whose economy is as backward as possible. Unfortunately there are no very good data for contemporary peoples in a pre-agricultural phase, nor even for those with the most primitive forms of agriculture. The data which was collected by the Japanese for the Chinese farming population of Taiwan about the end of the last century seems to be about the best available for the present purpose. The general level of mortality was extremely high, one-half of those born being dead before the age of 26. The fertility was also high with an average of about seven children per woman living right through the reproductive period. This fertility was sufficient to give a positive rate of increase over 1% per annum. At the time the influence of modem contraceptive methods must have been quite negligible.
28
W.
D.
HAMILTON
Specifically, the basic data used in our “male” were obtained as follows: (A)
MALE
AND
FEMALE
SURVIVORSHIP
and “female” computations
SCHEDULES
(lx)
Life tables for 1906 given in Barclay (1954), p. 172. (B)
FEMALE
FERTILITY
RATES
(&
FOR X =
3, 14, . . .)
The data collected by Tuan (1958) has been used. It refers to a local sample of women from a farming population whose fertile period centred about the same time as the life table was made. When combined with Barclay’s Z, these fertilities lead to a per annum rate of increase of 1.7 ‘$