Elements of Physical Biology

more will be said later, in discussing the physical basis underlying the survival curve. Survival Curve Data. Without having recourse to any refinements of mathematical analysis it is clear that a close relation exists between the survival curve of a given species and its rate of increase, 6 © Pa a oe Sei SST PSE he Ss ae pes ie (a CPE Sass Re ee Re ie ee Cae ae be eT im Me es ae LF Ss Es NS ae es Be ae Tle? co Ge Ca ae oe oe ee on ee et es ee a SS a a a (a a A a a eAwbeteletitetelats | intqed Aah |

The slope of the curve thus plotted measures the Force of Mortality. This slope is at first very steep, decreases to about the twelfth year of life, when it reaches a minimum (point of inflection), and then increases ag be expected, therefore, that the study of such survival curves for different species of organisms should have formed an essential part of quantitative studies in evolution. As a matter of fact, however, data for survival curves of organisms in their natural environment

are, for obvious reasons not easily obtained. It is difficult enough to make an accurate census of ages in a supposedly intelligent and responsible human community, where the codperation of the individual can at least in a measure be engaged. A census of ages in the population of field, forest and stream must indeed be a severe test of the ingenuity of any investigator that should approach such an enterprise. Not that it is utterly hopeless. Isolated data can and have been gathered. Every layman knows that the age of trees, for example, can be estimated at least approximately by the number of rings in a cross section of the trunk. The age of certain fishes can be gaged from tell-tale marking on their scales. Theage of certain birds can be told from their plumage. An estimate of the average age of herring gulls has been made, on this basis, by J. T. Nichols.2 But data of this kind are at best scant. The problem is more manageable in the case of domestic species, or of species in captivity. The most complete study of this character is probably the work of R. Pearl and 8. L. Parker with colonies of fruit flies (Drosophila) grown in an experimental ‘universe,’ a glass bottle containing a suitable quantity of banana juice. The survival curves thus obtained are singularly like those for the human species, when allowance is made for the difference in the total life span. The fruit fly’s allotted days, in the most favorable case, number about ninety, or, say, one day for each year of human life. In figure 19 are shown, plotted on the same diagram, three survival curves; one of these is that of the human population of the United States in 1910; the second is the survival curve for one of the numerous varieties of fruit flies.

For only one other organism, Proales decipiens (a rotifer), a life table and curve has been prepared, on the basis of observations by B. Noyes.‘ The life curve has been computed by Pearl and Doering’ and compared with that of man, on the basis of the life span * J. T. Nichols, in The Condor, vol. 17, 1915, p. 181; for other data on the longevity of animals see C. 8. Mince: The Prout of ees Growth and Death, 1908, pp. 226, 266; A. Weismann, Uber die Dauer des Lebens, Jena, 1882. ae also A. T. Masterman, Report on the scales of freshwater fish in relation to age determination. . M. Stationery, Office 1924.

measured from the point at which the force of mortality is a mimimum to the extreme end of life. Itis the curves thus obtained that are shown in figure 19. As will be seen, they exhibit a close analogy. Fia. 19. Logariramic SurvivaL Curves For Man, Drosopuina, AND Proautpes DecrPreNns Plotted according to centiles of life-span; the human life curve is plotted with the twelfth year of life (point of inflection, see figure 18) as zero of the time scale. After R. Pearl.

Influence of Age Distribution Upon Rate of Growth of Population. For our present purposes it is not sufficient to fix our attention on a group of individuals at the moment of their birth, and to observe the gradual diminution of this group by deaths; we must take in view an actual population, comprising individuals of all ages, and we must enquire into the effect of deaths upon such a group as this. If the survival curve for such a population were of the simple form

with y, the force of mortality, independent of a, it is evident that the age distribution in the population would have no influence upon the total death rate. For if individuals of all ages are equally susceptible to death, it is evidently immaterial how the population is constituted as regards age. The death rate per head, in such a population would, in fact, be simply equal to the force of mortality. The Stable or “Normal” Age Distribution. But in the populations with which the biologist and the vital statistitian deals, the force of mortality varies very decidedly with the age, and it might therefore be supposed that any discussion of the rate of increase of a population of organisms must fully take into account the age distribution. This supposition, however, involves an assumption, namely, the assumption that the age distribution itself is variable. Now, in point of fact, the age distribution is indeed variable, but only within somewhat restricted limits. Certain age distributions will practically never occur, and if by arbitrary interference, or by a castastrophe of nature, some altogether unusual form were impressed upon the age distribution of an isolated population, the “irregularities’”” would tend shortly to become smoothed over. There is, in fact a certain stable type of age distribution about which the actual age distribution varies, and toward which it returns if through any agency disturbed therefrom.’ The form of this distribution, in an isolated population (i.e., with immigration and emigration negligible) is easily deduced, as follows:

If we denote by N (¢) the total number of the population at time t, and by N (#) c(a) da that fraction of the total whose age is comprised, at time ¢, within the limits a and a + da, it is evident that the individuals of this element of the population are the survivors of the B(t—a)da persons born at times (t—a). Hence if p(a) is the survival factor, we have If we denote by D (¢) the total death rate at time ¢, we have, evidently We are supposing that we are dealing with a population in which the survival factor is constant, i.e., independent of t. If c(a) also is independent of ¢, the integral in (17) is evidently a constant and we have

where d is the death rate per head. On the other hand, since c(a) is independent of the time, we have Brief reflection shows that the value of c(a) for zero age is the birth rate per head. Thus we have where r is the ‘“‘natural rate of increase” of the population. We have further The normal age distribution (23) is of fixed form in the sense that, once established, it perpetuates itself. More than this, it is also stable, in the sense that, if disturbed by a temporary change in the conditions of life (e.g., war), it will spontaneously return upon restoration of normal conditions.

A rigorous proof of this stability of the age distribution (23) cannot be briefly given, and for details of such proof the reader must be referred to the author’s publications in the journal literature.* The general character of the proof may, however, be indicated. If the original population has any arbitrary age distribution, such as that represented by the more heavily drawn curve in figure 20, by judicious trimming we could reduce the population to the normal distribution represented by the lower curve tangent to the heavy curve; or, we could, by filling in gaps, supplement the population to fit the upper tangent normal distribution curve. The trimmed down population, having the normal age distribution, would always retain it. The same is true of the supplemented population. The actual population will therefore always lie between the trimmed and the supplemented. Moreover, it can be shown that at intervals of about one hundred years (the span of life) new tangent curves can be drawn to the actual distribution curve, the new tangents lying between the old. Thus the two tangent curves ultimately approach until they coincide, and then, necessarily, the actual distribution curve lying between them coincides with them also. That is to say, the actual conforms with the normal age distribution.’

It must be understood that the normal or stable form of age distribution represents merely a broad type, toward which actual age distributions will tend. However, the approach seems to be at times very close, as is shown by the figures in table 7 giving the observed and the calculated age distribution for England and Wales in decennium 1871-1880. A graphic representation appears in figure 21, which shows the observational data, plotted (in dotted lines) as a stepped curve. The corresponding figures calculated by the formula (13) are plotted in two ways, namely, first as a stepped curve (drawn in full), for comparison with the observational data; and also as a continuous curve. The latter brings out a feature that may be noted in passing, namely the fact that the curve of age distribution is very flat, roughly linear, over a wide range, from the

7An exceedingly interesting effort of early date to demonstrate the ultimate approach to geometric increase of the birthrate, independently of initial conditions (e.g. starting with a single pair of parents) is to be found in L. Euler, Recherches générales sur la mortalité. See also E. T. Gumbel, Jahresber. Deutsch Math. verein., vol. 25, p. 251. Fic. 20. Diacrams TO ILLUSTRATE PROOF oF STABILITY OF NoRMAL AGE Population of England and Wales 1871-1880, as an example of ‘‘normal’’ age distribution a, 4a Caleulated | Observed | Calculated | Observed | Calculated | Observed

Fiq, 21. “Srasuy’’? Aap DisTrRipuTION, AS EXEMPLIFIED BY THE POPULATION fifth to the eightieth year of life.8 This, of course, is a special feature of a human population when the natural rate of increase r lies in the neighborhood of a certain value. Minor fluctuations of the age distribution will not greatly affect the birth rate and the death rate. Since actual populations approximate the normal age distribution, i.e., that defined by equation (23), it seems permissible and it is certainly expedient to assume, in further discussion, that the normal age distribution is actually established; we may then virtually disregard the influence of the age distribution upon = the rate of increase of the component. This element is, as it were, automatically ruled out of further discussion, by the natural establishment of the normal age distribution;® a circumstance which is in so far fortunate, as we are here interested in the relation of the rate of growth to the more fundamental biological characteristics; the age-distribution appearing merely as an adventitious element complicating the relation, without being essential to the fundamental characterization of the species.

Demographic Relations in “‘Normal’? Population. It is worth while to note briefly in passing that in the case of a population in normal age distribution, many demographic relations assume a simple form. So, for example, the relation between birth rate per head b and death rate per head d is here given by the formula’® which, to second order approximation, is reducible to the simple form 8 Compare J. Brownlee, The Use of Death Rates as a Measure of Hygienic Conditions, Report to Medical Research Council, London, 1922, pp. 36-37.

As an illustration it may be mentioned that in the same population (England and Wales, 1871-1880) which has already been referred to, the relation (26) is found by actual computation to be, numerically, The observed values of 6 and d, together with those computed by the formulae (22), (23), are shown in table 8. It will be seen that the agreement is good. The relation (26) between b and d lends itself readily to graphic representation by means of a diagram involving only straight lines. This is shown in figure 22. The method is as follows.

Along OX mark off a length OP = Along OY mark off a length OQ = TL Complete the rectangle QOPM. Suppose we are now given b = 0.0337 It is required to find d Along OX mark off OV = 0.0337 Join VM and produce it to meet OY in W Read off at W on the scale of OY d = 0.0200 4U is proper fraction. It then follows from (26) that L is intermediate 1 LMC ee between 5 and 7) this circumstance has been noted by Bortkewitch, Mitt- A single equation, such as (25), connecting the two quantities b and d, is of course insufficient to determine their value. To do this requires a second independent relation. Such a relation is furnished by the following considerations:

The law of the fixed age distribution holds separately for each sex. Thus, if we denote N; the total number of the female popula- Fria. 22. DiAGRAM OF ReLaTION Brerween BirtH Rate per Heap b AND Death Rate pER Heap d IN POPULATION WITH STABLE tion, by b, the number of female births per unit of time and per head of female population, and finally by p; (a) the survival factor in the female population, then the number of females of an age between a and a + daisgiven by N,be™ p; (a) da. If these reproduce, on an average, 6; (a) females per unit of time, then the total number of female births per unit of time is evidently given by

where a, a2 are the lower and upper age limits of the female repro- = {eve (a)Bx(0 da (34) This is the required second relation” between b and d. The birth rate and death rate in a population of fixed age distribution are thus completely determined as soon as the functions p; (a) and 6; (a) are given. Aside from its direct interest as the second relation required to complete the determination of b and d in a population with fixed age distribution, equation (34) is also of value in enabling us to deduce certain secondary results, of which two will here be noted.

Rate of Increase per Generation. To derive our first conclusion from (25), we expand e~* under the integral sign. We find A little reflection shows that the first integral measures the ratio between the total number of births in two successive generations. Denoting this by R, and the second integral by S, we have a formula which connects the rate of increase per head per annum r and the rate of increase R in the total number of births in one generation, per generation. It will be seen that this relation is not one that could very well be foretold by any simple elementary or common sense principle. It is this relation that is implicitly involved in W. R. Thompson’s treatment of a problem in parasitology cited in Chapter VIII (page 87); at least, we must know this relation if we are to interpret his results in terms of the usual time unit, such as the year, whereas Thompson reckons in generations.

Effect of Selective Slaughtering. A second deduction from equation (34) follows from the fact that the integral extends only between the age limits of the reproductive period. In consequence of this the rate of increase, per head, of the population is independent of the form of the life curve (the survival factor) outside this period; thus the slaughtering of the ‘‘superannuated’’ members of a herd has no effect upon its rate of increase, though both death rate and birth rate are augmented (by equal amounts). The economic significance of this, as a means for raising the productive capacity of the herd, considered as a food factory, is self-evident.

It is true, as previously noted, that generally the influence of age distribution upon the rate of growth of a species is eliminated from discussion by the spontaneous establishment of the fixed or normal distribution. But the subject referred to in the preceding paragraph clearly shows that there are exceptions to this general rule. We have here an instructive example in which the influence of the age distribution upon the course of events is so fundamental that a discussion of the case which should fail to cover this feature would be lacking in an essential particular. Indiscriminate killing of one species by another, as practised by the untutored savage or the dumb animals, has the effect of reducing the ordinates of the life curve of the food species more or less along its whole extent. This is illustrated in figure 23. We may suppose that the life curve of some species, wild cattle, for example, is that shown in a full line, curve I, and that curve II represents the life curve for the same species after the habitat of the species has been invaded by a tribe of primitive hunters, too thoughtless or too unintelligent to regulate their depredations in accordance with anything of the nature of game laws. Curve III is the type of curve that would be produced by selective killing guided by intelligent control, as distinguished from random killing. The verticals erected at a and az represent the limits of the period of reproduction for the cattle. The slaughtering, in the case of curve III, takes place exclusively after the end of the period of reproduction. This is an extreme case, which, in practice, for obvious reasons, would be only distantly approached. For both in game preservation and in animal husbandry on the farm many factors have to be considered; not only the quality of the meat of animals at different ages, but the cost of raising and maintenance.

It is out of the question to restrict the slaughtering to the postreproductive period. The unnecessary male may, in fact, be slaughtered as soon as the cost of his upkeep exceeds the corresponding gain in marketable value of hiscarcass. Just whatisthe most advantageous time for his slaughter is a question in agricultural economics which has been discussed,among others by A. Gouin and P. Andouard. These authors compute that to bring up three calves, from weaning to the age of three and one-half years, requires about 33,000 kgm. of hay (or their equivalent); this quantity would suffice to bring

Fia, 23. DiAGRAMMATIC ILLUSTRATION OF INFLUENCE OF RANDOM AND OF SELECTIVE SLAUGHTERING UPON SuRVIVAL CuRVE OF BIoLoaicaL SPECIES seven head of cattle to the end of their second year, which would give a gain, in meat, of 40 per cent, as compared with three head brought to the age of three and one-half years. Intelligence as a Discriminating Agency. It is particularly worth while to note how in this connection human intelligence exerts its influence upon the course of physical events by substituting systematic selection in the place of the more haphazard, more random actions characteristic of the mentally less developed species.

These latter must, in many ways, remain dependent upon certain average manifestation, whereas man, in a corresponding situation, is able to discriminate and address himself directly to individual elements of which these averages are the expression in the gross. In the manifold interactions on a macroscopic scale that constitute the perennial struggle for existence on nature’s battlefield, the lower organisms are in many respects situated in a_ position analogous to man’s relation toward the ultra-microscopic elements of his environment, the molecules and atoms. These he can handle only in bulk, unable to avail himself of the distinctive features of this or that particular atom or molecule.

Just as our senses and our bodily members are inadequate for the task of handling individual atoms, so the mental or other discriminating powers of the lower organism are often inadequate to lead it toward any other than a rather random selection, and by no means an optimum selection, among the somewhat varied opportunities of self-service presented to it. However, though man does far excel the other creatures in this respect, the difference is, after all, one in degree and not in kind. Many, if not all organisms, possess in some degree the power of selection, are in some measure independent of pure haphazard. This introduces an altogether peculiar complication into the dynamics of systems comprising living organisms, a complication of which the statistical dynamics of molecular physics are free. Not only is the living organism capable of performing, on a macroscopic scale, exploits analogous to those which in the world of molecules are permitted only to such figments of the imagination as Maxwell’s demon; but this power of “cheating chance,’ as it were, is possessed in different degree by the several living organisms, and a dynamics of systems comprising living matter must necessarily take account not only of this faculty, but of the gradations in this faculty which have a large part in assigning to the several biological species their place in the scale of evolution. ‘This will require the development of special methods. It is essential that we bear in mind constantly the ultimate aim of our reflections. The transformations of matter, the change in its distribution among the components of the physical system in the course of evolution, are the first to strike the eye, and are properly the first to receive our systematic consideration, just because they are of more obvious and elementary character.

But our fundamental aim must ultimately be to gain an enlarged understanding of the dynamic relation involved, of the play of forces, the transformations of energy. : Kinetics of Intra-Group Evolution. It is not intended to attempt here a systematic discussion in any sense exhaustive of the course of events in intra-group evolution, that is to say, in the redistribution of matter among the several sub-types of which a biological species is composed. We shall however briefly note an enterprise, led more particularly by J. B. S. Haldane,“ to investigate the trend of evolution in a population in which selection is operating upon a character subject to Mendelian inheritance.

Case1. Haldane considers first of all the simple case of a species of constant total population, consisting of two types (phenotypes) A and B that do not interbreed, but react upon each other merely through competition in the same environment. Let the nth generation, counted immediately after fertilization, consist of types A and B in the ratio un: 1, and let the coefficient of selection be fk, 1.e., let (1—k) of B survive (to breeding age)'* for every one of A. Then the survivors (to breeding age) of the nth generation, and hence the first nwmbers'" of the (n-+1)th, will be in the ratio

16 The words in italics are here added to Haldane’s text, in accordance with the first paragraph of his paper, page 20. 16 The words in italics are here added to Haldane’s text, in accordance with his paper, page 23, line 6. 17 The meaning of this term is not altogether clear. It seems to refer to the total births in the (n + 1)th generation, or, in other words, to the (n-+1)th generation, counted at birth. This does not seem altogether consistent with the rest of the argument, notably the wording referred to in footnote 15. On the other hand the terms referred to in footnote 16 are vague unless reproduction takes place once and once only in the life of the individual, since in general there is not only one single age of reproduction, so that the fraction surviving to breed cannot be represented by any single number. It seems that it would be better to base the argument on the total number born in each generation, i.e., on the generation, counted at birth; and to regard the ratio R between the births in two successive generations as a function of 8 (a) and p (a), as set forth in Chapter VIII equation (24) and Chapter IX equations (35), (36). We would then start with equation (38) of page 123 as a fundamental assumption; beyond this the argument would remain essentially unchanged.

If uo be the original ratio of A:B in the Oth generation, then Un = (1 —k)“®u, (39) If we write y, for the proportion of B’s to the total population of A and B in the nth generation, then a cu 1 — Yo = Yotlo (41) Ste ets We co If we start with an equal number of births of A and B Yo=F (43) it tae Sed ans Ss If k is very small, i.e., if selection is very slow, then approximately Ya = 1 ga (45) or kn = log 1 Yn (46) In equation (45) we recognize once more the Verhulst-Pearl relation. It is seen that in the circumstances to which the discussion relates, the better adapted of the two types grows, along the typical S-shaped curve, at the expense of the less well adapted, ultimately displacing it entirely. Thisis quite in accord with what we should expect, since the total population A + B is constant, that is to say, it just holding its own. Any constituent of the population that falls below the average in adaptation, must therefore be unable to hold its own, and diminishes continually.

The curve representing graphically the change of the population—increase of type A at the expense of B, which is ultimately displaced entirely—is shown in figure 24 for the case k = 0.001, i.e., that 999 B’s survive to reproduce, for every 1000 A’s. In these circumstances 9184 generations are needed for the proportion of A’s to increase from 1 per cent to 99 per cent. Case2. Selection of a Simple Mendelian Character, with intermingling of dominant and recessive type. Haldane next takes the following case: