Elements of Physical Biology

As regards the formulation of the laws of evolution in form of a maximum or minimum principle, it should be remarked that one such principle follows directly from the fundamental equations of kinetics as set forth in Chapter VI. If we multiply the first of these equations by Xi, the second by Xe, and so on, we obtain where Q represents a quadratic form. The relation thus obtained is not of general utility in this form. However, by a linear substitution

where Ai, A2,. . . A, are the n roots of the characteristic equation for , the same d’s that function as exponents in the series solution of the original system of equations.’ Now it will be recalled that the condition for stability at the origin is that all the real parts of the roots shall be negative. But in that case the quadratic form Q’ is definite and negative. Hence the condition for stability at the origin can be expressed by saying that the quadratic form Q’ must be definite and negative; or, by saying that Q’ must have a minimum at the origin. And the law of evolution, near the origin, evidently is, according to (20), that Dé? continually decreases. (At points remote from the origin the terms of higher order, which have here been omitted, may cause increases in Z£.)

The chief interest of the minimum principle here indicated lies in its analogy? to certain theorems in dynamics and thermodynamics, for which reference must be made to the literature, in particular to P. Duhem, Traité d’Energétique, 1911, vol. 1, pp. 460 et seq.; F. Michaud, Ann. de Phys., 1921, vol. 16, pp. 148 et seq. 8 For the sake of simplicity the argument has here been presented in the form in which it appears when all the roots ) are distinct and real. For a detailed discussion of the conditions of stability when some of the roots A are multiple or complex see E. Goursat, Cours d’Analyse, 1915, vol. 3, pp. 31-43.

® The analogy to the dynamical cases treated in the reference cited becomes particularly plain if we bear in mind that where the bracketed exponent (2) denotes the symbolic square, in be ie —)i — —— laced byy——— ee) is replaced by OE’ and the product de Og; is replaced by DEE; The condition that the form so defined shall be negative, is that the determinant shall be negative, and also all determinants derived from it by striking out the last p lines and the last p columns.

In the present case the same condition, can be expressed in simpler form to the effect that Ai, Ae, . . . An must all be negative. But it is worth while, in order to bring out the analogy, to note also the more complicated general form of the condition. Since the struggle for existence is chiefly a struggle for subsistence, a careful comparative account of the food of various competing species and genera at different places and seasons and at all ages of the individual . . . . cannot fail to throw much light upon the details, causes and effects of the struggle.—Forbes.

Equilibrium Condition in More Particular Form. The fundamental relations of statics are derived immediately from the corresponding equations of kinetics by substituting in the latter the value zero for the several velocities. This has already been noted with regard to the equations of kinetics in their most general form. In somewhat more particular form, useful in common numerical applications, we have a condition for equilibrium derived from the system of equations (14) of Chapter X

It should be noted that these formulae hold equally well if the masses are measured in ordinary units (e.g., pounds) or if they are measured in “head of population,’’ with the proviso, of course, that the coefficients uj, vi, are in each case expressed in corresponding units. The equation (1) expresses the fact that, for each component, the total inflow is just balanced by the total outflow, so that nowhere in the system is any accumulation of mass going on. This clearly implies that, unless there is complete equilibrium, the matter in the system must be in circulation, it must be going through one or more cycles. Such cycles are, indeed, very characteristic features in the scheme of nature.

Numerical Illustration. We may, by the way of illustration, apply the formula (3) to the equilibrium between the several biological species comprised in a life-bearing system. For obvious reasons numerical data are most readily available for man and the species directly under his control. So, for example, we may let X; represent the mass (or number) of a human population, and X; the mass (or number) of a population of sheep serving as food for that human population. In the United States in 1918 the consumption of mutton (or lamb) per head of the population per annum was 5.417 pounds. This is not strictly an equilibrium ration, since our population is increasing. However, the difference between this and the equilibrium ration is probably small. In our example we will therefore put

Again in the United States in 1918 the number of sheep slaughtered per annum was 23.22 per cent of the standing herd. Hence 1 According to the Year Book of the Department of Agriculture, 1920, p. 759, the number of sheep slaughtered under Federal inspection in 1918 was 8,769,498. According to R. Pearl, The Nation’s Food, 1920, p. 61, this represented 77 per cent of all the sheep slaughtered in that year, so that the total number slaughtered was 11,370,000. The total dressed weight of these, according to the Year Book, p. 826, was 562,214,000 pounds, which makes the average of one sheep carcass 49.45 pounds.

The standing herd of sheep in 1918, according to the Year Book, p. 701, was 48,603,000 on farms, or, adding a correction for animals not on farms, say 48,963,000. The percentage of animals slaughtered in a year out of the standing herd, was therefore 23.2216 per cent. For a review of various estimates of the output of herds of cattle, sheep, swine, etc., the reader is referred to a paper by R. H. Rew in the Journal of the Royal Statistical Society, 1902, vol. 65, p. 666.

The actual standing herd of sheep in 1918 was 48,963,000 head of which 90,000 head furnished mutton for export, the United States not being a self-contained system. It is to be noted that in this example the products yi ui and a; Vi are more easily ascertained than the individual coefficients y u,a,v. Inasmuch as these coefficients, in the formula (3), appear only in these products, it is not necessary, for the purposes of this example, to ascertain the values of the coefficients separately.

The example of the equilibrium between a human population and the national herd of sheep, cited primarily for the purpose of illustrating the equilibrium equation (3), incidentally brings out some other points that may be noted in passing. We meet here a pointed suggestion of economic factors entering into play in the processes which wearestudying. For the coefficientsa y, u,v, have obvious economic relationship. So, for example, v;, the proportion of sheep slaughtered per annum (in a stationary state of the system), is of the nature of interest on the standing herd, which latter, in turn, is of the nature of capital. The gross interest rate of 23.2 per cent is, of course, greatly diminished, in effect, by the extensive accessories, representing a further investment of capital, and by the general running expenses required, in addition to the mere herd, to produce, transport, and market the ware. On the other hand, certain secondary products (e.g., wool) add their quota to the returns on the invested capital.

Again, the coefficient \i1, which measures what fraction of the total consumption by the component S; (human population) is derived from S; (sheep), is clearly a factor of economic significance. Mutton is typically a commodity of the replaceable type—beef, pork, fish, etc., furnishing ready substitutes. In such case as this the factor vir will be elastic, capable of assuming, in ready response to slight changes in general conditions, a whole range of values from zero up. In the case of less readily replaceable commodities the coefficient will be of more rigid habit.

The full significance and the precise nature of the relation between the biological and the economic characteristics of a system must form the subject of special considerations to which we shall find ourselves led inevitably later, in our efforts to gain insight into the dynamics of life-bearing systems. At this juncture it may not be amiss to indicate in preparation of a viewpoint to be more fully developed later, that if economic factors force themselves upon our notice primarily inthe

consideration of systems comprising a human population, this is not because the operation of economic stresses is peculiar to human aggregations, but only because these stresses find their ready numerical expression and measure in such communities; owing to the development of a system of social codperation and division of labor, coupled with a very special mechanism of adjustment by ‘economic exchange,”’ which is peculiar to man. For though not a few other species, bees, ants, etc., display a social organization in some respects perhaps superior to ours, their organizations make use of other expedients than the transfer of ownership through a universal medium of exchange, in bringing about the allocation, to each individual, of his share in productive effort and in product.

For obvious reasons our information regarding the interdependence of the several biological species and other components of our lifebearing system is most complete and most exact in so far as it relates to species under human cultivation, species that contribute, as producers, to our political economy. However, this does not mean that we are wholly cut off from all information regarding the life balance of other species. Two sources, two methods of observation furnish us with data on this subject, namely, first, biological surveys, and second analyses of the stomachs of sample specimens. A third method would consist in establishing experimental systems comprising several species of organisms and making periodic censuses by direct count, after the manner of the work of Pearl and Parker with a single species (Drosophila). There is here an attractive field open for research. Perhaps the readiest, though not the most interesting approach to this problem would be the study of mixed bacterial growths, say along the lines followed, for a single species, by H. G. Thornton, Annals of Applied Biology, vol. 9, 1922, p. 241.

Biological Surveys. Biological surveys, supplemented by estimates depending more or less on personal judgment, are aimed to give us some degree of quantitative description of our world by investigating both the number and the variety of living organisms. A perfect biological survey would enumerate the several general and species found in the locality examined, and would furthermore give us ameasure of the extent of each species, either in numbers orin some other suitable terms. It would give usa species of ‘“General Demology” of our globe. Needless to say, in this matter we are very far from having attained perfection. The best that can be done is to

give rather crude estimates, based, in the most favorable instances, on counts or observations made with some degree of care, but without pretense of great precision. As to the number of species, some interesting figures are given by J. A. Thomson.? On the small island of Britain alone 462 different birds have been observed; the total number of living species of birds he estimates at not less than ten thousand. Of vertebrates he quotes an estimate by H. Gadow.’ The total number of recent species this author puts at 24,241, as follows:

UIA hos ae ecln Denes ek, wk Saws tc ati acta oe heat 2,702 | BG Es eae SA Se Reni arts am io Te Cae ee ae eee ee ae oe 9,818 PRED OS ener ee ele re s 4,5 ite ee creed te ei oleisrih hime oe 3,441 RING MAMA. Metis Mae Soe IoC ene ania en Len tie ee 925 IETSSEL CH Sarai asta ct tee IE Sieve Seen Oe Rede is vie os Una ee me ao 7,328 PPM VONvey UC DRATON yon sel. enge hist oe fal ecto nie code ee 27 The vertebrate élite, however, forms but a small minority in the scheme of nature. It has been intimated that, if the present order of things should come to a term, the supremacy would, as likely as net, pass from the crowned vertebrate Homo sapiens to the now despised, presently perhaps to be feared, creeping thing, the insect. Indeed, it has been pointed out that were it not for the relentless internecine warfare which its members carry on among themselves, we should very soon find ourselves driven out of house, home and granary by the insect pest. Even as it is, though the largest insects barely exceed, individually, the size of some of the smallest vertebrates, yet, as D. Sharp remarks, ‘‘the larger part of the animal matter existing on the lands of the globe is in all probability locked up in the form of insects.”’ He estimates the number of insect species that have been definitely named, at 250,000, and suggests that this is only about a tenth of the total. The number of plant species has been estimated at 200,000. Darwin records the finding of 20 species in a patch of turf four feet by three.

As to the numerical strength of the several species, here again some telling figures are given by J. A. Thomson.‘ At the spring maximum of the Rotifer Synchoeta there may be about three millions to a square yard of lake. At the summer maximum of the slimy Alga Clathrocystis ceruginosa there may be 500 millions to the square yard; at the autumn maximum of a well-known diatom Melosira varians, which has a summer maximum as well, there are about 7000 millions to the square yard, so that the waters of the lake form a veritable living soup. . . . . In an ordinary sample from a warm part of the Atlantic and from a depth of 500 metres (which is the most densely populated as far as plants go), there are likely to be about 5,000 plant-cells to a liter; but there may be as many as a quarter of a million.

Elsewhere Thomson tells us that in the midst of a swarm of fish at spawning time in the Norwegian fjords a boat may be so densely packed in among the mass of fish that an oar stuck upright into the swarm remains standing for an appreciable time after the hand relinquishes its hold.® Examination of Stomach Contents. The second method by which information has been gathered regarding the interdependence of biological species consists, as already stated, in examining the contents of the stomachs of sample specimens. This method has been applied particularly to birds and fishes. The results of such an analysis of the feeding habits of the common crow and of the starling are strikingly brought to view in the accompanying charts figures 31 and 32, reproduced by courtesy of the Department of Agriculture from Bulletins 868 and 1102. Such a chart does not, of course, yield any direct information regarding the relative abundance of the several species upon which the crow feeds, but it does give us at least an indication of a resultant compounded of that relative abundance and a number of other factors, such as the preference or selective tastes of the crow, the greater or less degree of protective characters with which nature has endowed the various species exposed to attack, ete.

The converse problem, also, has been investigated, namely the extent to which different species of birds participate in the destruction of one selected noxious insect. So, for example, Bulletin 107 (1914) of the Department of Agriculture lists 45 different species of birds that were found to have fed upon the alfalfa weevil. A similar study by H. C. Bryant® was carried out during a grasshopper outbreak in ’ For an account of bird censuses in the United States see M. T. Cooke, Bulletin No. 1165, Bureau of Biological Survey, U.~S. Department of Agriculture,

California, when “in the infected areas the grasshoppers were computed to number from 20 to 30 per square yard.’’ Bryant’s results are shown, in part, in tabular and diagrammatic form in figures 33 and 34, reproduced from his original publication. Jy =] | Fig. 32. SEASONAL ANALYSIS OF THE SromacH CONTENTS OF STARLING The same author has also given us a classic in his extended study ‘“‘A Determination of the Economic Status of the Western Meadow Lark in California.” This paper contains among other things a

detailed bibliography up to 1913, and a historical survey of the methods employed and the investigators who have labored in the TOTAL DESTRUCTION OF GRASSHOPPERS PER SQUARE MILE DAILY, 120455 Fig. 33. Comparative Darty DestRUCTION OF GRASSHOPPERS BY SEVERAL Species oF BrrRps Fia. 34. Spasonat Foop Hasits or THE Mpapow LARK After H. C. Bryant field. | The results obtained by Bryant exemplify the proverbial voracity of birds. Young birds require about one-half their own weight in food each day. In the course of a year the average meadow

ep pores say, 4 ounces, consumes about the following quantities of food. oy RS, CT oa Ie Pahe ate re PS rH, 0477 © ye Pee ete ly KRY eas yf? ey Maly VP YsgVtartala-d*| wen Con Nene Ree Seok ee a &2@ am \ &/ EES SAS Crd WR ee 9 ea% ave tots an Verads SVNQNT PP - yas aN Me EGA RARE AES CAS TR Cogs raw rer)prtlh & Pinwea rARmiktl B ear er arkerns? a7 ?ee\ - KRDRP OE ace Beetey A \ me!) Coe Net SE Corte Ve he dee a ie ( are Pr AIO IN Sn PaVvQd iS o& 7% Peresgs asemstcQeV¥as VU) sore Bay CU BAIN NC YCP US OPP wat AV DENBS A VVELS OVE Ly of veer Fi eg AOI er tetten Rartieleh et ee oe ee 5 eee 66S = oP

Since an adult bird weighs about 4 ounces, this meansthat it consumes on an average about 24 times its weight of food in a year. Dr. Bryant remarks: “If we consider that there is an average of one meadow lark to every two acres of land available for cultivation (11 million acres) in the Sacramento and San Joaquin Valleys, and that each pair of birds raises an average of four young, it takes over 3434 tons of insects each day to feed the young birds in the valley alone.” His findings regarding the seasonal changes in the food of the meadow lark are summarized in a number of charts of which one is reproduced in figure 38. An illustration of the voracity of birds, and their destructiveness to insects is also seen in figure 35 (from Bulle-

tin 107 of the Department of Agriculture) showing the stomach contents of a Brewer’s Blackbird, for this bird was found to have gorged upon 374 larvae, 65 pupae and 3 adults of the alfalfa weevil. Observations on the feeding habits of young English sparrows are recorded by E. R. Kalmbach in Bulletin 107 of the United States Department of Agriculture. This author remarks (p. 54): From a series of five observations it appears that the parent English sparrows visited their nest on an average about once every 53 minutes, or a little more than 11 trips an hour. The four adults captured had as food for their young 2 kernels of wheat; 17 alfalfa weevil larvae; 1 ground beetle, 9 weevil larvae and a caterpillar; and 28 weevil larvae, respectively. Three other adults taken in the fields had food for nestlings in their bills. This amounted to 18 weevil larvae and an aphid in the first, 5 larvae in the second, and 3 coccinellid larvae, 13 weevil larvae, and 2 pupae in the third.

Though this is rather heterogeneous assortment, it would appear that 15 larvae of the weevil or their equivalent in bulk of other insects would be a fair estimate of an average amount of food brought in at each trip by adult birds. In fact, it is certain that the material brought in frequently greatly exceeded this amount. Allowing 15 larvae at each trip and 11 trips per hour, these birds would bring in 165 larvae per hour. Then, assuming that the young were being fed for 12 hours each day, a conservative estimate, we would have a total of 1980 larvae consumed by one brood in one day. Straw-thatched sheds containing upward of 100 nest holes, both old and new, are frequent, and it is not uncommon to find farmyards where this number of nests are occupied. There are also ample nesting sites about the other buildings and in the ever-present Lombardy poplar, cottonwood, or box elder. Such a colony of birds would devour a daily total of 198,000 larvae, or an equivalent bulk in other food. As the young birds remain in the nest for at least 10 days and are probably fed several days longer by the adults, they will have eaten food equivalent to the bulk of 1,980,000 larvae during their nestling life.

Intra-Species Equilibrium. As has been remarked on a previous occasion, it is not intended, in this volume, to take up the discussion of the evolutionary changes within the confines of a species. Passing notice may, however, be given to the fact that the equilibrium within a Mendelian population has been discussed by G. H. Hardy’ and by R. C. Punnett’ and latterly by J. B. S. Haldane.® (See p. 122.) Aquiculture is as susceptible to scientific treatment as agriculture; and the fisherman, who has been in the past too much the hunter, if not the devastating raider, must become in future the settled farmer of the sea, if his harvest is to be less precarious.—W. A. Herdman.

From what has already been set forth the direct economic importance of studies in general demology should be sufficiently clear, if this term be used to denote the quantitative study of the population of the several species of organisms living together in mutual interdependence through their food requirements, feeding habits, and in other ways. In no other field, perhaps, has the study, from this angle, and under this economic impetus, been so systematically undertaken, as in the biology of aquatic species. On the one hand the threedimensional extension of the systems here involved (as distinguished from the essentially two-dimensional spread of land species over the earth’s surface), facilitates, in certain respects, the operation of sampling (by the use of the dragnet) and counting; on the other, the close relation of such investigations to the practical problems of our inland and our ocean fisheries has furnished alike the economic occasion and the financial support for work on an extended scale. The methods employed have by this time developed into a more or less standardized technique. The dragnet, already referred to, and the stomach and gills of fish, acting, as it were, as natural dragnets, themselves caught within the collector’s man-made dragnet, are among the principal accessories in this field of investigation. L. H. Tiffany recommends particularly the gizzard shad as a convenient collector and sampler of Algae.'—Contrasting these natural samplers with man-made contrivances he remarks: