Elements of Physical Biology

To give a touch of concreteness to the discussion at this stage a very simple example may be given to illustrate how several equilibria may occur in a life-bearing system. A perfectly screened dwelling may be kept indefinitely free from flies. This is a condition of equilibrium, but of unstable equilibrium; for if only a few flies gain access, presently these will breed, and the room will become inhabited by a population of flies whose number will depend on the amount of food present, on the measures taken to combat the pest, etc. Unless these measures are very active, the flies will not be wholly exterminated, but the population, will attain some approximately steady number (for a given season). There are, then, in this case, two possible equilibria; one with a zero population of flies, the other with some positive number of fly population.

The equilibria in nature, involving countless species, are of course much more complicated in character, but the general principle is the same; and we must expect that in general a variety of different equilibria are possible, some unstable and some stable. For the further discussion of the equations (1) it is now desirable to express these relations, not in terms of the masses X, but in terms of the excess x; of each mass X; over its corresponding equilibrium value,

the parameters P, Q being omitted, for the sake of brevity. Ex- panding the right hand member by Taylor’s theorem we obtain the system of equations (abies ihe Amit + Gnet2 +. © «+ Gnntn+ Gants + Gnwtit2 + Gnete +.. . A general solution of this system of equation is where the G’s are constants, of which n are arbitrary, and \1, de . A, are the n roots of the equation? for \ It is seen by inspection of the solution (8) that if all the \’s are real and negative, 2, 2%, . . . x, all approach zero as ¢ increases toward infinity, since e-> = 0. But according to (5), as the x’s approach zero, the X’s approach their equilibrium values C.

In this case, obviously, equilibrium is stable, since, by allowing , sufficient time to elapse, we can always make the system approach as near as we please to X; = Cj, for all values of the subscript 7. Precisely similar conclusions hold if some or all of the \’s are complex, and the real parts of all the \’s are negative. If all the roots X are real (and negative),? each term in the solution (8) diminishes continually and approaches zero asymptotically as ¢ approaches infinity.

If all the constants G are of the same sign, the sum of the series also will, evidently have a similar type of approach to equilibrium. If some of the G’s are positive, others negative, there may be a species of irregular oscillations, the mass X of the component rising sometimes above its equilibrium value C, sometimes falling below it. Ultimately, however, the equilibrium is approached from one side only, since ultimately the term containing the numerically smallest \ will predominate over all other terms.

If some of the roots \ are complex, the solution will contain truly oscillatory terms, since the exponential function, for complex exponents, assumes the trigonometric form In this case there will be regular oscillations about the equilibrium position; in general these oscillations will be damped, that is to say, their amplitude will diminish, so that equilibrium is approached more and more closely, but always with oscillation—the equilibrium is approached from both sides at once, so to speak, the oscillations persisting forever, though on a diminishing and ultimately vanish- | ing scale.

These conclusions are the analytical confirmation and extension of an inference drawn by Herbert Spencer‘ on qualitative grounds: 3 The analytical condition that the real parts of all the roots ) shall be negative is given by Hurwitz, Math. Ann., vol. 46, 1895, p. 521. See also Blondel, Ann. de Physique, 1919, pp. 117, 153. It might be noted here that a necessary though by no means sufficient condition, evidently is that the absolute term in D (A), that is to say D (O), shall be positive when n is even, and negative when 7 is odd; for this absolute term is equal to the product of all the roots d.

4 Herbert Spencer, First Principles, Chapter 22, Section 173. Every species of plant and animal is perpetually undergoing a rhythmical variation in number—now from abundance of food and absence of enemies rising above its average, and then by a consequent scarcity of food and abundance of enemies being depressed below its average . . . . amid these oscillations produced by their conflict, lies that average number of the species at which its expansive tendency is in equilibrium with surrounding repressive tendencies. Nor can it be questioned that this balancing of the preservative and destructive forces which we see going on in every race must necessarily goon. Since increase of numbers cannot but continue until increase of mortality stops it; and decrease of number cannot but continue until it is either arrested by fertility or extinguishes the race entirely.

It will be observed, however, that our analysis enables us to be considerably more specific in distinguishing several modes of approach to equilibrium, and in indicating the particular conditions under which each occurs. A point that here deserves particular emphasis is that, in order to make these distinctions and indications, it is by no means necessary to have a complete knowledge, or even very extensive information regarding the functions F. Only the coefficients of the first (linear) terms in the Taylor expansion of these functions enters into the determinant D(A), and only these need therefore be known to draw the requisite conclusions regarding the stability and mode of approach of equilibrium. Furthermore, Spencer’s commentary relates specifically to the case of species related to each other in certain particular ways (food, enemies, etc.); the analysis here given is framed on perfectly general lines and covers any sort of interrelation, interdependence of the components S;, S. . . . S,. Certain particular interrelations will be duly considered in the course of the further development of the theme. Here it may be well to draw attention once for all to the fact that there is nothing whatever to restrict the application of the principles and methods set forth to systems in organic evolution. Indeed, a typical example to which these reflections apply is the case of a chain of elements in the course of radioactive transformation.®

It will be seen later, in considering a concrete example, that the equilibrium equation (12) may yield zero or negative roots for C. A zero root is simply interpreted; if the equilibrium to which it relates is a stable one, this means that the species in question will become extinct®. It is unfit, is unadapted ultimately’? to survive under the existing conditions’ (as defined by the parameters P, etc.). A negative value of C may also signify that the species is incapable of survival under the existing conditions. Masses cannot assume negative values. As soon as any component passes through zero it ceases to function in the system, whose history is henceforth represented by a new set of equations in which this component does not appear.

These conclusions must in one respect be accepted with a certain caution. Since there may be several equilibria, a species may be incapable of existence in the neighborhood of one such equilibrium, but might nevertheless succeed in maintaining itself in the neighborhood of another equilibrium. Whether such cases occur in practise may be left an open question. Our analysis suggests this possibility. 6 This conclusion does not apply, of course, if the equilibrium with X = O is unstable, as in the example of a fly population cited above.

7It may, however, persist for a time, and, it may be, for a long time. An instance in point is furnished, outside the field of organic evolution, by a chain of elements in radioactive transformation. Although here the ultimate equilibrium is one with a single survivor, namely the last link in the chain, yet for millions of years the several products exist side by side in constant ratio though in slowly diminishing amount. At the head of such a chain is always found a ‘“‘parent substance” which is the longest-lived in the chain. This is not accident. 1t is easily shown that if at some prior period this substance was preceded by a more rapidly decaying pre-parent, this latter must have disappeared in the course of the ages. It is totally unfit, unadapted, even for a temporary equilibrium, under present conditions. (See A. J. Lotka, Phil. Mag., August, 1911, p. 354.)

8 It should hardly be necessary to point out that adaptation is purely relative. If arithmetic, mensuration and weighing be taken away from any art, that which remains will not be much—Plato. Law of Population Growth. It will add concreteness to the present exposition to consider at this point a numerical illustration. The simplest possible example of numerical application of the equations set forth in the preceding pages will be one in which there is only a single variable X. The fundamental system of equations then reduces to a single equation

A case in point arises when for any reason one particular biological species or group grows actively, while conditions otherwise remain substantially constant. This seems to be essentially what has occurred in certain human populations. It is true that, in their growth, they have carried along with them a complicated industrial system or group, comprising both living and non-living elements. We may, however, look upon the number of the human population itself as a sort of single index or measure of the growth of the group as a whole.!

1 This is an example in which certain of the variables X are connected by equations of constraint. So, for example, the number of head of cattle Nec has for many years past, in the United States, been about six tenths of that of the human population, Nh so that we have an equation of constraint or, putting the average mass of cattle at 1000 pounds per head, that of ahuman being at 100 pounds (an average to include all ages) Similar equations of constraint apply to the other species of domesticated animals and plants, so that the mass of each can be (approximately) expressed in terms of the single variable Xn.

Applying to this case the equation (1) and the general method set forth in the preceding pages, we are led to consider first of all the equilibrium equation This equation, obviously, has a root at X = O, for at least one female is required to start the growth of a population. Expanding by Taylor’s theorem we shall therefore have for F a series lacking the absolute term,? thus Furthermore, the equation (2) will have at least one other root, since there must be some upper limit to the growth of the population. The simplest case satisfying this condition is that in which the right hand member of (3) terminates at the second degree term, 1.e.,

oP = aX + bX? (4) The characteristic equation’ for is in this case simply A-a=O (5) and the solution of (4) therefore is X = Gyett + Gye2*t + Guye®tt. 2. (6)4 Substituting this in (4) and equating coefficients of homologous terms we find 8 Corresponding to equation (9), Chapter VI. ye 4 The series (6) is divergent for large values of ¢ if a is positive. But the expression (9) for the sum of (6) remains a solution of (4). In formula (9) either G, or the origin of time is arbitrary. Asimplification can be effected by so adjusting the origin of time that

the symbol ¢’ denoting time reckoned from the origin indicated.’ The equation (9) can also be written in anotherform. If we denote Population of United States. Formula (12) has been applied by Pearl and Reed® to the population growth of the United States.’ The calculated curve for the number N of the population fits the observed data over a long period of years (1790 to 1910) with remark- 5 This result can also be obtained by direct integration of (4) in finite form. The process given above has here been followed to illustrate the general method of the solution (8), (9), of Chapter VI.

7 Measured, in this case, by the increase in the number N of persons. This is evidently, in first approximation at any rate, proportional to the total mass X of the population. able faithfulness, as will be seen from table 2 and the graph shown in figure 4. Numerically the formula (12) here takes the form and the time ?’ (in years) is dated from April 1, 1914 (t’, being negative for dates anterior to this). This epoch is one of peculiar interest. It represents the turning point when the population passed from a progressively increasing to a progressively diminishing rate of growth. Incidentally it is interesting to note that if the population of the

Results of fitting United States population data 1790 to 1910 by equation (14) United States continues to follow this growth curve in future years, it will reach a maximum of some 197 million souls, about double its present population, by the year 2060 or so. Such a forecast as this, based on a rather heroic extrapolation, and made in ignorance of the physical factors that impose the limit, must, of course, be accepted with reserve. Stability of Equilibrium. The equilibrium at X = 0, i., with total absence of human population, is evidently unstable, since \ = a and a is an essentially positive quantity, its numerical value being, for the population of the United States, 0.0313395. The second equilibrium, corresponding to the saturation point, is evidently at

a -— a = and here it is easily found by the substitution x = X + ? Fig. 4. Tar Law or Poputation GROWTH FOR THE UNITED STATES ACCORDING To PEARL AND RexEp The lower S-shaped limb corresponds to the actual approach to equilibrium from below. The upper limb represents the presumptive course of events for a diminishing population approaching equilibrium from above. Experimental Populations. Pearl has also fitted the same formula to the population of a number of countries, but the range covered by the available observation in these other cases is less extended, so that there is less opportunity for comparison with observed figures. Of particular interest is an application of the same formula, also by Pearl, to an experimental population of fruit flies (Drosophila). In this case practically the entire range of the S-shaped curve defined by equation (12) is realized, and a glance at the plot in figure 5 shows that the agreement of the observed figures (represented by small circles) and the calculated curve is exceedingly satisfactory. Still closer is the agreement in the case of bacterial cultures studied

Fia. 5. GrowrH oF A PopunaTION OF DROSOPHILA (FRUIT FLIES) UNDER ConTROLLED ExPERIMENTAL CONDITIONS, ACCORDING TO PEARL AND by H. G. Thornton’ (Annals of Appl. Biology, 1922, p. 265), whose observations are set forth in table 3 and are shown by the small circles in figure 6; the theoretical curve to fit these points, as computed here in the laboratory, is shown in the fully drawn line. As will be seen the agreement is excellent. This is due in part to the fact that the figures plotted represent the means of a number of individual cultures.

A very particular interest attaches to this example, inasmuch as it forms, as it were, a connecting link between the law of growth of a population, and the law of growth of the individual. A colony of unicellular organisms, regarded as a whole, is analogous to the body of amulticellular organism. Or, to put the matter the other way about, a man, for example, may be regarded as a population of cells. We need not, therefore, be greatly surprised, if the growth of the multicellular organism should be found to follow a law similar to that exhibited by populations. And in point of fact, as will be shown a little further on, this expectation is in not a few instances fulfilled.

_ Diminishing Population. It may be noted here that in one respect / the formula (12), while particularly simple in form, is of more restricted scope than (13). The former defines the characteristic S- | shaped curve that appears in the graphs of actual populations shown ) in figure 4. But formula (13) gives a more complete definition of a | curve composed of two limbs; one of these is the S-shaped curve already considered. The other is a steeply descending arc, shown in the upper portion of figure 4. This portion of the curve has not been

realized in any recorded population. It represents the computed course of events if the population initially exceeds its equilibrium Fia. 6. Grows or A Bacrnrtau Coitony (B. DenprRorpss) Observations by H. G. Thornton Growth of Individual Organism. Although, strictly speaking, the growth of the individual organism is a subject properly belonging to the field that has here been termed ‘‘micromechanics”’ (Chapter V), yet, in view of the very close analogy which has been found to exist, in certain cases, between the law of growth of a population, and that of the individual, it may be noted here that the formula (9) has also

been applied by T. B. Robertson’ and by Wo. Ostwald? and others, to the growth of the individual organism. An example of such application is shown in figure 7, which exhibits the growth day by day, of male white rats according to observations by H. H. Donaldson" and computations by T. B. Robertson. The fully drawn curve represents the calculated values of the mass of the rats (in grams) at different ages. The circles indicate selected observed values, namely, those which diverge most widely from the computed values. It will be seen that up to about the one hundredth day the agreement is good. Above this there is no agreement worthy of the name.

Another example, and one in which the computed curve fits the observed values with remarkable agreement, is the growth (in height) of sunflower plants as studied by H. 8. Reed and R. H. Holland.?? The curve to fit these observations has been recomputed by the method of least squares by Dr. L. J. Reed, who has very kindly placed his results at the author’s disposal. They are shownin figure8. The observations on which they are based are shown in table 4. It will be seen that the fit is practically perfect through the whole range of observations.

In practice we are not usually given the differential equation (4), but data corresponding to points on the integral curve (12). We then have the problem of determining from these points the characteristic constants of the curve. The detailed working out of the prob- * Archiv fiir die Entwickelungsmechanik der Organismen, 1907, vol. 25, p. 4; 1908, vol. 26, p. 108; The Chemical Basis of Growth and Senescence, publ. Lippincott, 1923. T. B. Robertson also quotes A. Monnier, Publications of Inst. of Botany, Univ. Geneva, 1905, which in turn refers to Chodat as having recognized the analogy of organic to autocatalytic growth. The idea is rather an obvious one, which probably has oceurred to many. The earliest reference noted by the writer is L. Errera, Revue de l’Univeristé de Bruxelles, 1899-1900, May issue. Something very similar is found in Ostwald, Vorlesungen tiber Naturphilosophie, 1902, p. 342. These last-mentioned lectures were published in 1902, but were actually delivered somewhat prior to that date. The writer recalls that Ostwald referred to the matter in his lectures on General Chemistry in 1902, and probably others will recall similar references on earlier occasions,

10 Die Zeitlichen Eigenschaften der Entwickelungsvorgiinge, Leipzig, 1908. lem may well be left to the reader, after pointing out the following interesting property of the curve (12). Taking reciprocals and writing A for a/b, we have Fig. 7. GRowtH oF Rat Accorpina To H. H. Donaupson anv T. B. Circles indicate only those observations that diverge most widely from the calculated curve. where ¢, is the time corresponding to the point of inflection of the S- curve. Thusif we plot A é — 1 against ¢ on logarithmic curve paper, we shall obtain a straight line diagram. ‘This is shown in figure 9 for the same data (growth of sunflower) which have already been

Fig. 8. GrowrH oF SUNFLOWER SEEDLINGS AccorDING TO H. S. REED AND R. H. Hotzanp; Computep Curve By L. J. Rnzep Growth in height of sunflower plants (H. S. Reed and R. H. Holland;* computed values by L. J. Reed) The same data as in figure 8, but plotted in logarithmic diagram, with ordinates as indicated in the legend. exhibited by another method in figure 8. Taking now three equidistant points of time ¢; f: fs and the corresponding ordinates £1, &, &,1t is readily shown that