Elements of Physical Biology

The remarks of W. R. Thompson relative to this may also be quoted: Recent studies on the utilization of entomophagous parasites seem to show that the rdle of these auxiliaries of man finds its maximum effectiveness when the noxious insect has increased in numbers to the point of a plague, one or more of the factors of natural equilibrium having somehow failed to act as a check. The expansion of the noxious species then automatically produces an increase in the number of parasites; generation

Fig. 14. Coursz or Parasitic INVASION oF Insect SpeciIES, ACCORDING TO Lorxa; More Exact TREATMENT after generation this number increases at the expense of the host, until it first equals, and presently surpasses the number of the host species, and finally almost annihilates the host; but then comes a moment when the parasite population, having grown to excess, largely disappears for the lack of food ° L. O. Howard, Bureau of Entomology, Technical Series No. 5, 1897, p. 48. It will be noted that the analysis given by W. R. Thompson fails to give any indication of this oscillatory process.

In this elementary discussion the terms of second and higher degree in N, and N2 in equations (27) have been neglected, as in (29). When they are taken into account it is found that, in general, the system of closed curves in figure 13 is replaced by a spiral winding about the second equilibrium point (see fig. 14). The process is still oscillatory, but, in general, it is the nature of a damped oscillation. For detail of this more exact treatment the reader must be referred to the original literature.!°

Annihilation of One Species by Another. The preceding example was suggested by W. R. Thompson’s paper in the Comptes Rendus. Another case, leading to equations of the same form, had been previously treated by the writer, namely the following: A species S, feeds on a species S;, which, in turn feeds on some source presented in such large excess that the mass of this source may be considered constant during the period of time under consideration. Then we have the obvious relation

destroyed by| | matter elimi- (37) Sq per unit of nated from S; time per unit of The first term in the right hand member, Mass of newly formed Si per unit of time, evidently must vanish with X,, and we shall therefore write this term b,X,, where b1 will, in general, be a function of both X,and X2. Similarly, and for the same reasons we shall write for the third term, excretory matter, etc., eliminated from S; per unit of time, d,X;. Contracting these two terms into one we shall write for b:X, — d,X, the single term 7,X:. The middle term, mass or S; destroyed per unit of time by So, must evidently vanish with either X, or X2 separately. We shall accordingly write this term kX,X2, so that the equation now appears dx

For the species S,; which feeds on S; we have a corresponding relation Mass of newly Rate of informed S2 per Mass of Sz crease of X2| junit of time destroyed or per unit off (derived from{ |eliminated per (39) (time Si as food inunit of time gested) which, for reasons precisely analogous to those set forth above with reference to equation (38), we shall write a = KX\X_ — dX2 (40) It will be seen that these two equations are identical in form with those discussed in the preceding example, and it is therefore unnecessary to repeat the analysis there given. But one point invites attention, and will naturallylead us on presently to the consideration of a case in three dependent variables, which offers certain features of special interest.

Let us leave aside all other considerations, and restrict our enquiry here to the following question: Is it possible, under the circumstances represented by our equations, for the hostile species S; to exterminate completely the species S; upon which it feeds? This would, of course bring in its train also the extermination of S.. dX, Xz (KXi — dz) is satisfied by the particular solution X, = O for all values of X2 (42) Hence if we plot the integral curves of (41) in rectangular codrdinates, the axis of X> is itself one of the integral curves. Now if n, k, K, and d; are single-valued functions of X,, and Xz, as it is reasonable tosuppose that they are, then any point in the X,X» plane is traversed qx. is uniquely determined by (41). It follows that no integral curve can cross the axis of X, and therefore X, can never fall to the value zero,

it can be zero only if it had that value from the beginning. The food species cannot, therefore be exterminated by the predatory species, under the conditions to which our equations refer. This argument fails, however, at the origin, where the derivative dX,/dX» assumes the indeterminate form O/O. But in the neighborhood of the origin, where second degree terms are negligible, we have, essentially, from which it is seen that in the positive quadrant the integral curves always slope downward from left to right near the origin. They cannot, therefore, in the positive quadrant, cross the axis of X_ at the origin, any more than at any other point, and the conclusion is now fully established, that the species S. cannot exterminate S; under the conditions here considered.

A word of caution, however, is perhaps in order. Although S2 cannot exterminate S;, it may so reduce the latter in numbers as to render it very vulnerable, and liable to extinction from those other influences which have deliberately been ignored in the development of the equations. Case of Three Dependable Variables. A singularly interesting conclusion is reached when we enquire what may be the effect of introducing, into the system discussed in the preceding section, a third species, which also serves as food for S2, so that this now has two sources to draw upon, after the pattern

One would perhaps naturally suppose that this alternative and additional source would in some measure protect S; from the attacks of Sz. But we shall see presently that this is not necessarily the case, that, in fact, in certain circumstances, the introduction of the third species S; may bring about the extermination of S;, from which, as we have seen, S, is naturally protected by the diminution of growth forced upon S: when this species too greedily consumes its sole source of food.

The equations for this case are similar to those for the preceding, except that we must add a third equation for the species S3 Jacl Pas Sees dt = T3A3— X2X3 (44) and add a term to the equation for S, to represent the consumption of the species S; as food by S:2, so that this equation becomes dX, XX, 1 ee A oe a and, if X; is sufficiently large, the projection of the integral curves upon the X,X2 plane will slope wpward from left to right in the positive quadrant. Such an integral curve may therefore cut through the X2X; plane, that is to say, the species S1 may be reduced to zero. Similarly, near the axis of X; dX, iA £4 3 T3 dX, X:KX,—d,

and hence an integral curve may cut through the X2X,, plane, thus reducing X3 to zero. This observation is of practical interest. It has been pointed out that in sea fisheries the accompanying presence of a common fish may cause the extermination of a rarer species which, were it present alone, would be protected by its very scarcity, since this would make fishing unprofitable. But the more abundant fish continues to render a balance of profit from the trawling operations, and thus the rarer species, so long as any of it remains, is gathered in with the same net that is cast primarily for common species.

Replaceable and Irreplaceable or Indispensable Components. The last two examples present an illustration of a point which calls for brief comment. So long as the species S: has only one source dX : of food, Si, it is to be observed that rg becomes negative as soon as X; is zero (see equation 40). In this sense S, is an essential component of the system, relatively to So, i.e., it is indispensable for the growth and even the mere continued existence of S:; On the other hand, when two or more sources, such as S; and S3 are provided, the vanishing of either X; or X; singly does not bring : : dX:

with it a negative value of a The components S,; and S; can more or less effectively replace, act as substitutes for, each other. When the feeding species (S2 in the example) is the human species, the facts indicated above find their expression in economic terms. It is an elementary fact of common knowledge that among the varied materials which the human race requires for its growth and sustenance are many that are more or less readily interchangeable. So, for example, a deficiency in the wheat crop may be in some degree met by increased supply of potatoes or other starchy food; or a deficiency in beef may be compensated by increased production in pork. On the other hand, there are certain requisites that are irreplaceable, and therefore absolutely essential. The most obvious example of such is our supply of oxygen.

In terms of our general analysis these facts would be expressed somewhat as follows. If we survey the various components whose masses X}, X2 . . . X, appearin the function F; dX; im re VCR, Age os eoetad we may effect a classification by first of all dividing them into two classes, namely those that adversely affect the rate of growth of Xj, that is to say, those for which 0 dX; dX; dt and those that promote the rate of growth of X;, those for which 2 aX OX, dt

Among the latter components, those favorable to the growth of Xj, there is a special class distinguished by the following property: aa X, is the mass of a component of this special class, then 7 q 18 invariably negative as soon as Xy is zero, no matter what may be the masses X;, X2... . of the other components. Xx is indispensable for the growth and the sustenance of X ee) © an essential of irreplaceable component, relatively to Xj. The ground upon which we are here treading is evidently close to the biological basis of economics. A detailed analysis of the relations involved belongs to the domain of the dynamics of lifebearing systems, and will in due course be considered in its natural place.

Limiting Factors. In general the several components that promote the growth of the component of S; will be presented in varied abundance. If one essential component (or a group of components which jointly are essential) is presented in limited amount, any moderate increase or decrease in the ample supply of the other components will have little or no observable influence upon the rate of growth Fj of S;. An essential component presented in limited supply thus acts as a check or brake, as a limiting factor, upon the growth of Sj. The significance of such limiting factors seems to have been first pointed out by J. Liebig:"! ‘‘Der Ertrag (des Bodens) ist von dem im minimo in ihm enthaltenen Nahrstoff abhangig.”’ And again:

Fiir die Wiederherstellung der Ertrige der durch die Cultur erschépften Felder durch Stallmistdiinguug ist die Zufuhr von allen den Nahrstoffen welche das Feld im Uberschuss enthalt, vollkommen gleichgiiltig, und es wirken nur diejenigen Bestandteile desselben giinstig, durch welche ein im Boden enstandener Mangel an einem oder zwei Nihrstoffen beseitigt wird. Limiting factors not only set certain bounds to the growth of the components to which they are thus related, but are competent also to give rise to the phenomenon of “moving equilibrium” the discussion of which is reserved for a later section dealing with equilibria generally, under the heading of Statics.

This chapter may fittingly be concluded with table 6, which exhibits the principal modes of interdependence of biological species, and summarizes the analytical characters of these, as discussed above. sisorquihis joowoporshy gq UIST}ISVIV = = f susorgufis yonzirA (sdoxo (090 “Qinay pyr uO Burpady a5 pavyoio “3*9) soroeds Zurpooy S[VUIIUB) dDUE4SIxe 04RIB setoeds pooy jo Apoq uo sor Aq pozearyyno saroeds pooyr -dos spvo] so1oeds Zurpoa,y 10 UI S@AT] Sotoeds Surpoay -eds on Ayd ee mr a am -o1des pus (SME snavydoideg SIVTUIABIG (speurtue oured 0A1900401d tH -—____] wey) uoly Jepun owed “3'a) | (A090 (Aouoy | ~B4TFTNO YM WOTPBATITND JNOGIM yoytm “3"9) u01ry prea °3'0) uoy | “BATIINO = GEM ~BATFTND JNOYFTM Bute] BUrPy ie + al PATPOIIIG wopuByy yi SouUINSUOD sar L oT _! i syonpoad | -ods Surpaoy aq SUIp9e} JO 40v 94} UT = OJSBM Jo syonpord pny SAT] 0} sonuTy sotoods Zurpeoy Aq WOTzVeZ119., -asn Jo AOBILT -uod so10eds poo wy Per seroeds poo = Le 5 | apts setoeds 1aqy}0 so1deds 1ay}0 qoyytea uo 4youeq uorjoniysep “aya ‘Ajddns poo

— ee ar ee ae SO aa + —_ —_ _ ae Xe = %p uorjouosd penny 3081}8 popis-ouCQ eae s fae eee wor} yedur00 od ug qoort] pextyl yoortpuy oywurpsoqng Elegant intellects which despise the theory of quantity are but half developed.—A. N. Whitehead. The Form of the Growth Function F. The fundamental equations (1) of Chapter VI express in a very general, and for that reason somewhat colorless way, the interdependence of the several components of evolving systems of the kind here under discussion.

In the special cases with one, two and three dependent variables that have been presented as examples, the particular form and the concrete meaning of the functions / appearing in these equations has been illustrated for these specific instances, without any attempt at systematization from a general standpoint. It is desirable now to make a somewhat detailed analysis of the functions F in their more general aspect. A natural step to take is, first of all, to split up the function F into a positive and a negative term; that is to say, to express the rate of increase of the mass X; of the component S; as the balance, the surplus, of the mass U; added to that component per unit of time, over the mass V; eliminated therefrom per unit of time. Thus

Growth of Aggregates. Among the components of systems of the kind in which we are here mainly interested, a particularly important type are those built up of a large number of essentially similar units or individuals. Such are the aggregates of molecules that constitute the components (chemical elements and compounds) of the systems with which physical chemistry is concerned; such, also, are the aggregates of individual organisms that constitute the biological species, the component population groups, of which are built up the systems in which organic evolution takes its course.

In the case of an aggregate of this kind, if N is the number of individuals, and m their average mass per head, we have If the average mass per individual, mj, is constant, the second term of the right hand member in (8) drops out, and we have simply, This holds strictly for aggregates of similar molecules, for example, whose masses are all equal and constant. It will often hold with close approximation (as will be set forth in greater detail shortly) for populations of living organisms of one species.

dt naturally appears, after the manner indicated above, as the balance of the number of newly formed individuals B; per unit of time, and the number D; eliminated per unit of time. When these symbols refer to a population of living organism, B; is the (total) birth rate and D; the (total) death rate per unit of time. We have, then, the lower case letters b, d denoting birth rate and death rate per head per unit of time. Combining (4) and (5) we have

Demographic Functions. The quantities B, D, or b, d lend themselves to further analysis in terms of more fundamental characteristics of the aggregate. In a qualitative way everyone is familiar with the manner of the elimination of individuals from a population by death: some are carried off in infancy, some in childhood, adolescence, and maturity, until the remnant is finally called in old age. Quantitatively this fact finds expression in an actuarian’s life table, or the corresponding life curve, of which some examples are shown in figures 15 to 17. Starting with some large number, say a million, of newly born individuals, counted at birth, if we follow this sample batch of population through life, we find them thinning out, at first rather rapidly in infancy and childhood (steep part of curve on left): then more slowly (more gentle slope) in mid-life; and faster again as the natural term is approached. At any particular age a there are thus left, out of the original batch, a fraction p(a) of survivors. The fraction p(a), which we may speak of as the survival factor,! is a measure of the probability, at birth, that a random individual of the batch shall reach age a, under the conditions under which the data assembled in the life table were collected. The value of p(a) for every age of life depends, of course, on the general conditions of life in the population. It therefore varies in different localities and at different epochs, as illustrated in figures 15,16, 17. When it is desired to bring out the fact that p(a) depends on the time, it may be written p(a, #). But as a rule the change with time is not very rapid and we may often consider p(a) as a function of the age a alone. We shall do this in the present analysis, which relates to a population in a selected locality under essentially constant conditions of life.

If we denote by N, the survivors to age a, out of an original batch N, counted at birth, we have The coefficient «, thus defined is termed the force of mortality at age a. From its definition it is clear that it measures the death rate per head in a population composed entirely of individuals of age a. 1 The function p (a) is that commonly denoted by 1, in actuarial notation and tabulated in the principal column of a “Life Table,” Fic. 15. Somp Historica, Human Survivau Curves, EXHIBITING AN Evouvu-

The improvement is probably rather one of general hygiene than of man’s physiological constitution. The earlier figures are of very doubtful accuracy. The data on which these curves are based are naturally more reliable than those of the older life tables shown in figure 14, and exhibit very clearly the upward trend of the average length of human life in recent decades. Fig. 17. SurnvivaL Curves ror Dirrerent Countries, SHOWING INFLUENCE or Locat ConpiTions upon Lenatu or Human Lire. Arrer GLOVER

A particularly simple case is that in which y, is independent of a, the force of mortality is independent of the age, or is the same at allages. We then have by integration of (10) Such a simple life curve as this is not to be expected in a species of living organisms. It implies that the individual does not age, that his chance of living another year is just as good at ninety years of age as at fifty or at ten or at five; he can die, as it were, only by accident; he is perpetually young. Survival curves of this form do occur and play a significant réle in the aggregates of atoms and molecules which the chemist and the physicist make it their province to study. The atoms of an element in radioactive transformation, for example, are picked off, one by one, according to a law of this form, and so are the molecules of a chemical compound decomposing by a monomolecular reaction. If, in such a case, the plot the function p(a) is drawn on logarithmic paper, evidently a straight line is obtained, since

The force of mortality is here seen as the (constant) slope of the curve representing log, p(a). In the more general case it is also often convenient to plot p(a) on a logarithmic scale. This has been done in figure 18 for the United States survival curve shown in figure 17. It is seen that the curve thus obtained is at first convex toward the axis of a, but soon becomes concave toward that axis and then remains so to the end. The significance of this is, of course, that the force of mortality is very high in infancy, decreases in early childhood, until it reaches a minimum about the twelfth year of life; and finally increases continually to the end of the life span.

The detailed analysis of the human survival curve is a matter of interest not only to the student of evolution, but also to the guardian of public health and to the insurer of human life, the actuarian. The essentially practical requirements of these has led to a highly developed technique, amounting virtually to a separate branch of science, in the preparation and analysis of life tables. Of this phase of the subject no more needs to be said here, since there is a voluminous special literature available. Only the more strictly biological aspect of the matter is for us here of immediate interest, and of this