Elements of Physical Biology

where M denotes the geometric mean of £, and £; and while mdenotes their arithmetic mean. Thus the constant A can be determined from three suitably chosen points on the curve (after smoothing graphically if need be). The other constants then follow easily. Autocatakinesis. Both Robertson and Wo. Ostwald draw attention to the similarity between the law of growth (12) and the law of formation of a chemical substance by autocatalysis under certain conditions. The analogy is interesting, but must not be taken too seriously, inasmuch as in the one case the rate of growth is determined by ordinary chemical influence, in the other (organic growth) by a complicated combination of factors both of chemical and of | physical character. Rather more to the point seems to be the suggestion made above in connection with the law of growth of bacterial colonies, that, the body of a multicellular organism being a “‘population’”’ of cells, it is not altogether surprising that it should be found to follow the law of growth of a population.

A suggestion of terminology by Wo. Ostwald is worth noting. He proposes that growth of any kind, in which the substance or structure itself acts as nucleus for the formation about it of further quantities of the same substance or structure, be broadly termed autocatakinetic growth, the narrower terms autocatalysis of autocatalytic growth being reserved for that particular kind of autocatakinesis which is chemical in character. This is a useful suggestion, as it leaves the term autocatakinesis quite neutral and free from any implication or restrictions as to the mechanism by which the growth takes place.

Life is a system of relations rather than a positive and independent existence—G. A. Sala. Interdependence of Species. The case of two dependent variables, X,, X2, which we now approach, is of interest as the simplest example exhibiting the relation between interdependent species. This relation can take on a variety of forms. Most fundamental, perhaps, is that form of interdependence (1) in which one species S, serves as food to another species Se, so that, in this sense, S; becomes transformed into S:, thus

All animal species, and many plants also, thus derive their substance from other species on which they feed. And several different types of this form of interdependence are observed. In the first type, (a) the organism S, kills S; outright in the process of feeding upon it. We might term S. in such a case an episite | of S,, in contradistinction from the second type, (6) in which S» lives on S, without killing it outright, being parasitic upon Sj. The host is in most cases more or less injured by the parasite, and all pathogenic organisms fall into this class. For this reason guantitative epidemiology appears as one of the special branches of the general subject under consideration here.

A third type (c) of interdependence, is that in which S; is saprophagous or saprophytic, feeding upon the cadavers of S; after death from other causes; or, S2 may live on waste products discharged by S,. In contrast to episites and parasites, saprophagous species are presumably beneficial rather than injurious to the host. species, since they function as scavengers. Still another type, (d), of interdependence is that of symbiosis, in which S, and S; live in partnership which, as a rule, is in some degree mutually beneficial. Man and his domesticated animals and plants are obvious examples of this

In addition to these types (1a) to (1d), another large group of cases (2) are those in which two or more species compete for a common food supply. In their general form the fundamental equations of kinetics for the case of two dependent variables are We may note, first of all, as a general rule, the following observations regarding the coefficients a in the several types of interdependence enumerated above: la. S» lives on S, by killing S; outright (episitic type). In this case, evidently S: unfavorably influences the growth of S;, while, on the contrary, S; is advantageous to the growth of Ss, so that

1b. S2 parasitic upon S;. Here az < O, ax > O, as in case Ja. Ic. S, saprophytic upon §;, or living upon waste products of S;. Here we may expect that ai. = O, aa, > O. Id. Sz feeds on S,, but at the same time cultivates it in symbiosis. Here Ay. > ne ay; > O. The characteristic equation for two variables is a1—X Qs Certain general conclusions follow immediately. So, for example, if d21, G2 are both of the same sign, as in the case of saprophytes

and of symbiosis, the quantity under the radical is necessarily positive, , and hence both roots for \ are real. The oscillatory type of approach | to equilibrium is here excluded. In the case of two species of the | type (1a) or (1b), the episitic or parasitic type, on the contrary, | the possibility of oscillations is indicated, under conditions where the equations (1) are applicable. Martini’s Equations for Immunizing Diseases. Numerical applications of the case of two dependent variables are not easily obtained. Of concrete examples in general terms (with unknown or very imperfectly known values of the constants involved) several are to be found in the literature. The simplest of these is a case for which the equations are given, without solution, by Martini in his Berechnungen und Beobachtungen zur Epidemiologie der Malaria (Gente, Hamburg, 1921, p. 70), namely, the case of the growth of an endemic disease of the type that confers acquired immunity upon persons who recover from it (e.g., measles, scarlet fever, etc.) Martini writes

u = fraction of the population affected and infective i = fraction of the population not available for new infection (i.e., immune or already affected) (1 — 7)= fraction of the population available for new infection p = fraction of the population newly affected per unit of time q = fraction of the population of affected population that ceases to be so, per unit of time, by recovery or by death fraction of ‘unavailable’ population that loses immunity or dies per unit of time infectivity (a proportionality factor)

Martini puts the newly affected population, per unit of time, jointly proportional to the infective population uw and to the population available for new infection, (1 — 7), so that from which it is seen that the equilibrium near the origin is stable if and only ifa < q There is a second equilibrium at Since, however, 7 can in reality assume only positive values, this equilibrium has a meaning only if a > gq, i.e., just in the case in which the equilibrium at the origin is unstable. Hence we conclude that if a < qg the equilibrium at the origin is the only possible one, and is stable, so that the disease will die out. In the other alternative, a > g, the second equilibrium has a real meaning, and we can develop a solution in series u—U=Gi rest + Giaett*+.. . | t— I = G'sre* + G'sne**+ . . i

We need not here consider the case (a — g) < O, for, as pointed out above, in this case the second equilibrium is meaningless. But when a > gq, it is seen from (14) that the real parts of the two roots will then in any case be negative, so that the equilibrium, if it exists at all, is stable. Furthermore, there will be two real and distinct, two real and coincident, or two complex roots, according as In the last-mentioned case equilibrium will be approached by a series of oscillations above and below the final state of equilibrium, so that a series of ‘epidemic’? waves will appear, a feature which has an obvious interest in connection with the type of disease here discussed.” The oscillations thus occasioned by the factors duly taken into account in this elementary analysis may in practise be enhanced by factors here neglected, such as varying virulence of the disease, exhaustion of susceptible population, seasonal influences, etc. The last-mentioned, however, will in general tend to produce a separate series of waves whose period will have no relation to that of the oscillations derived above.

m ; For the special ce = 1, Prof. G. N. Watson has given a complete solution, which may be consulted in the original. It is to be noted that this case cannot, according to the analysis here presented, lead to oscillations, since the condition for oscillations according and cannot be satisfied, as the square of a real quantity is necessarily positive. The Ross Malaria Equations. A system of equations has been established by Sir Ronald Ross‘ to represent, under certain

2Compare J. Brownlee, Investigation into the Periodicity of Infectious Diseases, Public Health, vol. 25, 1915, p. 125. 4 Sir Ronald Ross, The Prevention of Malaria, second edition, 1911, p. 679. This volume also contains a bibliography. For a detaile ddiscussion of the Ross malaria equations see A. J. Lotka, Am. Jour. Hygiene, January Supplement, 1923. conditions, the course of events in the spread of malaria in a human population by the bites of certain breeds of mosquitoes infected with the malaria parasite. These equations are of the same form as Martini’s equations discussed above, and inasmuch as Ross’s malaria equations have been very fully treated by the writer in a separate monograph,° their detailed study may here be omitted. It will suffice to reproduce from this monograph one of the curves representing the course of events, the presumptive growth of malaria in a human population, as defined by the differential equations of Sir Ronald Ross.

Fia. 10. Curve or Growra or Enpemic MALARIA ACCORDING TO Str Ronatp Ross’s Equations The upper (S-shaped) curve relates to the particular case in which the initial malaria rates in the human and the mosquito population stand in the ratio which they have at equilibrium or are both small. The lower curve represents the course of events when the initial malaria rate is 4.2 per cent in the human population, and 1.4 per cent in the mosquito population (The zero of the time scale is arbitrary.) Ordinates are malaria rates (human) expressed as fraction of unity. (Reproduced from A. J. Lotka, Am. Jour. of Hygiene, January Supplement, 1923.) ;

It will be observed (fig. 10) that the curve shown consists of two parts, the one an S-shaped limb which, in point of fact, is very “nearly identical in type with the Verhulst-Pearl population curve. The second limb ascends very steeply, almost vertically, and finally bends to join the S-shaped limb. The meaning of these curves is as follows: If a small nucleus of malarial infection is introduced into a (constant) population of human beings and mosquitoes, both being previously free from such infection, then the growth of malaria in the human population will follow the course represented by the S-shaped limb. It will be seen that the process is a rather leisurely one, its essential completion occupying about ten years, according to Ross’s figures. (Strictly speaking it is never quite complete in a finite time.) Seasonal effects are here disregarded.

On the other hand, if at the start there is already present a certain malarial rate in the human population, and also in the mosquito population, then, in the case here depicted, there is for a time a very rapid increase of malaria in the human population, until, in the brief space of about two months, the S-shaped curve is reached. After that the course of events is the same as in the first case. In practise, in temperate climes, we can expect only short sections of the S-shaped curve to be realized, owing to the interruptions of the seasons. No data are available for a numerical comparison of these results with observed conditions. Close agreement is not to be expected, as the Ross equations refer to a rather highly idealised case, a constant population both of men and mosquitoes. The latter could be even distantly approached only in the tropics. There is room here for further analysis along more realistic lines. It must be admitted that this may lead to considerable mathematical difficulties. The case of periodic seasonal influences is perhaps the one that promises to yield most readily to mathematical treatment.

The Ross malaria equations are a typical example of equations affected with a lag, owing to the period of incubation. Fora detailed discussion of this feature the reader must be referred to the author’s monograph published by the American Journal of Hygiene. An Example in Parasitology. An interesting and_ practically significant case of inter-group evolution, of conflict between two species, has been made the subject of an analytical study by W. R.

Thompson.’ He considers a host species, numbering initially n individuals, and a parasite species, initially p individuals. Ona number of simple assumptions, for which the reader must be referred to the original papers, he develops the following formula for the fraction a of the host population attacked, in the ¢ generation, by parasites where a is the ratio of the “reproductive power’’ of the parasite to that of the host, the reproductive power being measured by the number of eggs deposited per female. It is assumed that only one egg is deposited in each host. Putting at = ¢rt (18) and

Thompson’s formula becomes, after a simple transf ormation, ne a—1l ~ Ke-tt—1 which will be recognized, once more, as the equation of the law of simple autocatakinetic growth. According to the value of a and K several different cases may arise, whose graphs are shown in figure 11. It should be observed that In the special case that a=1 the formula (20) becomes indeterminate and a is then given by Thompson gives a number of numerical examples exhibited in table 5. From these examples and from this formulae he concludes that the invasion of the host species by the parasite may at first

Fria. 11. Coursn or Parasitic Invasion or INSECT Sprcips AccORDING TO proceed only very slowly, and that nevertheless, after a certain time, the increase may become very rapid. From a practical standpoint this is important to observe, since it implies that the first effect of “sowing” the parasite among a species of insect hosts, with a view to destroying them, may be quite discouraging, and that this must not be taken as indicative of ultimate failure. This

Percentage (100c) of host species attacked by parasite in t* generation, according to W. R. Thompson p=initial number of parasite species. n=initial number of host species. parasite host The figures in the columns headed 1, 2, 3, ete., denote the values of 100 @ in the Ist, 2d, 3d, ete., generation. Fie. 12. IncrEastine DIFFUSION-IN-TIME OF SuccESSIVE GENERATIONS IN observation is especially significant since in practise one must often be satisfied with the introduction of a relatively very small number of parasites. So, for example, in using the.parasite species

Liparis dispar, commonly about one thousand individuals have been sown. Supposing that there were one thousand million hosts, and that the parasite reproduced twice as fast as the host, it would require, according to Thompson’s calculations, about 19 generations to exterminate the host; and then, even to the sixteenth generation, only 10 per cent or less of the host species would be attacked. Thompson’s formula is open to certain objections. Its derivation seems to involve the assumption that each generation of the parasite is coextensive in time with the corresponding generation of the host. Furthermore, the use of a generation as a sort of time unit is unsatisfactory, because a generation is a very diffuse thing, spread out over varying lengths of time. This is easily seen by considering the progeny of a batch of individuals all born at the same time t=O. If we call this the Ot generation, and if a, a, are respectively the lower and the upper limit of the reproductive period, then it is clear that the next generation, the first, will extend over the interval of time from a; to az, the second from 2a; to 2a2, the nt from na, to naz, an interval which will ultimately become very large as n increases, as shown by successive intercepts between the two sloping lines in figure 12.

Less objectionable, perhaps, is the fact that Thompson’s formula is expressed in terms of the “rate of multiplication per generation” of the two species. This term is not as clear as it might be. From the context it appears that it refers to the number of eggs deposited per female. This number is closely and simply related to the ratio R of the total births in two successive generations. If an individual of age a reproduces, on an average 8 (a) individuals per unit of time, the ratio FR is evidently given by

The relation of this PR to the rate of increase r per head of the population is not altogether obvious and cannot be expressed in simple form. For a population with fixed age distribution, it will be shown in Chapter IX that r is given by In view of the doubtful features in Thompson’s formula which have been indicated above, it appears desirable to attack his problem in quantitative parasitology from another angle. We may do this by following the general method which has here been set forth and exemplified.

Treatment of the Problem by the Method of Kinetics. Let N; be the number of the host population, 6; its birthrate per head, (the deposition of an egg being counted a birth), and d, its death rate per head from causes other than invasion by the parasite. Let kN,N; be the death rate per head due to invasion by the parasite, in the host population, the coefficient k being, in general, a function of both N, and No», the latter symbol designating the number of the parasite population.

The birth of a parasite is contingent upon the laying of an egg in a host, and the ultimate killing of the host thereby. To simplify matters we will consider the case in which only one egg is hatched from any invaded host. If an egg is hatched from every host killed by the invasion, then the total birthrate in the parasite population is evidently kNiN»2. If only a fraction k’ of the eggs hatch, then the total birthrate in the parasite population is evidently kk’ N,No, which we will denote briefly by KNiNe. Lastly, let the deathrate per head among the parasites be d,.. Then we have, evidently

where 7; has been written for (b, — dj). Regarding the function k, we shall now make the very broad assumption that it can be expanded as power series in N, and No, thus It will be convenient first of all to consider an approximation.’ If the coefficients 8, y, etc., are sufficiently small, we shall have, for values of N;, N2 not too large, essentially where M is an arbitrary constant of integration. Expanding by Taylor’s theorem we find By giving successively different values to the arbitrary constant M’ a family of closed curves is obtained for the plot of (34) in rectangular coédrdinates, as indicated in figure 13. In the neighborhood of the origin, where terms of higher than second degree are negligible, (34) reduces simply to

7 It will be observed that the ‘‘diagonal’’ terms in the linear part of (27), (29) are lacking, i.e., there is no linear term containing N; in the first equation of (27) and none containing N, in the second. This gives rise to an exceptional case to which the general solution given in Chapter VI is not applicable. The course of events represented by these curves is evidently a cyclic or periodic process, corresponding to a circulation around the closed curves. The period of oscillation, near the origin,’ is given by

Fic. 13. Coursm or Parasitic Invaston oF INsncT SpEcIns, ACCORDING TO Lorka; ELpMENTARY TREATMENT This finding accords well with the observation made by L. O. Howard: With all very injurious lepidopterous larvae . . . . we constantly see a great fluctuation in numbers, the parasite rapidly increasing immediately after the increase of the host species, overtaking it numerically, and reducing it to the bottom of another ascending period of development.

8’ The purely periodic solutions have been discussed by the author in Proc. Natl. Acad. Sci., 1920, vol. 7, p. 410. The writer, however, at that time overlooked the existence also of the other types of solution, and also stated that the period of oscillation is independent of initial conditions. This is an error which he takes the present opportunity to correct. The expression given by him loe. cit. for the period of oscillation holds only in the neighborhood of x =y=0. See also note 10 below.