Elements of Physical Biology

For the present, without making here a closer analysis of the conjugate parameters a, p, (area, rent or population pressure) it will suffice to point out that the mere existence of the relation ae 20 according as pj 2 p (8) dt < AREAS may give into our hand the means of drawing certain conclusions regarding the behavior of the system, quite independently of the entimate nature of these quantities, a, p. In illustration of this it is only necessary to refer once more to the discussion of the principle of Le Chatelier in Chapter XXII.

Distant Analogy to Gas Law. Again, still without any searching analysis of all the physical implications of the parameters a and p, we may note certain facts regarding the relation. which connects the conjugate parameters a, p, Just as pressure p and volume v, in physico-chemical systems, are connected by the relation em G(P, Xi, X2, shaw!) (5) This latter, in the simple case of a gas takes the form v=m ee (6) Let N be the total number of a (human) population, a the area occupied by it, and Ni the total income of the population. Let p be the rent per unit area; and let the population spend a fraction R’ of its income on rent. Then evidently

p re ree R'% (9) It will be seen that there is a certain analogy between the formulae © (8), (9) as applied to the relation between rent (population pressure) and area occupied by a population, on the one hand, and the formulae (6), (7) as applied to the relation between the pressure p and the volume v of a gas. The analogy is particularly close if the income per head 7, and the factor R’ are constant as a changes. The former may be true, approximately, if the inhabitants of area a derive their income from some extraneous source quite independent of this area. It will not be true, even in rough approximation, if the inhabitants derive their income from the produce of the area a itself. It is not the author’s intention to emphasize unduly mere analogies. Nevertheless, the one here presented seems worthy of passing notice. Compare also E. Woodruff, Expansion of Races.

The actual relation between population density and rent or land value is a subject of great economic interest. Research in this subject is in progress under the auspices of the Institute for Re- search in Land Economics, Madison, Wis., but has not, at this time of writing, matured to published results. Law of Urban Concentration. An empirical law of urban concentration was pointed out some years ago by F. Auerbach (Petermanns Mittheilungen, 1923, p. 74). Arranging in order of magnitude the cities of a given country, he found that the product of population and ordinal number (rank) was approximately constant. Thus, plotting rank against population, he obtained a hyperbolic curve, or, plotting the product of these two quantities, he obtained, roughly speaking, a straight line. For some of his curves the reader must be referred to the original publication.

The illustration (fig. 60) shows the graph obtained by plotting the logarithm of the population of the cities of the United States (1920) against the logarithm of their respective rank. In the higher ranks only every fifth city has been plotted. It will be seen that, excepting cities of rank 4, 5 and 6, the plot does approximate quite closely to a straight line. The slope of this line, however, is not exactly unity, as demanded by Auerbach’s law, but 0.93, so that the

actual law of urban concentration in the United States, in 1920, was, within the limits indicated, given by Hundred Largest Cities in the United States, 1920, According to Rank Fra. 60. Law or URBAN CONCENTRATION Graph obtained by plotting as ordinates, on logarithnic scale, Population of United States Cities, and as abscissae the corresponding Rank (in order of magnitude), also on logarithmic scale. It may be left an open question how much significance is to be attached to this empirical formula. We shall meet with a similar relation in Chapter XXIII in dealing with Willis’ theory of Age and

The Biological Background of Population Pressure. In the preceding paragraphs we have accepted it as a fact that there is at any rate some degree of relation between population density and the “desire for expansion’? which finds expression, in the human species, in the willingness to spend a certain fraction of the income upon ground rent or its equivalent in interest and taxes upon real estate. It is not proposed to attempt here any searching analysis of the precise physical significance of this desire for expansion. But that it is ultimately referable to very cogent biological and physical factors is self-evident. The degree of crowding of a group of organisms affects both its death rate (mean length of life) and also its rate of reproduction, as well as, no doubt other significant vital functions. As Pearl and Parker? remark, in their paper on the influence of population density upon rate of reproduction in Drosophila:

It has long been known that degree of crowding of organisms in a given space, or the density of the population, has an influence upon various vital processes of the individuals composing the population. In the matter of growth Semper’ and before him Jabez Hogg® showed that volume of water apart from food and other conditions has an influence upon the rate. This subject has again been studied recently by Bilski.!° Farr!! showed that there is in man a definite relation between population and death rate. This old work of Farr has recently been gone over carefully and confirmed by Brownlee.!2 Drzwina!’ and Bohn show that a particular concentration of a toxic substance, just lethal for a single individual in a given volume of water (working with such organisms as infusoria, planarians, hydra, tadpoles, etc.), will

be sub-lethal if several individuals are present in the same fixed volume of water. Influence of Population Density on Rate of Reproduction. Pearl and Parker“ have determined experimentally the relation between rate of reproduction and the density of a population of Drosophila (fruit flies) in a universe of constant volume (glass bottle). This ‘IX. Semper, Animal life as affected by the natural conditions of existence. Fourth Edition, London, 1890.

relation is found to resemble in form Farr’s law for the death rate D in a human population of density d: log D= log a+k log d (10) in which a and k are constants. Pearl and Parker found Fic. 61. RevLATION BETWEEN Rate oF REPRODUCTION IN DRosopHILA (FRUIT Fiy) AND Density or Marep PoPpuLATION The circles indicate observed values; the drawn-out curve is computed according to equation (11). After Pearl and Parker. where y denotes the number of imagoes per mated female per day, and x denotes the mean density of the mated population (measured as flies per bottle) over a sixteen-day period. The results obtained by Pearl and Parker are exhibited graphically in figure 61, the small circles denoting observed values, and the drawn-out curve values computed by the formula cited above. It should be noted, however, that in Farr’s law the coefficient & of log d is positive, whereas in the Pearl and Parker formula it is negative; death rate in-

Fra. 62. RELATION BETWEEN Mpan Lunects OF LIFE AND PopULATION DENSITY The curve is a freehand smoothing of the observations indicated by the small circles. After Pearl and Parker. creases with population density, whereas rate of reproduction, in this series of experiments, was found to decrease as the density increases. Influence of Population Density on Duration of Life. The same authors have also investigated the relation between population density and duration of life in Drosophila. Their results are shown graphically in figure 62. The smoothed curve exhibits a very interesting feature. While crowding unmistakably diminishes

the average length of life, as one would naturally expect, the curve does not fall continuously from left to right, but has a maximum. The flies do not thrive best when they have the most ample space per individual; there is an optimum density which is best suited to their needs for company or for some other obscure factor supplied by a certain moderate amount of crowding. Willis’ Theory of Age and Area. As regards the influence of those topographic parameters which define the boundaries of the system, a special case arises during that period in the life of a body of organisms, when its spread has not yet extended to those boundaries. Certain aspects of the phenomena presented during this period of diffusion have been made the subject of a painstaking study, conducted with much originality of view, by C. J. Willis." One may not agree in all points with Dr. Willis’ conclusions, but the material of fact collated by him is in itself significant and of value. As Prof. W. Bateson in a review of this book remarks: “To have hit on a new method of investigating even a part of the theory of evolution is no common achievement, and that the author has done this cannot in fairness be denied.’’ Dr. Willis’ principal thesis is essentially this, that the area occupied by a biological species is a measure of its antiquity in evolution. To be more precise, and to quote the author’s own words:

“(The area occupied at any given time, in any given country, by any group of allied species at least ten in number, depends chiefly, so long as conditions remain reasonably constant, upon the ages of the species of that group in that country, but may be enormously modified by the presence of barriers, such as seas, rivers, mountains, changes of climates from one region to the next, or other ecological boundaries, and the like, also by the action of man and other

The thesis, thus stated, is ‘well hedged” with qualifications. Professor Bateson remarks: “Every evolutionist agrees that, apart 16 Age and Area: A Study in Geographical Distribution and Origin of Species; Cambridge University Press, 1922. For review see Nature, 1923, p. 39; Science Progress, January, 1923, p. 474. from disturbing elements, area is a measure of age7 . . . . We are, however, asked to believe that in practice this mode of estimating the age of a species is, on the whole, trustworthy; that endemic species and varieties in general can and must be for the most part accepted as new starters in evolution, and not as (remnants of) survivors.”

Dr. Willis supports his views by observations gathered chiefly in Ceylon and New Zealand. His evidence is not altogether convincing. Professor Bateson in the review already quoted remarks pointedly: ‘On any theory of evolution endemics (and rare species) must be in part novelties and in part relics; but why, apart from the theory of Age and Area, we should believe that endemics are in such great majority novelties I do not clearly understand, for though we know little of origins, we are certain that myriads of species have become extinct. It is surely contrary to all expectation that the process of extinction should be in general so rapid, and the final endemic phase so short that the number of species in that final stage should be so insignificant.” Referring to the same point A. G. Thacker remarks: ‘Species and genera do die out, and therefore there are diminishing ranges as well as expanding ranges.

Dr. Willis thinks that very few of the small range species are, in fact, decreasing.”” For a further critique of the theory of Age and Area the reader may well be referred to the two reviews already cited, while Dr. Willis’ own presentation of his case will be found expanded in detail in his book Age and Area, Cambridge University Press, 1922. Quite recently (I add this in correcting proof) Prof. G. V. Yule has lent his able advocacy to Willis’ theory in an extensive paper published in the Phil. Trans. Roy. Soc., 1924, vol. 213, Series B, pp. 21-87. Some of the assumptions underlying Professor Yule’s argument do not seem to commend themselves to the critical reader, in particular his supposition that

" “For this and other reasons Dr. Willis’ findings seem hardly competent to furnish a basis of attack against any otherwise established or suggested theory of evolution. On this point Dr. Willis himself does not seem wholly consistent, for though he advises in one place that it would be ‘wiser to abandon natural selection’ as the general principle that has guided evolution, in another place he admits that ‘nothing can come into lasting existence without its permission.’ This admission is all that any thoughtful adherent of any theory of evolution asks.’’ A. G. Thacker, The Dynamics of Distribution, Science Progress, 1923, vol. 17, p. 474.

the number of new species thrown is independent of the number of individuals. Turning from the theoretical and debatable portion of Dr. Willis’ contribution, to the factual material presented by him, we are confronted by a number of very remarkable relations, such as those represented graphically in figure 63. In this drawing ordinates represent the number of genera in the several natural families of plants and animals indicated, and the corresponding abscissae represent the number of species in each genus plotted. So, for example (fig. 63), of the family Compositae 1143 genera were noted, as follows:

446 genera of 1 species 140 genera of 2 species 97 genera of 3 species 43 genera of 4 species 55 genera of 5 species etc. An examination of these drawings clearly brings out the following facts: The monotypic genera, with one species each, are always the most numerous, commonly forming about one third of the whole group; the ditypics, with two species each, are next in frequency, genera with higher numbers of species becoming successively fewer. Set out graphically as in figure 63, the genera exhibit what Dr. Willis calls a “hollow curve” of frequency (in point of fact a hyperbola of the generalized type), and, as Professor Bateson remarks, there is no gainsaying the fact that these curves, though collected from miscellaneous sources, have a remarkable similarity. Perhaps more striking still is the relation established in figures 64, 65, and 66. The quantities plotted here are of the same character as in figure 63, but they are plotted on a doubly logarithmic scale, with the remarkable result that the graphs obtained are in close approximation straight lines. Thus if x denotes the number of species in a genus, and y denotes the number of genera comprising x species, we have

that is to say, the variables x and y are connected by a hyperbolic!* relation. It should be noted that this relation covers a wide variety of cases, including both plants and animals. The numbers (thus 446/1) at the beginning of each are the numbers of monotypes. By courtesy of Nature Fic. 63. Hypprsoric Curves OBTAINED BY PLOTTING AS ORDINATES THE NuMBER OF GuNnERA Havine 1, 2, 3, . . . n SPECIES, AND AS ABSCISSAB THH NUMBER n OF SucH SPECIES The last curve above shows as ordinates the number of species of endemic compositae in the Galapagos Islands, as abscissae the corresponding areas over which such species have spread. After J. C. Willis.

18 Of the general type into which the conic hyperbola zy = const. falls as a special case. Fic. 64. RELATION BETWEEN NUMBER AND SIZE OF GENERA OF ALL FLOWERING PLANTS, PLorrep LOGARITHMICALLY fe) 2 4 6 8 r0 12 4 r6 18 2 log (N° of species) By courtesy of Nature Fig. 65. RELATION BETWEEN NUMBER AND Size or Rupracwan, PLoTrep LoGARITHMICALLY Dr. Willis’ interpretation of the remarkable curves obtained by him may be quoted in his own words (Nature, February 9, 1924, De Lis):

If species of very limited area and genera of one species (which also have usually small areas) are, with comparatively few exceptions, the young beginners in the race of life, and are descended in general from the species of wider dispersal and the larger genera, and if the number of species in a genus is, broadly speaking, a measure of its age, the idea at once suggests itself that a Fic. 66. RELATION BETWEEN NUMBER AND Size oF GENERA OF CHRYSOMELID Brpties, PLrorrep LoGARITHMICALLY

given stock may be regarded as ‘‘throwing’’ generic variations much as it throws offspring, so that the number of genera descended from one prime ancestor may be expected to increase in geometric ratio or according to the law of compound interest. The number of species descended from one ancestor might be expected to follow the same form of law with a more rapid rate of growth. On such a very rough conception it is found that the form of frequency distribution for sizes of genera should follow the rule that the logarithm of the number of genera plotted to the logarithm of the number of species gives a straight line.

It follows from the conception stated that the excess of the slope of the line over unity should measure the ratio of the rate of increase of genera to that of species. The slope should always, therefore, lie between the limits 1 and 2, for a slope of less than unity would have no meaning, and a slope exceeding 2 would imply that generic variations were more frequent than specific variations. Hitherto no exception has been found to the required rule. One group of fungi tested (Hymenomycetineae) gave a line with a slope very little exceeding unity (1.08), but the figures found for flowering plants lie between the narrow limits 1.38 and 1.64, with an average of about 1.438. Snakes and lizards both give a figure very near 1.50, and the Chrysomelidae about 1.37.

For further details the reader must be referred to Dr. Willis’ book Age and Area, and to a paper by E. 8. Pearson, in Biometrika, August, 1923, vol. 15, of which the following passage, part of that author’s conclusions, may be quoted: There is no doubt that these principles represent a certain aspect of the process of evolution, but I believe that Dr. Willis has stressed their importance beyond the limits which the evidence of observation will bear. They cannot explain everything, and we have seen that in many cases results which we are led to predict with their assistance are scarcely borne out, while in other cases recurrent distributions can also be accounted for on different hypotheses.

Climatic Parameters. The state of a physico-chemical system is commonly described in terms of its temperature (in addition to pressure, etc.). Without attaching any deep significance to the analogy, it may be remarked that systems in which organic evolution is under way commonly require, as essential parameters to define their state, the statement of sundry quantities that describe climatic conditions, such as temperature, humidity, precipitation, light, etc. The influence of these upon the course of events must form an essential part of the study of evolution in life-bearing systems.

The investigations of the influence of temperature upon metabolic processes—an approximate doubling of reaction velocities for every 10°C. rise in temperature—we shall here note only very briefly as belonging rather to the field of biochemistry than to the more strictly ecological studies in which we are here interested. More in the line of our immediate interests are certain data gathered in oceanographic researches, which have already furnished us with much illustrative material. The influence of light, temperature, and

CO, concentration is discussed by G. W. Martin in his paper al ready cited.!® He remarks: Plants on land receive the full benefit of the sun’s rays as we know them. Plants living under water receive only a portion of the rays that reach the land. Part of the light that strikes the water is reflected, and the part that penetrates the water is gradually absorbed in passing through that medium, the red and yellow rays first, the blue and violet last. This differential absorption is reflected in the curious and well-known distribution of marine algae according to color—the green kind growing in shallow water, the brown in an intermediate zone, and the red in the deepest water, although there are, of course, numerous exceptions to this general rule of distribution. Another property

Fra. 67. SonuBILITY OF CARBON D1oxIDE IN WaTER, EXPRESSED IN VOLUMES or CO, Mrasurgep AT NORMAL TEMPERATURE AND PRESSURE, of light is that it is refracted by water, and the greater the angle at which the rays strike the water, the greater will be the refraction. In the tropics, where the rays are practically vertical, the amount of refraction is insignificant, but in high latitude, where the rays strike the water at a sharp angle, the refraction is marked, as a result of which the rays are bent into a more nearly vertical direction, thus increasing their penetration in depth, and partly compensating for the unfavorable angle at which they strike the water. Helland Hansen was able to show that in the Atlantic Ocean south of the Azores, on a bright summer’s day, light is abundant at a depth of 100 meters, still including at that depth afewredrays. At 500 meters the red rays have completely disappeared, but blue and ultra-violet rays are still plentiful, and may be detected at 1000

meters, but have completely disappeared at 1700 meters. It is not probable, however, that under the most favorable conditions photosynthesis may be carried on at depths greater than 200 meters. Temperature is less directly important in the sea than on the land since there is no great danger of injurious extremes being reached. Indirectly, its importance lies in the fact that carbon dioxide is much more soluble in cold water than in warm (see fig. 67) and it is probably this, rather than the direct influence of temperature, which accounts for the fact that the most luxurious development of plant life is in the colder waters of the earth.