and structure of semiconduc- in the "normal" semiconductors considered I have tried here to put the emphasis position here. on those basic properties that are com- tors on the one hand and parameters 11. P. Fielding, G. Fischer, E. Mooser, J. Phys. mon to all semiconductors and that such as energy gap and charge-carrier Chem. Solids 8, 434 (1959). 12. H. Winston, Phys. Rev. 94, 328 (1954). distinguish them from other solids. It mobility on the other. 13. S. Geller and J. H. Wernick, Acta Cryst. 12, is interesting as well as surprising to 46 (1959). 14. U. Winkler, Helv. Phys. Acta 28, 633 (1955). see how the many and various semi- References and Notes 15. P. Junod. E. Mooser, H. Schade, ibid. 29, conducting compounds are all governed 193 (1956). 1. A. H. Wilson, Proc. Roy. Soc. (London) 16. E. Mooser and W. B. Pearson, Phys. Rev. by the same simple chemical and struc- A133, 458 (1931). 101, 492 (1956). 2. , ibid. A134, 277 (1931). tural rules. These rules present a chal- 7a, 744 (1952). 17. , J. Chem. Phys. 26, 893 (1957). 3. H. Welker, Z. Naturforsch. G. Busch, F. Hulliger, U. Winkler, Helv. lenge to the theoretician, who has yet 4. The zinc blende structure can be considered 18. a superlattice of the diamond structure. Phys. Acta 27. 195 (1954). to interpret them in a rigorous way. 5. E. Mooser and W. B. Pearson, Progress in 19. J. Appel, Z. Naturforsch. 9a, 265 (1954). They present a challenge also to the Semiconductors, vol. 5 (Heywood, London, 20. E. Mooser and W. B. Pearson, J. Phys. Chem. in press). Solids 7, 65 (1958). experimentalist because they introduce 6. C. H. L. Goodman and R. W. Douglas, 21. G. Busch, E. Mooser, W. B. Pearson, Helv. and un- Physica 20, 1107 (1954). Phys. Acta 29, 192 (1956). him to large families of new 7. 0. G. Folberth and H. Pfister, Semiconduc- 22. E. Mooser and W. B. Pearson, Acta Cryst. explored semiconducting materials. And tors and Phosphors (Interscience, New York, 12, 1015 (1959). 1958), p. 474; I. G. Austin, C. H. L. Good- 23. F. Laves, Theory of Alloy Phases (American the challenge is all the greater since it man, A. E. S. Pengelly, J. Electrochem. Soc. Society for Metals, Cleveland, Ohio, 1956), is to be expected that, as our knowledge 103, 609 (1956); C. H. L. Goodman, Nature p. 124. 179, 828 (1957). 24. U. Dehlinger, Theoretische Metallkunde of semiconductors and their properties 8. E. Mooser and W. B. Pearson, J. Electronics (Springer, Berlin, 1955). increases, the chemical and structural 1, 629 (1956). 25. Since the principal quantum number of the 9. C. H. L. Goodman, J. Phys. Chem. Solids 6, valence shell of an atom coincides with the rules will be reflected in at present 305 (1958). number of the period in which the atom 10. While the mechanism of electronic conduction stands, one readily finds from Fig. 3 that, for largely unknown but much-sought-for in molecular compounds is not fully under- example, n = 4 for AlSb and n - 4.75 for relationships between the chemical com- stood at present, it certainly differs from that CuInTe-2.

rate of change of N, the number of elements in the population, is given by dN==YoN-Oo N = ao N dt (1) where ao = yo - Oo may be called the Doomsday: Friday, productivity of the individual element. Depending upon whether ao : 0, inte- gration of Eq. 1 gives the well-known 13 November, A.D. 2026 exponential growth or decay of such a population with a time constant of l/ao. At this date human population will approach infinity In reality, alas, the situation is not if it grows as it has grown in the last two millenia. that simple, inasmuch as the two param- eters describing fertility and mortality may vary from element to element and, Heinz von Foerster, Patricia M. Mora, Lawrence W. Amiot moreover, fertility may have different values, depending on the age of a par- ticular element. To derive these distribution func- Among the many different aspects tion and has to be excluded from the tions from observations of the behavior which may be of interest in the study population count ("death"). of a population as a whole involves the of biological populations (1) is the one Under conditions which come close use of statistical machinery of consider- in which attempts are made to estimate to being paradise-that is, no environ- able sophistication (3, 4). the past and the future of such a popu- mental hazards, unlimited food supply, However, so long as the elements lation in terms of the number of its and no detrimental interaction be- live in our hypothetical paradise, it is elements, if the behavior of this popula- tween elements-the fate of a biologi- in principle possible, by straightforward tion is observable over a reasonable cal population as a whole is completely mathematical methods, to extract the period of time. determined at all times by reference to desired distribution functions, and the All such attempts make use of two the two fundamental properties of an fate of the population as a whole, with fundamental facts concerning an in- individual element: its fertility and its all its ups and downs, is again de- dividual element of a closed biological mortality. Assume, for simplicity, a fic- termined by properties exclusively at- population-namely, (i) that each ele- titious population in which all elements tributable to individual elements. If ment comes into existence by a sexual behave identically (equivivant popula- one foregoes the opportunity to de- or asexual process performed by an- tion, 2) displaying a fertility of /0 off- scribe the behavior of a population in other element of this population spring per element per unit time and all its temporal details and is satisfied ("birth"), and (ii) that after a finite having a mortality Oo = l/tm, derived The authors are members of the staff of the de- time each element will cease to be a from the life span for an individual ele- partment of electrical engineering, University of distinguishable member of this popula- ment of tm units of time. Clearly, the Illinois, Urbana. 4 NOVEMBER 1960 1291 with a general account of its develop- choice of a is a simple linear depend- the living conditions beyond those ment over long stretches of time, the ence of the form found in a "natural setting." problem reduces to solving Eq. 1, ex- The human population may serve as cept that N, yo, and 9o have to be re- a = ao-al N (2) a typical example, as evidenced by its placed by appropriate mean values which, inserted in Eq. 1, results after steady social build-up during historical (y, 9) taken over several generations, integration in what demographers pre- time, its vigorous urbanization in recent over all ages, and over all elements. fer to call the "logistic growth curve," centuries, and its extensive development The difficulties encountered in es- displaying a "sigmoid" shape, if N is of the means of mass communication tablishing the distribution functions for plotted linearly against linear time in recent decades. y and 9 from observations of the be- (6; 7, p. 67). The choice of this par- Since a, the productivity, reflects in havior of the population as a whole ticular function is usually justified by a sense the living standard of the popu- should not be confounded with the our general observation that populations lation, one is tempted to hypothesize predicament which arises if one drops do not grow beyond all measures but that the productivity of populations the fictitious assumption that the ele- settle down to a stationary value N., comprised of elements capable of mu- ments are all thriving in a hypothetical which is given at once for a = 0 from tual communication is a monotonic paradise. While the former difficulties Eq. 2 as N. = ao/ a,. Furthermore, rea- increasing function of the number of can be overcome by "merely" develop- sonable fits of the resulting function elements. Tentatively, let a be a weak ing the appropriate mathematical ap- have been observed with actual bio- function of N: paratus to cope with this intricate logical populations-for example, fruit problem, the difficulties in the latter flies in milk bottles (8), bacterial col- a = ao Nl" (3) case are of a different kind, since now onies in petri dishes (7, p. 71), and so where ao and k 1 are later to be de- the fate of the population is not any on. termined from experiment. Inserting longer solely dependent upon the two Regardless of whether or not the Eq. 3 into Eq. 1, and integrating, yields, intrinsic properties of the elements- simple expression given in Eq. 2 is still with the integration constant deter- their fertility and their mortality. Haz- valid if the mechanisms of the inter- mined (t = ti . . . N = N1) at once the ards in the environment, competition action between environment and pop- desired dependence of N (t): between elements for limited food sup- ulation are analyzed more carefully, ply, the abundance of predators or there seems to be strong evidence that, N=N1 to -t ) (4) prey-to name just a few factors- for instance, in sexually reproducing may all act on either mortality or species the advantages of having mates where the characteristic date to replaces fertility or on both, and in the absence more readily available in larger popu- a collection of constants: of further insight into these mechan- lations is more than counterbalanced isms, Eq. 1 becomes obsolete and noth- by the disadvantages resulting from a to = II + a_ N1 -1/ (5) ing can be said about the long-term stepped-up competitive situation if development of our population. The more and more elements have to strug- For obvious reasons, to shall be called since it is on that usual way out of this predicament is gle for existence in a finite environment. "doomsday," date, to devise plausible arguments which In other words, the general idea that t = to, that N goes to infinity and that the clever annihilates will link the two intrinsic properties the productivity may decrease with an population itself. If r = to - t of our elements with some of the increase in the number of elements has "dooms-time" (that is, characteristics of the environment, in undoubted merits. the time left until doomsday), Eq. 4 the hope that the linkage is adequately can be rewritten as N = K/T7. This described and also that one has picked form is listed below together with two those attributes of the environment Coalitions other relations easily derived from Eqs. which are most relevant in studying the 3 and 4. population under consideration (4, 5). However, what may be true for ele- N = K/Tl(6) ments which, because of lack of ade- a = kIr quate communication among each (7) Environmental Influences other, have to resort to a competitive, A tp = (1 - P-1/k) T (8) (almost) zero-sum multiperson game where The usual approach in trying to ac- may be false for elements that pos- k count for the environmental influences sess a system of communication which K=( (9) is to make the productivity a in Eq. 1 a enables them to form coalitions until monotonic decreasing function of the all elements are so strongly linked that In these equations the constant K rep- number of elements N. Since, in an the population as a whole can be con- resents the fundamental constants ao and environment of given size, N is also a sidered from a game-theoretical point k as seen in Eq. 9; in Eq. 7 the produc- measure of the density of the popula- of view as a single person playing a tivity is given as a function of dooms- tion, it is easy to see that increased two-person game with nature as its time and increases more and more density may in many cases reduce the opponent. In this situation it is not ab- rapidly as one approaches doomsday; probability of survival for an individual surd to assume that an increase in ele- Eq. 8 expresses the time intervalY tp be- element-for example, where increased ments may produce a more versatile fore a population which has N elements density aggravates mutual competition and effective coalition and thus not at time 7 will have pN elements. If or improves availability of elements only may render environmental haz- p = 2, one speaks about the "doubling for predators. A typical and popular ards less effective but also may improve time" of the population, and it may be 1292 SCIENCE, VOL. 132 worth while to note that in this popu- taken of the fact that, if (and only if) tion of the communicability of ele- lation, A tp, the "p-folding time" is a an appropriate value for to has been ments will to some extent lose its linear function of dooms-time, in strong established, experimental results should meaning. Thus, if one desires to cal- contrast to exponentially growing pop- appear on a straight line with negative culate the date of the emergence of a ulations, where these intervals are fixed slope k in a double logarithmic plot. hypothetical "Adam"-that is, N =1 for all times: A tp (1/ao) lnp, l/ao For convenience, the abscissa is -one finds it about 200 billion years being the time constant of the growth marked on the lower margin in his- ago. Even astronomers in their wildest process. torical time t and reads from right to speculations have not yet come up with left, while on the upper margin, from an age of the universe which would left to right, dooms-time T is indicated. approximate this figure [current esti- Human Population Similarly, on the left margin of the mates - 24 billion years (13)]. abscissa the number N of elements is In order to check whether or not the recorded, while the right-hand margin hypothesis expressed in Eq. 3 has any gives the global population density Optimists versus Pessimists? merit at all, we took the human world n in elements per square mile; this population as a test case, since it was value is simply obtained by dividing It is hoped that the preceding ex- felt that the most reliable long-range the number of elements by the area pose will add some fuel to the heated data on the development of a popula- A of all the lands of the earth: A = controversy about whether or not the tion comprised of communicating ele- 5.27 X 10' square miles. For compari- time has come when something has ments may be found in the history of son, some density estimates for 1958 to be done about population growth men. The use of estimates of the world are indicated. control. This controversy has divided population rather than of populations From inspection of Fig. 1 and con- those elements of the population under of certain geographical regions elimi- sideration of the small root mean consideration who profess to show nates to a certain extent the influence square deviation of 7 percent, it may some interest in human affairs into of local fluctuations and migration. A be seen that, even without making such two strictly opposed camps (14): the bibliographical search produced 24 es- generalizations as led to Eq. 6, Eq. 11 optimists, who see in the population timates (see 9-11) of the world popula- seems to serve as an adequate empirical explosion a welcome expansion of their tion, ranging over approximately 100 formula for representing most of our clientele, be it consumers of baby generations from the time of Christ recorded data on human population goods (15), voters, or devoted souls (t = 0) almost to the present (t = 1958). growth, covering a time interval of (16), and, on the other hand, the pes- These estimates were carefully checked about two millenia. In the light of the simists, who worry about the rapid with respect to their independence, and interesting singularity supposed to oc- depletion of the natural resources and those which were suspected of being cur at t = to A.D. 2027, the question the irreversible poisoning of our bio- merely cross references in the literature arises as to the reliability of an extrap- sphere (14, 17). While the optimists were eliminated from the statistics in olation beyond a time t* < to. adhere to the thesis that no matter how order to avoid improper weighting. It requires only simple calculations fast the population is growing, food The method of least squares was em- to show that if Charlemagne had had technology and the industrial sciences ployed in order to extract from the Eq. 6, with the evidence he could have will easily keep pace with the develop- data the three values to, K, and k; the had with respect to the world's popu- ment and thus will maintain the ele- following values were obtained: lation, he could have predicted dooms- ments of the human population-at day accurately within 300 years. Eliza- least for some generations to come-in to = A.D. 2026.87 + 5.50 years (lOa) beth I of England could have predicted a perfect state of economic and indi- K = (1.79 + 0.14) X 101" (lOb) the critical date within 110 years, and vidual health, the pessimists prefer to k = 0.990 + 0.009 (lOc) Napoleon within 30 years. Today, paint the future of mankind in not The root mean square deviation for however, we are in a much better po- quite the same rosy colors by pointing all points considered is approximately sition, since we are required to extrap- to the increasing growth rate of the 7 percent. olate our evidence only 4 percent population while assuming that indus- With these values Eqs. 6, 7, and 8 beyond our last point of observation: trial and scientific development will become, with p = 2: we can predict doomsday within ap- proceed at a much slower pace. Hence, proximately 10 years. the pessimists anticipate that further N= 1.79 X 10"/T'99 (11) Although it is always fascinating to rapid increase in the population density a =0.99/Tr per annum (12) imagine one's future fate, the possi- will be accompanied by a deterioration A t2=0.445 -T years (13) bility of deriving some fun by extrapo- in human dignity, and they see the And finally, through Eq. 9, with Eqs. lating our function into the past should ultimate fate of the human race as a lOb and 1 Oc, we obtain for ao, ao = not be overlooked. We find that 1 mil- mere vegetation of the individual on 5.5 x 10-12 lion years ago the world population the edge of existence, if no measures Figure 1 is a graphical representa- was about 200,000 individuals, and 12 are introduced to keep the world popu- tion of the accepted data together with million years ago, not more than per- lation under control (18). the theoretical function (Eq. 6) for haps 15,000 of Hurzeler's Abominable When we refer to our population which values of Eq. 10 have been em- Coal Men (12) populated Tuscany. If growth curve as given in Eq. 11 and ployed. By using logarithmic axes for we wish to extrapolate much further in Fig. 1 and remember the premise the number N of elements, as well as into the past we must be prepared to under which it was derived, it is ob- for dooms-time T, advantage has been find inconsistencies, since the assump- vious that the optimist's viewpoint is 4 NOVEMBER 1960 1293 r

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Fig 1. World population N (left scale) and world population density n in elements per square mile (right scale) observed (circles), calculated after Eq. 11 (solid line) and projected by different authors (triangles) as a function of historical time t (bottom scale), and of dooms-time T (top scale). The numbers associated with each point are references. 1294 SCIENCE, VOL. 132 correct: man has always been able to areas of life where the activity of 2. H. Von Foerster, in The Kinetics of Cel- all lular Proliferation, F. Stohlman, Jr., Ed. develop the appropriate technology to the individual has influenced his own (Grune and Stratton, New York, 1959), p. feed himself, or he has always pro- 391. kind. 3. J. V. Grauman, in The Study of Populations, duced the appropriate population to There is no doubt that it will be ex- P. M. Hauser and 0. D. Duncan, Eds. master his technological tasks. This (Univ. of Chicago Press, Chicago, 1959), traordinarily difficult to establish a con- pp. 557-574. can be conjectured from the relatively trol mechanism, a "peoplo-stat" so to 4. J. Neyman and E. L. Scott, Science 130, 303 (1959); , in "Population Studies," small deviations which actual popula- speak, which would keep the world's Cold Spring Harbor Symposia Quant. Biol. tion counts show as compared with population at a desired level. The im- 22, 109 (1957). 5. A. J. Lotka, Elements of Mathematical Biol- calculated values, in spite of the fact portant point to note here, however, ogy (Dover, New York, 1956), p. 57 ff.; that during the last two millennia men is that it is of secondary importance J. M. Reiner, in Fundamental Formulas of Physics, D. H. Menzel, Ed. (Prentice Hall, underwent several fundamental techno- to find out what this level should be. New York, 1955), p. 733 ff. logical revolutions. Thus, we may con- The primary problem consists of find- 6. P. W. Frank, in The Study of Populations, R. M. Hauser and 0. D. Duncan, Eds. clude with considerable confidence that ing means to keep it constant, whatever (Univ. of Chicago Press, Chicago, 1959), the principle of "adequate technol- this level might be. This means that, p. 661. 7. A. J. Lotka, Elements of Mathematical Biol- ogy," which proved to be correct for if a particular N*, supposed to remain ogy (Dover, New York. 1956). over 100 generations, will hold for at constant, is chosen, obviously dN/dt 8. R. B. Vance, in The Study of Populations, R. M. Hauser and 0. D. Duncan, Eds. least three more. Fortunately, there is must vanish, or a -> 0; hence, -y - (Univ. of Chicago Press, Chicago, 1959), no need to strain the theory by undue lI/ tm. p. 297. 9. In the calculations the year 1958 was not further extrapolation, because-and Since the tendencies today do not considered. here the pessimists erred again-our point in the direction of observable 10. Encyclopedia Americana (Americana Corp., New York, 1955), vol. 22, p. 367. great-great-grandchildren will not starve efforts to reduce the mean life span, 11. H. Hansen, Ed., The World Almanac and to death. They will be squeezed to death. ti, of human individuals-on the con- Book of Facts (New York World-Telegram Corp., 1960), pp. 257, 265. In view of this uncomfortable pic- trary, we see a steady increase in this 12. J. Hurzeler, report on Oreopithecus, New ture it is clear that, while the pessi- value-it is clear that our peoplo-stat York Times 1959, sec. 4, E9 (8 Nov. 1959). 13. A. Sandage, report on age of the Milky mists, one way or another, are "Mal- has to control the fertility y*, and has Way, New York Times 1959, sec. 4, E9 to maintain it at the level l/tm. Today, (3 Jan. 1960). thusians by profession," the optimists 14. H. B. Mills, Jr., Trans. Illinois State Acad. must be "Malthusians at heart," hop- this means cutting the birth rate to Sci. 52, 5 (1959). about half its present value or, in 15. D. Seligman and L. A. Mayer, Fortune ing that at some time, somehow, some- 1959, 94 (Feb. 1959). thing will happen that will stop this other words, cutting the size of an av- 16. Genesis 1:28; 9:1; 13:16; J. K. O'Loane, ever-faster race to self-destruction. erage family to just a little above two Science 130, 1302 (1959). 17. L. C. Cole, Sci. American 198, No. 4, 83 children. Tomorrow, of course, it will (1958); F. Osborn, Population: An Inter- be more difficult, since-as we have national Dilemma (Population Council, New York, 1958); P. B. Sears, Science 127, 9 Population Servo seen-the gap between birth rate and (1958). death rate is widening every minute. 18. R. A. Piddington, The Limits of Mankind (Wright, Bristol, England, 1956); Science But in a highly communicating so- Among the suggestions that have 129, 882 (1959); ibid. 128, 1562 (1958). is no need to been advanced for meeting this prob- 19. R. C. Christopher and T. Griffith, Time ciety there invoke good 75, No. 2, 22 (1960). old Malthus again, who may cite this lem-legislation, heavy taxation of 20. Population Bulletin No. I (United Nations, or that environmental factor whose families that have more than two chil- New York, 1951), pp. 1-3. 21. A. M. Carr-Saunders, in World Population abundance or depletion may curb ex- dren, cancellation of tax deductions, (Oxford Univ. Press, New York, 1937), cess productivity. There is no need to and so on-space travel has been pro- pp. 30, 42. 22. W. F. Willcox, in Studies in American wait until an external mechanism in- posed recently as an alternative (19). Demography (Cornell Univ. Press, Ithaca, fluences human activity. Since today It is only unfortunate that no re-entry New York, 1940), p. 43. 23. C. T. Lewis, Ed., Harpers Book of Facts man's environment becomes less and permit to earth can be given these (1906). less influenced by "natural forces" and space-trotters. 24. Statistical Yearbook, 1940-41 (League of is more and more defined by social Nations, Geneva, 1941), p. 18. References and Notes 25. Encyclopaedia Britannica (Encyclopaedia forces determined by man, he himself Britannica, Chicago, 1959), vol. 18, p. 232. 1. This study was sponsored in part by the 26. H. Hansen, Ed., The World Almanac and can take control over his fate in this National Institutes of Health, Bethesda, Md. Book of Facts (New York World-Telegram matter, as well as he has done in almost (grant No. C-4044). Corp., 1958), p. 265.

4 NOVEMBER 1960 1295