Darwinian selection for self-limiting populations

J. theor. BioL (1975) 55, 415--430

Darwinian Selection for Serf-limiting Populations MARY B. WILLIAMS

Biomathematics Program, Institute of Statistics, North Carolina State University, Raleigh, North Carolina, U.S.A. (Received 27 December 1973, and in revised form 13 November 1974) The rarity of Malthusian overexploitation has long puzzled naturalists. Nicholson has given an important proximate explanation of how population size is limited, but his discussion of density governing traits does not tackle the ultimate question of how populations acquired these traits which prevent overexploitation. This paper shows that Darwinian theory can explain the evolution of density governing limitation mechanisms which prevent overexploitation; they are a predictable by-product of selection for adaptation to the early growth phase of a population with recurrent population crashes. In this situation Darwinian selection, instead of maximizing the number of offspring of the ultimately favored allele, maximizes the number of its descendants in the comparable portion of the next growth cycle. I. Introduction

Malthus's (1798) famous essay may be taken as stating the first scientific theory of the natural regulation of population numbers. But field data do not support his theory; natural populations usually are not regulated by starvation brought about by their overexploitation of their environment. The two most important modern theories are stated in Nicholson (1933) and Andrewartha & Birch (1954). McLaren (1971) provides a recent review of the state of the theories. Most of the work in this field concentrates on elucidating the mechanisms that actually limit the population to a size that prevents overexploitation. Indeed most of the work is on the environmental component of the mechanisms; as Ayala (1968) points out, little attention has been devoted to the genetic component of the organism-environment interaction that results in limitation. The present paper concerns the evolutionary processes which are responsible for the fixation and subsequent maintenance in the population of the genes involved in a density governing limitation mechanism that prevents a population from overexploiting its environment. Wynne-Edwards (1963) (see also Williams, 1971), has suggested that group selection working in opposition to Darwinian selection on individuals is the evolutionary process responsible for fixing limitation mechanisms; many 415

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M. B. WILLIAMS

authors (see especially Lewontin, 1970) have pointed out that group selection cannot account for the ubiquitous regulatory mechanisms for which WynneEdwards invokes it. Chitty (1967), (see also Pimentel, 1968) has suggested that, in animals with cyclic changes in population size, the upper and lower limits of the cycle are controlled by a balance, continually maintained in the population by selection, between the numbers of two different morphs; his mechanism has not been sufficiently explicitly stated to allow an evaluation of whether it could actually control the population size. (One confirmed prediction of his theory, that cyclic populations would show gene frequency changes, would also be predicted for any populations containing genes with density dependent fitness (Charlesworth & Giesel, 1972).) The evolutionary process given in the present paper differs from both of these; it is a hypothesis concerning the evolution of the genetic component of the density governing traits on which Nicholson'S theory of regulation is based. With one exception the mathematical model used in this paper is essentially the same as that used by Roughgarden (1971), and generalized by Charlesworth (1971); the exception is that my model includes an assumption that population crashes are related to the population density, while Roughgarden's population crashes are caused by seasonal changes and are unrelated to the population density. Roughgarden and Charlesworth focus on situations in which the density governing traits are maintained in the population at frequency less than 1; these situations will not lead to selflimitation. I will focus on situations in which the density governing traits are fixed in the population. 2. Introduction to the Model

If r(N) is the intrinsic rate of increase at density N, the fitness function

f(N) is 1 + r(N). In this model r(N) may be negative for some N. Let m be an allele with the density governing (i.e. monotone decreasing) fitness function from" Let n be the allele that is fixed in the population when m appears. Let Emax be such that if the population size reaches or exceeds Em~ there is a population crash. Let fn,(N) be a monotonic function such that: (1) there exist N < Ema x such that (2) fro(N)= ½(fm=(N)+fa,(N)) (3) fro(N) intersects fa~(N) at Ni (4) for N < Nt, fa~,(N) > fan(N) (5) for N > N,, fmra(N) < fan(N).

Nfa,(N) >

Ema x

Conditions (4) and (5) imply that the frequency of m will increase when N < Ni, and will decrease when N > Nt.

SELECTION FOR SELF-LIMITATION

417

Condition (I) ensures that the population containing n suffers repeated overexploitation crashes. For simplicity assume that each crash reduces the population size to Emin. Then the size repeatedly cycles from Em~n to EmaxIf Em~n > Nt, n will be superior throughout each cycle and m will have no opportunity to spread. If Emax < N~, m will always be superior and will be fixed. If Emtn < Ni < Emax, then the percentage of m increases during the first part of each cycle and decreases during the last part. Let k~ be the first generation in a particular cycle and k2 be the last. Denote by N(g) the population size in generation g. Then the frequency of m will be higher at k2 than at k 1 if and only if the geometric mean of the fitnesses of m in those generations is higher than the geometric mean of the fitnesses of n; that is, if

( ~2=~kz fmm(N(g)))I/k2-k~+l > ( ~_~_2 fmm(N(g)))I/k2-~'+~ Abbreviating this inequality by 1-Ifmm > 1-If, n, the possible fates of m can be characterized by the following conditions: (a) For all cycles I-[from < I-Ifnn' In this case m will remain in the population close to its mutation frequency. (b) For all cycles I~[~r~ > l~-f~*" In this case m will spread and ultimately n will remain in the population close to its mutation frequency. Simulation IH below is an example of this. (c) There exists percentages Pm and P~ such that for all cycles in which Pm(kl) < P ' , I-[ mm > 1--[fn~' and for all cycles Pro(k2) < P~. In this case m will spread until Pro(g) /> P " Its behavior thereafter will be investigated below; simulations I and II are examples of this, and section 4 examines the probability of fixation for such alleles. The mathematical techniques available necessitate the division of the investigation of the behavior of m into two parts. (1) Will m ever reach a high frequency in the population? (2) Will it finally be fixed ? Since depends on is a recursive function and no general analytical techniques adequate to deal with such functions are available; consequently the first part of the problem will be investigated using computer simulation. The question concerning fixation will be investigated using traditional mathematical techniques.

fmm(N(g-1)),fmm(N(g))

N(g)

3. Simulations

The mathematical model which translates the above model into a computer program is given in detail in Appendix A. Briefly, the simulations allow us to follow selection between two alleles when population size changes in such a way that each allele is superior for a part of each population cycle.

418

M. B. WILLIAMS SIMULATION I

For the first computer run of this simulation from(N) = 2"1--0"0001N, = 1"5, Em,~ = 11,100, and Eml, = 500. The trait corresponding to m will be a self-limiting trait at this Em~; if m is fixed, the population will regulate around N = 11,000 and will never exceed N = 11,025. Figure 1 shows the changes in Pm, the frequency of m, in the early generations. The peak in each cycle corresponds to the generation just after the population size at which the two fitness functions intersect was reached.

f.,(N)

x 25

o

0

X

x

IO,O00

x

8,

X

2O x

o

X

x 0

x x

x

g X

I0

~

x

x

x

o

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x

:~

8000

x

6000 o

o

X

5

0

0

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0

0

0

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I

40

45

0

0

~, 4000

x x

~ _o ~

I

50

2000

0 I 0

55

0

I 60

,,,

500

Generolion

FIo. 1. Changes in mutant percentage (X) and in population size (O) during early generations of selection, where homozygote and heterozygote fitness functions are those given in Table 2. A population in which this mutant is fixed will self-stabilize. Below that size from > f . . so Pm increases; above that size from < f . . so the Pm decreases. The low point in each cycle corresponds to the generation when the population size reached Emax and crashed; after the crash the population is again at a size at which from > f . . . Table 1 shows the actual numerical results for generations 63 through 70. This figure illustrates the necessity, when the fitnesses vary systematically over the generations, of viewing selection as maximizing the number of descendants a suitable number of generations distant. When selection can be viewed as maximizing the number of offspring, the selective fate of an allele can be determined by comparison of the arithmetic mean of the fitnesses of individuals bearing that allele with the arithmetic mean for competing alleles. In this cycle, however, the arithmetic mean of the fitnesses of the 3517 mm

63 64 65 66 67 68 69 * 70

Generation

500 823 1361 2250 3680 5851 8812 (12,211) 500

Population size

35 73 148 294 553 960 1454 (1780) 73

No. of mm individuals

2.050

2"050 2'018 1'964 1'875 1"732 1"515 1"219

Fitness of mm individuals 195 344 602 1038 1747 2820 4251 (5764) 236

individuals

No. of mn

1"775

1"775 1.759 1"732 1"687 1"616 1.507 1"359

Fitness of mn individuals 270 406 611 918 1380 2071 3107 (4668) 191

individuals

No. of a n

1"5

1"5 1"5 1'5 1"5 1'5 1"5 1"5

Fitness of nn individuals

26"5 29"8 33"0 36"1 38"8 40"5 40"6 (38.2) 38"2

genes

~ mutant

Results of calculations for generations 63 through 70 in the one mutation path. The asterisk indicates the before-crash numbers in the crash generation

TABLE l

420

M. B. WILLIAMS

individuals is 1.488, while the arithmetic mean of the fitnesses of the 8763 nn individuals is 1.5. The geometric means are 1.739 for rnm and 1.5 for nn. This disparity, with the arithmetic mean giving an incorrect prediction of the selective fate and the geometric mean giving the correct one, is also found in the other cycles pictured in Fig. 1. If K is any positive integer, selection is maximizing the number of descendants K cycles later. Figure 2 shows the changes in the frequency of m during later generations of selection. The cycle to cycle increase pictured in Fig. 1 has stopped and the frequency of m has begun to oscillate between 0.97 and 0-99. The reason for this change is that when the frequency of m is high the rate of increase I00

99 xXX e8,

98

x X X X x X

X

X

xXX

X

X

X

X

X

X

X X

97 8 96

95 i 260

! 265

I 270

I 275

I 280

Generation

F1o. 2. Growth of mutant percentage during later generations of selection. (Later stage of process graphed in Fig. I.) of the population is close to the rate of increase of m; consequently the population is at numbers below 6000 (where m is superior) for fewer generations when the frequency of m is higher than when it is low; it is also at numbers above 6000 for more generations, giving n more chance to increase its frequency. The oscillation will continue forever between 0"97 and 0"99 unless random factors eliminate the few remaining n. A second computer run with E m a x = 11,027, and all other conditions as given above, showed that with this lower Emaxthe oscillation frequencies are between 0.992 and 0.996. This simulation shows that, in selection between an allele with a selflimiting fitness function and an allele with a density independent fitness function, the self-limiting allele can spread through the population almost to fixation.

421

S E L E C T I O N FOR S E L F - L I M I T A T I O N SIMULATION II

F o r this simulation f=m(N) = 2' 1 - 0"0001N, f..(N) = 1" 9 - (0"0000667)N, Ema~ = 11,100 and Emi, = 500. Figures for this case corresponding to Figs 1, 2, and Table 1 would be essentially the same. The process begins with a cycle to cycle increase which continues until the frequency of m is about 0-91, the frequency thereafter oscillates between 0.90 and 0.92. A second computer run with E ~ x = 11,027 and all other conditions as given above, resulted in oscillation frequencies between 0.965 and 0.975. A third computer run with E,,~x = 11,027, Eml, = 200 and all other conditions as given above resulted in oscillation frequencies between 0-990 and 0.995. This simulation shows that, in selection between alleles m and n for which from is self-limiting and f,n is density governing though not self-limiting, m can spread through the population almost to fixation. SIMULATION III

F o u r computer runs followed the changes in frequency of m for different pairs of fitness functions. The functions tested were: (I) f l = = ( N ) = (2) f2mm(N) = (3) famm(N) = (4) f4m=(N) =

1.6-0.0000167N v e r s u s f , , ( N ) = 1.5 1"7 - 0"0000333N versus f t ram(N) = 1"6 - 0"0000167N 1"8 - 0"00005N versus f2mm(N) = 1"7 - 0"0000333N 1"9 - 0"0000667N versus f3mm(N) = 1"8 - 0"00005N.

These functions are graphed in Fig. 3. For these runs Emax = 10,100 and Emi. = 500. N o n e of these from would be self-limiting at this Em,x. The

1'8 1.7 1.6 1.5

.-= I'0 kt..

}

1

[

t

I

1

I000

5000

5000

7000

9000

I 1,000

Fro. 3. Density governing functions used in simulation I l L

422

M.B.

WILLIAMS

numbers at which the from would stabilize the population are: 47,904 for flm=, 25,525 forf2m, 18,000 for f3 m and 14,243 f o r f , mm. For these cases the geometric mean for the left member of the pair was greater than the geometric mean for the right one for all cycles until n disappeared from the population. In each of these cases the more strongly density governing function was Darwiniauly selected for and deterministically fixed. Simulations I and II investigate the possibility of a self-limiting trait being selected for against a non-limiting trait. This simulation investigates the possibility of successive fixations of successively more strongly density governing traits. (Note that the successive fixations need not be of different alleles at the same locus. Once an allele is fixed, its fitness function can be regarded simply as the fitness function of the population.) This simulation shows that a population can become more and more strongly density governed. It also indicates that one probable result of selection in a population subject over a long period to frequent population crashes is the fixation of many density governing genes. 4. Fixation

Consider the case of a self-limiting gene in competition with a nonlimiting gene: that is, Nfmm(N) < N* < Ema, for all N but there is an N' such that N'f,,,(N') > Ema,,. Then if the n allele is in the population it continues to increase after the population size reaches N*, when the m allele has stopped increasing. Thus regardless of how high Pm can become at the most favorable point in the cycle, before the population reaches Em~x and crashes P, will have increased to at least (Emax - N*)/Em~ x; therefore for all k2, Pm(kz) < 1 - (Em~,- N*)/Emax. In fact, it is possible to get an even better upper bound on Pro(k2). For well-behaved functions f , m a n d f . , it is possible to find the maximum Pro(k2) such that N(k2)>t Em~,. For the functions from(N)= A - B N and fm,(N) = C - D N , which were used in the simulations, this maximum is found in Appendix B. The maximum depends onf,,m, f , , , and Em~x,but not on E,,i,. On the other hand, whether the maximum is ever attained does depend on Emi,, since, e.g. if Emi. is too large m will always be inferior and will not spread at all. Table 2 shows some values for this maximum for fitness functions used in the computer simulation. The maximum crash frequency actually attained in the computer runs was fairly close to the theoretical maximum; runs with smaller E,,~n would have produced actual maxima closer to the theoretical maxima.

SELECTION FOR SELF-LIMITATION

423

TABLE 2

Maximum possible crash percentages, Pro(k2), for two values of E,,ox for f,m(N) = 2.1 - 0.0001N versus increasingly close f,,(N)

fan(N) = f.n(N) = fnn(N) =

1"5 1"9--0"0000667N 2"0--0'0000834N /nn(N) = 2"05--0"0000917N

F-~ax = 11,027

F~ax = 11,100

0"9992365 0'9979569 0"9958658 0"9882930

0"9806690 0"9262897 0'8584499 0"7279904

Now notice that when the crash frequency of m is close to the maximum, the crash frequency of n may be very small. In the computer runs the crashes are assumed to affect m and n equally; thus the frequencies of m and n after a crash were the same as the frequencies in the population which exceeded Emax. But this will be exactly true only in a computer simulation; in actual situations random factors will cause some changes in the frequencies during the crash. When the frequencies are not too small these random factors can be ignored since the changes will be balanced either by other random changes or by the homeostatic nature of the situation, which tends to bring the frequencies back within a fixed oscillation range. But when the frequency of n is very small one of these random changes may force it to zero, and once this happens m is fixed. Let S = 2Eml,. Then the probability of n being lost during a crash is [P~(k2)] s. The closer Pro(k2) is to 1, the larger this probability is; the smaller S is, the larger this probability is. The probability of n being lost during one of t successive crashes is 1 - { 1 - [Pm(k2)]s} t. For large t, small S, and Pm close to 1 this probability is close to 1. Unless something else happens to prevent population crashes, 't will be large. Therefore, if there are many potentially self-limiting traits that have spread through their populations, the probability that some will be fixed by random factors is very large.

5. Biological Mechanism These results show that self-limitation may be evolved as a non-adaptive by-product of selection for adaptation to the early growth phase of a population cycle. The self-limitation trait may be the result of one gene or many genes. Any density governing limiting trait may be evolved in this way; as the following three examples indicate, the biological nature of the trait is virtually unrestricted. First consider territorial systems: taking the defence of a home site as primitive, we must explain the expansion of the defended area and the

424

M. B. W I L L I A M S

refusal of landless animals to reproduce. Let the territory be of any size up to the maximum useful to the owner. The selective advantage to the owner in defending it, minus the cost of defence, results in an overall advantage to defending a somewhat smaller territory. The animal that responds to defenceof-territory threats by searching elsewhere for a territory and not reproducing until he finds one will have, in low density generations, a fitness advantage over the animal that stays and fights for the territory or which reproduces without a territory. In high density generations the animal which looks elsewhere for a territory probably won't find one, so the animal which stays and fights will have a selective advantage. Therefore a look-elsewhere trait will have a density governing fitness function, and genes facilitating this behavior can be selected for in a population with frequent population crashes. Now consider dependence on a particular type of nesting site. Suppose there is a series of mutations which increase the strength of the urge to use a nesting hole instead of a relatively unprotected nesting site; this urge could begin as a slight preference, progress to an urge demanding an extensive search for a suitable nesting hole before settling in an unprotected site, and end in individuals who refuse to reproduce if nesting holes are not available. In an environment with a limited number of nesting holes each such allele would be density governing, and these alleles can be selected for by the described process. Lastly, consider regulation by a trace element. Suppose there is a mutation to an allele which greatly increases the efficiency of a particular enzyme by incorporating a copper ion in it. Assume that the available source of copper is a plant which is eaten by all animals in the population and that the total amount of available copper is close to that consumed by the population at N "
S E L E C T I O N FOR S E L F - L I M I T A T I O N

425

Darwinian selection maximizes the number of offspring, while (as Fig. 1 and the discussion of Table 1 show) in this case Darwinian selection maximizes the number of descendants in the same stage of the next cycle. The self-limitation resulting from such a single fixation could be destroyed by a single appropriate mutation. But the results of all three simulations illuminate an evolutionary process that would produce a self-limitation trait that could not be easily destroyed. In populations with frequent crashes: (1) Density governing traits will be selected for as a by-product of selection for adaptation to the early growth phase. (2) These selected density governing traits range from traits which would self-limit far above the then attainable population maximum down to traits which would self-limit below the then attainable population maximum. For any particular amount of low density advantage, the higher the selflimitation point of the trait the greater the probability of its being fixed in the population. (3) All of the population traits which are affected by density (e.g. foodgetting techniques, mate-finding techniques, predator-avoiding techniques) will have been selected to be efficient in the normal range of population sizes, and many of them will be progressively less efficient outside that range. (4) Therefore genes which improve fitness at low population density and are deleterious at densities above the attainable population maximum will accumulate in the population. [This situation is closely analogous to the situation postulated by Hamilton (1966) concerning the evolution of senescence.] (5) The factors noted in (3) and (4) make it probable that any population which has been kept within a certain density range for a long period of time will have many density governing genes which wilt become deleterious above that range. What would happen to such a population if a drastic climatic change (e.g., the end of a glacial epoch) significantly increased Em~~? The population size would increase until the combined deleterious effects stabilized the population. ]-Note that Calhoun's (1962) crowded rats exhibited a multitude of deleterious effects, though in the protected experimental environment these did not prevent increase beyond the normal density. Also Snyder's (1961) apparently self-regulating woodchucks exhibited several deleterious effects at their stable density.] Now the question is whether selection would soon eliminate the deleterious genes, destabilizing the population so that it again overexploits its environment. If only one gene were involved this would be expected. But if this population has been selected over millenia for adaptation to a certain range of population density, a great many coadapted

426

M.B. WILLIAMS

density governing genes are involved. All of these genes must be removed from the population in order for the self-limitation trait to be destroyed. Mayr (1963), in his discussion of the founder principle, showed one way in which a major reconstruction of the gene pool could be accomplished; the traditional, consciously oversimplified, picture of genes being replaced one by one, independently, shows another. Let us now consider the possibility reconstructing the gene pool so as to remove the self-limiting trait. First note that the trait will not be removed as a result of selection in a founder population; each of the genes involved, although deleterious at high population density, is advantageous at low population size and thus will be selected for during the period of reconstruction. Nor is it likely that the trait would be lost randomly; some of the many density governing genes might be randomly lost, but the probability of enough being lost to destroy the trait is very small. So a population could undergo several major reconstructions of its gene pool without losing the self-limitation trait. Now let us consider the probability of loss of the trait due to the process of independent replacement of individual genes. Each gene must be replaced by an allele which not only raises the efficiency at high population sizes of the character it primarily affects but is also at worst only mildly deleterious for the other characters which it affects. Since these genes have been in the population sufficiently long to have acquired a web of coadaptive interactions with the other genes in the population, the probability of a mutation to such all allele is about as low as the probability of an advantageous mutation for an average gene. But a many-gene self-limiting trait has additional sources of deleterious side effects of an otherwise advantageous mutation: every increment in the attained population size caused by the fixation of such a gene causes a decrement in the fitness of every density governing gene in the population. Furthermore, the fixation of such an allele would probably cause an increase in the number of deleterious genes, since (1) the higher attained population size would cause genes that had been slightly advantageous to become deleterious and (2) the fact that the probability of fixation had been greater for genes with a higher limitation point implies that each increment ill the attained population size causes more genes to become deleterious than the previous increment had. These factors show that the removal of a limitation trait is a process which at best very slowly increases the size at which the population is limited until a size is attained at which overexploitation is possible. The population may, in fact, be on an adaptive peak with respect to population size; since it is clear that any increase in size necessarily causes many deleterious effects, the possibility that there are no sufficiently advantageous size increasing mutations cannot be ignored.

SELECTION FOR SELF-LIMITATION

427

These considerations do not show that such a self-limitation trait could never be destroyed, but they do show that the genetic inertia opposing such destruction is very strong. Therefore the trait, once it is entrenched, will tend to remain in the population. A glacial epoch might provide an environment in which these genes are likely to become entrenched in the population; in the less rigorous environment of the following interglacial they would limit the population below the size at which the environment is overexploited. Since the earth's history shows many glacial epochs during which such a trait could become entrenched, since the trait once entrenched is unlikely to be lost, and since our own epoch is an interglacial, this evolutionary process can explain the existence of contemporary species with density governing self-limitation mechanisms. CONTEMPORARY CYCLIC POPULATIONS The functions used in the computer simulations ensure an essentially stable population size once the genetic mechanism is fixed in the population. Many contemporary species exhibit cyclic changes in population size; presumably many of them lack any internal control mechanism, but Chitty (1967) has suggested that population size is controlled in some of these species by natural selection acting on genetically different morphs within each cycle. Since the model presented in this paper concerns cases in which control is exerted by genetic mechanisms which are fixed in the population, Chitty's model is not a special case of my model. However, it is probable that my model could be extended to account for fixation of a genetic mechanism with a fitness function with lag terms which would prevent the population size from increasing far enough to cause environmental destruction but which would cause the population to undergo cyclic changes in size. Further computer simulations, using fitness functions with lag terms, will be necessary to determine whether cyclic changes can be subsumed under the model presented here. 7. Conclusion

During a period in which a population is subjected to frequent population crashes there will be selection for density governing traits. Some of these traits can actually limit the population in the environment in which they were selected; others pre-adapt the population for self-limitation in an environment which can sustain significantly more individuals. The limitation mechanism generated by these preadapted traits has a genetic structure which makes its destruction by natural selection very difficult. Therefore the •r . n .

:28

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rarity of contemporary species which overexploit their environment can be explained as the result of a change from a very rigorous environment (e.g., during a glacial epoch) to a much less rigorous environment. This work was supported by National Science Foundation Grant GU-1590. I would like to thank an unknown referee who pointed out the analogy to the evolution of senescence and thus stimulated a train of thought which greatly improved the paper. REFERENCES ANDREWARTHA,H. G. & BIRCH,L. C. (1954). The Distribution and Abundance o f Animals. Chicago: University of Chicago Press. AYALA,F. (1968). Science, N. Y. 162, 1453. CALHOUN(1962). Scient. Amer. 206, 139. ~tARL~WORTn, B. (1971). Ecology 52, 469. ~Lr.SWOa'rH, B. & GI~EL, J. (1972). Am. Nat. 106, 402. CalTTY, D. (1967). Proc. ecol. Soc. Aust. 2, 51. HAMILTON,W. (1966). J. theor. Biol. 12, 12. L~.woNaTr%R. (1970). Annual Review o f Ecology and Systematics. Palo Alto, California: Annual Reviews, Inc. MALTHUS,T. (1798). An Essay on the Principle o f Population. London: Johnson. MA~, E. (1963). Anhnal Species and Evolution. Cambridge, Mass.: Harvard. McLAREN,I. (ed.) (1971). Natural Regulation o f Animal Populations. New York: Atherton. NICHOLSON,A. (1933). J. Anhn. EcoL 2, 132. Pt~mN'r~, D. (1968). Science, N.Y. 159, 1432. ROUCHOARDEN,J. (1971). Ecology 52, 453. SNYDER,R. (1961). Proc. natn. Acad. Sci. U.S.A. 47, 449. WlLLtAMS,G. (ed.) (1971). Group Selection. New York: Atherton. WVNNE-EDwAaDS,V. (1962). Animal Dispersion in Relation to Social Behavior. London: Oliver and Boyd.

Appendix A Let N ( g ) denote the size of the population in generation g. Let Nnn(g), N m ( g ) , and Nmm(g) denote the number of, respectively, nn, mn, and mm individuals in generation g. Let 2fn,(N(g)), 2fm,(N(g)), and 2fmm(N(g)) denote the average number of gametes surviving to adulthood produced by, respectively, nn, mn, and mm individuals in generation g. Then the number of m gametes produced is 2Nmm(g)f~,m(N(g))+Nm(g)fmn(N(g) and the number of n gametes produced is 2 N , , ( g ) f n n ( N ( g ) ) + N m , ( g ) f m n ( N ( g ) ) Assuming random mating, the numbers of nn, mn, and mm individuals in the next generation is given by the Hardy-Weinberg principle. Let E~ax denote the maximum population size the environment will support and Emin denote the size to which the population crashes if the population size exceeds Em~. In the following equations square brackets,

SELECTION

FOR

SELF-LIMITATION

429

[ ], will denote the "greatest integer in" function; it is used at crashes to weed out fractions of organisms. If P=(g) is the percentage of the m allele in generation g and k~, k2, and k 3 are arbitrary constants, the following equations provide a recursive definition o f Pro(g) : N.,m(0) = k~,

N.,.(0) = k~,

N . . ( 0 ) = k3

S(g) = Nmm(g- 1)fmm(N(g- 1 ) ) + N m . ( g - l)fm.(N(g- 1)) + N . . ( g - 1 ) f . . ( N ( g - 1)) Pro(g) = (Nmm(g- 1)fmm(N(g-- 1))+(½)Nmn(g- l)fm.(N(g- 1)))/S(g) N(g) = S(g) = Emi.

if S(g) < Ema~ if S(g) >~ Emax

Nmm(g)

(Pm(g))2N(g) = [(Pm(g))2N(g)]

if S(g) < Emax if S(g) >>-Emax

N.,.(g)

= 2Pro(g)(1--em(g))N(g)

if S(g) < Em~,

= [2Pm(g)(1-Pm(g))N(g)]

if S(g) >>.Em~x

=

N..(g) (I -P~(g))2N(g) = [(l-Pm(g))ZN(g)]

if S(g) < Em~, if S(g) >>.E ~ ,

In the examples used in the paper, from(N) = A - BN, fm.(N) = (½)(from(N) +f..(N)), and f . . ( N ) = C - DN.

Appendix B For the functions from(N) = A - B N and f . . ( N ) = C - D N , we wish to find the maximum frequency of m during the crash generation, k2, of a cycle. Since k2 is a crash generation, N(k2) > Emax. The maximum Pro(k2) will occur for the smallest such N; therefore set N(k2) = Emax. Let r = Pro(k2 - 1) and s = N ( k 2 - 1 ) . Then Ema. = N(k2) = Nmm(k2 - 1)from(S)+ Nm.(k2- t)fm.(S) + N..(k2 -- l)f..(S) = r2s(A - Bs) + (1 - r)rs(A - Bs + C - Ds) + (I - r ) Z s ( C - Ds). Solving for r, we get, Ema~- s( C - Ds) r = s ( ( A - C) - (B - O)s)"

430

M.B.

WILLIAMS

T r e a t i n g r as a f u n c t i o n of s, dr ds ( - C + 2 D s ) s ( ( A - C) - ( B - D)s) - (Emax - -

s(C

D s ) ) ( ( A - C) - 2(B - D)s)

--

s2((A - C) - (B - D)s) 2 T o m a x i m i z e we set the derivative equal to zero, which implies: 0 = ( - C + 2 D s ) s ( ( A - C ) - ( B - D ) s ) - (Emax - s ( C -

Ds))((A - C) -- 2(B-

= s 2 ( D ( A - C ) - C ( B - D)) + s ( 2 ( B - h ) E m ~ ) -

h)s)

(A - C)Em=.

Therefore r is m a x i m u m when

8=

- 2 ( B - O)Ema ~ + x / ( 2 ( B - O)Em~) 2 + 4(D(A - C) - C ( B - D ) ) ( A - C)Em~x 2(D(A - C) - C ( B - D))

S u b s t i t u t i n g this s into the e q u a t i o n for r, we find r as a f u n c t i o n o f these parameters. F r o m this we can calculate Pm(k2) f r o m : Pm(k2) =

sr

2fmm(S) + (¼)(1 -- r)rsOemm(S) + f ~ . ( s ) ) .