J. theor. Biol. ( 1977) 69, 543-560
Genetic and Phenotypic Models of Natural Selection DAVID G. LLOYD Botany Department, University of Canterbury, Christchurch, New Zealand (Received 5 July 1977) The following theorem is proposed: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness, provided that certain modes of inheritance apply; in particular, all genotypes specifying the same phenotype must have the same average fitness. If these conditions of “uniform fitness” patterns of inheritance are not met, particular genetic models of natural selection should replace an analysis of phenotypes. If the conditions are met, an analysis of the stationary conditions when the phenotypes have equal fitnesses permits quantitative statements about the outcome of selection without recourse to genetic models. Phenotypic analyses of natural selection are illustrated by models of sex ratios in plants, sexual versus asexual reproduction in plants, and parental investment by animals.
1. Introduction In genetic models of natural selection, genotype and gamete fitnesses are specified and the outcome of selection is determined in the first instance in terms of genotype and allele frequencies. Even when they are of greater interest, phenotype frequencies are obtained secondarily from the genotype frequencies. Genetic models of natural selection established the hereditary basis of evolutionary change and sponsored the modern theory of evolution. Phenotypic models of natural selection reason that fitter phenotypes (those contributing on average more genes to the next generation) inevitably increase at the expense of less fit phenotypes until the contrasted phenotypes are equal in fitness or one phenotype completely replaces the other. The verbal arguments of Charles Darwin and A. R. Wallace, although not couched in such terms, were of this kind. More recently, phenotypic models investigating adaptive strategies have commonly assumed that the outcome of selection can be predicted solely from a knowledge of phenotype fitnesses and that the particular mode of inheritance of the phenotypes is unimportant. 543
544
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There appears to be some uneasiness about the assumption that models of adaptive strategies arc applicable to all patterns of inheritance of the phenotypes considered, however, as genetical models are sometimes added to strategic ones, e.g. Murphy, 1968. The present paper distinguishes modes of inheritance under which evolutionary change ceases (stationary conditions are reached) when the fitnesses of two phenotypes are equal. This property provides a simple method for obtaining fully quantitative phenotypic models of natural selection. Although phenotypic models of natural selection contain no information on genotype or allele frequencies or rates of approach to selection targets, they are nevertheless often valuable, as their recent popularity testifies. They are applicable to a whole class of genetic models, whereas genetical arguments have no validity beyond the particular mode of inheritance that is specified. And since they omit genetical considerations, phenotypic models can consider more complex ecological situations. As a result, they are more suitable than genetic models when the primary interest lies in the effects of phenotypic parameters on fitness rather than in their genetical basis. Certain circumstances such as partial self-fertilization are difficult to incorporate into genetic models of selection (Fisher, 1941; Jain & Workman, 1967). Phenotypic models avoid these difficulties. A number of phenotypic parameters can be readily handled in phenotypic models involving mixed self- and crossfertilization, as illustrated below (and in Lloyd, 1975 and in press: Charlesworth & Charlesworth, in press). 2. Genetic Conditions for Phenotypic Analyses After cross-fertilization a zygote receives one set of genes from each gamete and parent, but after self-fertilization both sets of genes are derived from the same parent. If a fitness measure is to be applicable to selffertilization (or inbreeding generally), it must therefore be defined in terms of the relative genetic contributions to the gametes or genes making up the offspring rather than as the number of offspring themselves. Fitness is defined here as the average number of genes which a genotype, phenotype or allele contributes to the next generation. Consider two phenotypes, P, and PI, which differ to any extent in any set of attributes affecting their fitnesses. The two phenotypes may occur at any The phenotypic differences are relative frequencies in one population. determined by two alleles at one or more controlling loci, i.e. by aI, a2 ; b,, b,, etc. Selection at only one locus, a, is considered initially. By definition, the frequency of the fitter allele increases from one generation to the next at the expense of the less fit allele. But phenotypes are not directly inherited.
MODELS
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NATIJRAL
545
SLLEC~IIOU
The combination of genes into genotypes intercedes between the phenotypes of successive generations, so it does not follow that selection of the fitter allele necessarily increases the frequency of the fitter phenotype, even in the absence of interlocus interactions. The changes in phenotype frequencies that accompany selection at controlling loci must be established for each mode of inheritance of the phenotypes. The average fitnesses of alleles a, and a2 are designated WI and W2, and the average fitnesses of P, and P, are @(PI) and W(P2). We seek to delimit a class of genetic models for which it is true that the two phenotypes have equal fitnesses at stationary conditions, i.e. fo1 which W(P,) = iT(P,) when tj, = E2. (A)
UNIFORM
FITNESS
MODES
OF INHERITANCE
In “uniform fitness” modes of inheritance, all genotypes of the same phenotype have the same average fitness. That is, for the a locus, qa,a,-P,)
= iiqa,u,-P,)
= w(u,u,-P,)
= W(P,),
(1)
W(u,u, -P2)
= w(u,u~-P,)
= i&u,-P,)
= W(PJ,
(2)
and where alal -PI, etc. are all individuals phenotype one, etc. Then,
of genotype
alal
which
have
w, = ivl)f, 1+ ~(P2lfi2 fl ’ and w
=
Wdf* __--
I
+
W2)f22
2 f2
where the frequencies of a, in individuals and fi2, and the combined frequency .f2 = f21 +f22, where fi +fi = 1.
3
of P, and P, are respectively f, , of a,, fi = fi I +fi2. Similarly
Substituting
gives w,--zw,
=
(3)
Stationary conditions at the a locus are defined as the conditions under which selective forces cause neither allele to increase in frequency from
546
I). G.
l.l.OYI~
one generation to the next. These apply when and only when i?r = wz. From equation (3), this may occur when (fi rlf --fzllfi), which we call X, is zero. If x = 0, the two alleles are represented in the same proportions in P, and P,. This can happen only in the trivial case when locus a has no effect on the phenotypes or when the substitution of a, for a, causes a switch from P, to P, in some genetic backgrounds and causes the reverse switch from P, to PI in other backgrounds. This would in turn require such unusual gene interactions that it may usually be discounted. Hence in uniform fitness genetic models, unless an allele has opposite effects in different genetic backbackgrounds, if ti;, -I, = 0, ti(P1)-ii
= 0.
(4)
The two phenotypes have equal fitness at stationary conditions. Suppose now that a higher proportion of a1 representatives than of a, representatives occur in P,. Then x is positive, and from equation 3, if WI -cj, > 0, E(P1)-5(P,) > 0. (5) That is, the phenotype with the higher proportional representation of the fitter allele is always fitter. Moreover, the fitter phenotype, P,, also increases except in the unlikely circumstance that an increase in the frequency of a, is accompanied by a decrease in the frequency of the phenotype which it specifies by itself or in combination with other genes. In general then, if not invariably, the fitter phenotype increases. By parallel arguments, when x is positive and E1 - Wz -C 0, W(P1)-G(P,)
< 0,
(6)
and the fitter phenotype (P2) again generally increases. This property also applies when x is always negative. Similarly, for other loci (b, c, etc.) which participate in specifying phenotypes 1 and 2, the phenotypes have equal fitnesses when the alleles have equal fitnesses. Hence when evolutionary change due to selection ceases at stationary conditions, the phenotypes have equal fitnesses with the genetic models stipulated, regardless of the number of loci controlling the phenotypes and the frequency of recombination between them. But where factors such as linkage disequilibrium or genetic drift interfere with the response of a locus to the selection forces operating on it, evolutionary change may not cease when the phenotypes have equal fitnesses. The analysis above is confined to loci with two alleles and to two phenotypes. Dr Brian Charlesworth (Appendix) has provided a general proof, for n alleles at a locus with m phenotypes, that in equilibrium populations the fitnesses of the phenotypes are equal unless the number of phenotypes
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SELI:ClION
547
exceeds the number of alleles at the locus concerned, or special constraints are placed on the equilibrium genotype frequencies which are independent of the equilibrium phenotype fitnesses. (8)
VARIABLE
FITNESS
MODES
OF INHERITANCE
In many patterns of inheritance, such as those involving lethal or subvital genotypes, overdominance or epistasis, different genotypes of the same phenotype do not have equal fitnesses-equations (1) and/or (2) are not entirely valid. For such “variable fitness” modes of inheritance, w
#
~(Pl)fll
+wzM-12 __--
I fl
’
W(P,)]
[ I
and/or
and J,
-J,
#
[I(P,)
When W, - iV2 = 0, iifP,)-
-
iG(PJ # 0.
j-11 fi--
f21 j;
.
(8)
The stationary conditions when the alleles have equal fitnesses will not coincide with equal fitnesses of the phenotypes. Hence Darwin’s argument and the common assumption of strategic analyses, that characteristics associated with an advantage in reproductive contributions will increase in frequency until the fitter phenotype replaces the less fit or their fitnesses become equal, is not always correct. Consider, for example, the situation whlere dominant and recessive alleles at one locus specify contrasted phenotypic traits and the controlling locus also displays overdominance, as in severa melanic polymorphisms in Lepidoptera (Kettlewell, 1973). The three genotypes all have different fitnesses, and at equilibrium the average fitnesses of the two phenotypes are not equal. Despite the truism that natural selection operates directly on phenotypes and not alleles, phenotypic models of natural selection do not always give correct results for variable fitness patterns of inheritance. When genetic interactions that significantly violate the uniform fitness conditions are operating, phenotypic models of selection may not give even a close approximation to the outcome of selection. Genetic models of natural selection are then necessary.
54x
I>.
(C)
c;.
GENERAL
LLOYD
THEOREM
The limited independence of phenotypic models from genetic considerations can be expressed in the following statement: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness only if all genotypes specifying any one phenotype have the same average fitness and no allele controlling the phenotypes has opposite effects in different genetic backgrounds. This theorem of phenotypic fitness delimits genetical conditions under which one can define adaptive strategies in purely ecological terms without explicit reference to genetics. The theorem is generally limited to the fitnesses of zygotes, since equations (3) and (4) apply for one complete generation only to zygotes. If two phenotypes differ in their rates of survival between zygote formation and reproductive adulthood, selection will not necessarily cause the fitter adult phenotype to increase even with uniform fitness modes of inheritance. The theorem applies to a later stage of development only if there is no change in the relative fitness of the phenotypes between the zygotes and that stage. The theorem does not require any one controlling locus to be fully responsible for specifying the phenotypes. Consequently, it will still hold if the phenotypes are not entirely determined by genes but are partly elicited by environmental conditions. Selection at a locus will generally result in the fitter of two phenotypes increasing (with uniform fitness genetic models) if the locus has any influence on the phenotypes. Several aspects of the preceding analysis have already been verified in the case of gynodiecy, a sex polymorphism in plants. In gynodioecious popultions, the unisexual females contribute genes to the next sexual generation solely through ovules, while the more or less bisexual males contribute genes predominantly through pollen but also to a minor degree through ovules (Lloyd, 1976). First, for any value of the relative (d/q) seed fecundity of the sexes, the equilibrium sex ratio is the same for genetic models in which the female phenotype is determined by a dominant or recessive allele of one locus or is controlled by three different two-locus models (Lewis, 1941; Ross & Shaw, 1971; Lloyd, 1974a). All five genetic models are uniform fitness models and should therefore give the same sex ratios according to the above analysis. Second, a number of variable fitness models of gynodioecy involving either lethal sex genotypes (Lloyd, 1974~) or overdominance (Ho & Ross, 1973; Ross & Weir, 1975, 1976) give equilibrium sex ratios which differ from those of uniform fitness models and depend upon the particular genetic model operating. Third, the average fitnesses of male and female zygotes are equal at equilibrium with the uniform fitness genetic models but not with the
MODELS
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StLECTlUN
549
variable fitness models (Lloyd, 1974b, 1975; Ross & Weir, 1975, 1976; Charlesworth & Charlesworth, in press). Fourth, when differential survival of the sexes occurs, the average fitnesses of male and female adults are not equal at equilibrium for either class of genetic models (Lloyd, 1974b). The genetical restriction in the theorem of phenotypic selection proposed above does not operate in the process of group selection between noninterbreeding populations. Suppose that two phenotypes, P, and P, OCCUI at frequencies of 100% in different populations, and that they are determined by alleles a, and a2 respectively of a controlling locus. When inter-population selection occurs by differential extinction or multiplication, a daughter population invariably has the same allele and phenotype as its parent population. Hence the fitness of alleles coincides with the fitness of phenotypes, and the population with the fitter phenotype always increases in the mctapopulation regardless of the genetical basis of the phenotypic difference. 3. Analyses of Stationary
Conditions
The condition that contrasting phenotypes have equal fitnesses a~. stationary conditions with uniform fitness modes of inheritance provides a useful quantitative technique for analysing the outcome of selection whenever the fitnesses of the phenotypes can be expressed exactly. All that is necessary is for the difference between the fitnesses of two phenotypes to be written down. Then the direction of selection can be determined for uniform fitness genetic models under any combination of values of the parameters affecting fitness. The outcome of selection is obtained from stationary conditions when the difference in fitness of the phenotypes is zero. The contrasted phenotypes may occur in separate individuals in a population or in different populations for group selection models. If the phenotypic fitnesses in individual selection models depend on their frequencies in such a way that each phenotype has ;I relduced relative fitness whenever its frequency exceeds that at which the phenotypic fitnesses are equal, a stable polymorphism will result with equilibrium frequencies set by stationary conditions. One such polymorphism is considered below. The relative fitnesses of many other contrasted states are not frequency-dependent in this way, and the outcome of selection is that only one phenotype persists with any particular set of parameter values. The s~ond and third analyses investigate adaptive strategies of this nature. (A)
SEX
RATIOS
IN
PLANTS
In seed plants with two sexes, the equilibrium sex ratio can be obtained very simply for all uniform fitness modes of inheritance of the sex phenotypes. Plants of one or both sexes frequently produce a proportion of gametes
550
D.
G.
LLOYD
of the other sex and are more or less bisexual. In sexually dimorphic populations, males may be defined as plants transmitting their genes predominantly or exclusively through pollen and females as plants transmitting their genes predominantly or exclusively through ovules (Lloyd, 1974~). Let the ovule production of females and males be 1 and o respectively and the pollen fecundity of males and females be k and kl. The survival rates and fertilization rates are both 1 in females and respectively S and e in males. The frequency of adult females and males are p and 1 -p. The average fitnesses of seeds resulting from outcrossing and self-fertilization are 1 and i in both sexes. The proportions of seeds which are derived from random crossfertilization are ts among the seeds borne on males and t, among seeds borne on females. All parameters may vary independently of each other; this requires in particular that self- and cross-fertilization compete with each other (rather than self-fertilization preceding or following crossing, Lloyd, 1975 and in press) and that the pollen used in self-fertilization does not affect the efficiency of cross-fertilizations or vice versa. Then for the whole population : Total number of ovules cross-fertilized = pr:+ oetJ 1 -p). Total pollen production = pkl+ (1 -p)k. :, Probability of a pollen grain achieving cross-fertilization = Ho +oebU -PI k(1 -p fp2) * The fitness of male and the contribution the next generation. have equal chances female zygotes,
or female zygotes is the product of their survival rate of the adults via both pollen and ovules to zygotes of Assuming that the pollen grains of males and females of achieving cross-fertilization, the average fitness of
ZQ = I t,+2i(l-t,)+
M& + oetA3.- 141 k(l-p+pZ)
>’
and iGa = S At stationary conditions
when E3-iVp = 0, after re-arranging,
S[tsOe +2oei(l’ = (1 - I)S[t,oe +2oei(l-
td)] - [to +2i(lts)] - [tr +2i(l-
to)] +(Sto)] +(S-
&et8 l)(oets- tJ *
(9)
The ovule contributions of a plant are considered to be independent of the sex ratio, but pollen contributions increase as the frequency of females increases and more ovules are available. Pollen contributions are more
MODELS
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551
SELECTION
important for males than for females; hence the difference in fitness between the sexes (Zg-Vp) increases monotonically as the frequency of females increases. The stationary conditions therefore represent a unique and stable equilibrium when 0 < fl < 1 is specified by equation (9). The analysis allows the joint effects on the sex ratio of six parameters to be determined. Three sets of factors maintaining sexual dimorphism will be briefly considered here. If the sexes survive at equal rates and their ovules are fertilized at equal rates solely by cross-fertilization, then S = I, e = 1, rg = t,: = 1. and equation (9) reduces to : l-20+0z
P= 2(1-0)(1-Z)’
orl-fi=
l-21+01 2(1-o)(l-I)’
Sexual dimorphism can be maintained simply by the suitable advantage in ovule and pollen production rel;ative advantages are equivalent. !$econd, if the sexes survive at the same rates and their in equal numbers and fertilized at the same rates, S = only inbreeding effects and pollen production differ Equation (9) then reduces to
Sexual dimorphism
(10) two sexes having a respectively. Their ovules are produced I, e = 1, o = I, and between the sexes.
B = (fs-- t&l -2i) +(I - l)t, 2(1- I)($ - t&l - i) * is maintained at equilibrium, 0 < lj < I, if l-2i
fd
2-2i-Z
At stationary conditions, 2w, = few,. The conditions under which asexual reproduction is advantageous are not identical for individuals and populations :tince the “cost of meiosis” (Maynard Smith, 1971) does not appear in the population selection model. But in both individual and group selection models, asexual reproduction is more likely to be advantageous when the Snvule fertility and fertilization rates are low and the relative fitness of :asexually produced offspring (W./W,) is high. The known distribution of agamospermy suggests that these factors are significant in natural populations. The great majority of the sexually reproducing relatives of agamospermous plants have mechanisms promoting outcrossing (summarized in
554
n.
G.
LLOYD
Grant, 1971). These are more likely to experience a low level:of fertilization than are species which are regularly self-fertilized. Most agamospermous flowering plants are species hybrids or hybrid derivatives, and many have sterile or semisterile pollen (Grant, 1971). It has long been recognized that the principal adaptive advantage of agamospermy is to restore fertility to individuals that are sexually sterile as a result of polyploidy and hybridization (Darlington, 1939; de Wet & Stalker, 1974). Some agamospermous hybrids are sexually fertile, however. In these cases Grant suggests that agamospermy may stabilize a favourable hybrid genotype (one with high w,). Alternatively, e may be low. A number of additional factors could be added to the analysis of agamospermy to approximate natural situations more closely. The advantages and disadvantages of asexual reproduction parallel in some ways those of self-fertilization, although self-fertilization does not overcome meiotic irregularities as agamospermy does. Both ensure that a parent contributes two genomes to each offspring and both circumvent failures of crossfertilization. When agamospermy precludes sexual reproduction of the same ovules, its selective advantages are similar to those of self-fertilization which occurs prior to any opportunity for cross-fertilization. In other instances, sexual and asexual embryos compete in the same ovules. Detailed models of comparable “prior” and “competing” self-fertilization have been developed elsewhere (Lloyd, in press). They show that with certain interactions between parameters, mixtures of self- and cross-fertilization are selected under some conditions. The models elaborated for self- and cross-fertilization could be applied with some alteration to strategies for sexual and asexual reproduction. (C)
PARENTAL
INVESTMENT,
PARENT-OFFSPRING
SEPARATION
AND
SEXUAL
SELECTION
The final example examines the selective advantages of alternative behaviours on the part of an animal which has mated and produced one or more offspring. At any stage during the maturation of the offspring, a male or female parent may either (i) care for the current offspring, thereby increasing the offspring’s chances of surviving and eventually reproducing, or (ii) desert the current offspring and thereby increase the parent’s own chances of producing more progeny subsequently. In a stimulating paper on parental investment, Trivers (1972) proposed a series of semi-quantitative models exploring among other topics the circumstances under which either behaviour confers higher fitness on parents displaying it. Trivers defined parental investment as “ . . . any investment by the parent in an individual offspring that increases the offspring’s chance of
MODELS
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555
surviving (and hence reproductive success) at the cost of the parent’s ability to invest in other offspring”. He suggested that “At any point in time the individual whose cumulative investment is exceeded by his partner’s is theoretically tempted to desert”. Dawkins & Carlisle (1976) have objected to the latter view and conclude that “. . . a parent ‘deciding’ whether to dleselrt a child should ask the following questions. How closely related is this child to me? How likely is it to survive if I do, and if I do not, desert it? What proportion of my total future reproductive potential does this child represent? How much would it cost me to make a new child equivalent to this one? The parent should ignore its own previous investment in the particular child, except in so far as it affects the answers to these questions.” I Ipropose to show here, without entering into many ramifications of the subject, that the advantages of parental investment versus desertion can be readily quantified by an analysis of stationary conditions using parameters akin to those suggested by Dawkins and Carlisle. Two parental phenotypes are contrasted; “investors” who care for the n, current offspring over any specified period and “deserters” who separate themselves from the offspring at the beginning of this period. Parental investment over the period at issue increases the viability of each offspring, ui, by a factor, r, beyond that of deserted offspring, ZJ~.Deserters produce a number of later offspring, nld, that is greater than the number provided by investors, n,,, by a factor p. That is, I’{ = Ud(1+ ‘A), an d lZ[d = n,Jl +@. Parents are related to their offspring to a degree measured by Sewall Wright’s coefficient of relationship, r-normally I- in large biparental outbred populations (Hamilton, 1964). It is assumed for simplicity that both investors and deserters will treat later offspring in the same way as they treat their current progeny, and that the fitness functions remain the same. The fitness of the two phenotypes,
= ~cL%(l +cc)+n,,(l and
Then,
+cl)],
556 At stationary conditions,
D.
G.
LLOYD
wi - w, = 0 and a=--
Bnli
It, +?Z*i ’
(16)
or p=
41, + k> .
(17)
n/i
Equations (15) to (17) describe the extent to which investors are favoured by higher values of the investment benefit, a, by a greater number of offspring currently being invested in, by higher ratios of current: later numbers of offspring, and by a smaller cost of investment, B. The opposite factors favour deserters. The algebraic analysis points out that the cost and benefit components of fitness of the investment strategy must be formulated separately. Males and females may favour different strategies under certain conditions when any of the fitness factors show sex differences. The cost of producing a zygote is characteristically greater for females of a species than for males (Trivers, 1972). The number of subsequent offspring and the term nri/(n,+n1i) will accordingly be often lower for females, favouring an investment strategy in a wider range of circumstances than is the case for males. The cumulative investment by a parent up to any point may be important in deciding which strategy confers higher fitness, as Trivers (1972) suggested, but the effect of previous investment is indirect. Previous investment in a growing juvenile will also affect parental strategies through effects on the prospects of the juvenile if deserted. With increasing maturity of the offspring, a may be expected to diminish and the probability that desertion will be favoured increases. In most animal species which indulge in parental investment, the parent(s) ultimately enforce separation and attempt to raise another family; presumably a drops so low while the parents have prospects of raising more offspring that desertion is eventually favoured. In humans, however, investment commonly continues into the reproductive period of the children, while the parents refrain from producing more children. The advantages of investing in adult children (and in grandchildren) and in other members of nuclear and extended families compared with the gain in fitness from later children could be investigated further with the aid of the coefficient of relationship. The complex roles that investment in relatives other than children play in selecting age-specific fecundity and family structure warrant a detailed quantitative study. Many other factors could be added to the above analysis. The expected number of later offspring, for example, could be divided into a number of components-the probabilities of parental survival and of securing late1
MODELS
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mates and the fertility of later matings. The costs of later offspring or their via.bilities need not be the same as those of the earlier progeny. The effects of population structure including the mating system, interactions between parameters (as when o!is dependent on nJ, and the consequences of desertion of one mate on the strategy of the other might also be considered. It must be remembered however, that the parameters affecting parental investment and the behavioral traits necessary to carry out either strategy have complex efIects on aspects of biology other than parental investment. In being selected folr their total effects, the parameters controlling parental investment may therefore not achieve the “optimal” investment. In some conditions, the ofTspring receiving an investment may also subvert the parent’s interests (Trivers, 1974). 4. Discussion These examples demonstrate that analyses of stationary conditions can handle individual or group selection and can define boundary conditions for alternative strategies, mixed strategies or conditions in a polymorphism. The benefits of any character or activity can be treated as direct contributions to fitness, while the costs can be incorporated as reductions to the fitness contributions in other directions (cf. Williams, 1966). The effects of the behaviour of one class of individuals on the fitness of others can be assessed, as’ in the use of the theory of games to define evolutionary stable strategies in animal conflicts (Maynard Smith, 1976). For some topics at least, simple but plausible algebraic functions which exactly express the fitness of contrasting phenotypes can be readily obtained and lead to precise strategy predictions. It will not always be possible to obtain exact models incorporating several ecological factors, but ecologists and ethologists have perhaps been too willing to settle for imprecise semi-quantitative models when faced with the complexity of adaptive strategies. Analyses of stationary conditions are not directly applicable to enquiries into evolutionary processes involving forces other than selection (mutation, genetic drift, etc.) which require genetic specifications. On the other hand, analyses of stationary conditions are admirably suited to studies of interactions between conflicting selective forces, where the interest is predominantly in the effects of these forces on individual fitness rather than the genetical mechanisms providing the phenotypes. Phenotypic models of selection can treat a number of selective forces simultaneously. Phenotypic analyses are therefore particularly useful for situations which are too complex to be manipulated adequately in verbal models and for situations where genetic models introduce severe mathematical difficulties, as in consideration of the selective forces affecting self-fertilization (Lloyd, in press).
558
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Topics on the reproductive biology of plants and animals seem particularly amenable to precise strategic analyses. The equal contributions of male and female gametes to the gene pool of sexually reproducing populations place a useful constraint on the fitnesses of the sexes. The offspring resulting from sexual reproduction are comparable units, even when they vary in fitness. A number of quantitative parameters, including fertility and sexuality (gender) measures, the coefficient of relationship, inclusive fitness, reproductive value. and parental investment (when split into its components and quantified) arc available for strategic analyses of reproductive behaviour and other aspects of adaptive strategies. I would like to acknowledge the comments of Dr Brian Charlesworth, which greatly helped me to refine the ideas presented here. Drs B. DommCe, V. Grant, B. I. Hayman, R. A. Littler, R. Primack, M. D. Ross and C. J. Webb also provided helpful comments.
CHARLESWORTH, DARLINGTON,
REFERENCES B. & CHARLESWORTH, D. Am. Nat. (In press). C. D. (1939).Evolurionof Genetic Systems. Cambridge:Cambridge University
Press.
DAWKINS, R. & CARLISLE, T. R. (1976).Nature 262, 131. DE WET, J. M. J. & STALKER, H. T. (1974).Taxon 23,689. FISHER, R. A. (1941).Ann. Eugenics 11, 53. GRANT, V. (1971).Plant Speciation. New York: ColumbiaUniversityPress. HAMILTON, W. D. (1964).J. theor. Biof. 7, 1. Ho, T. Y. & Ross,M. D. (1973).Heredity 31, 282. JAIN, S. K. & WORKMAN, P. L. (1967).Nature 214, 674. KETEEWELL, B. (1973).The Evolution of Melanism. Oxford: ClarendonPress. LEWIS, D. (1941). New Phytol. 40, 56. LLOYD, D. G. (1974a). Heredity 32, 11. LLOYD, D. G. (19746). Heredity 32, 45. LLOYD, D. G. (1975).Genetica 45, 325. LLOYD, D. G. (1976). Theor. Pop. Biol. 9, 299. LLOYD, D. G. Am. Nat. (In press). MAYNARD SMITH,J. (1971).In Group Selection (G. C. Williams,ed.j, p. 163.Chicago:
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26, 1. 35, 21. Ross,M. D. & WEIR, B. S. (1976).Evolution 30,425. TRIVERS, R. L. (1972). In Sexual Selection and the Descent of Man 1871-1971 (B. Campbell,
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G. C. (1966).
G. C. (1975).
Am. Nat. 100,687. and Evolution. Princeton:PrincetonUniversityPress.
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APPENDIX BRIAN CHARLESWORTH
Ptvof of Stationary Fitness Theoremfor n Alleles ad m Phenotypes Let there be m phenotypes P,, P, . . . P,,, and ir alleles a,, nz . . u, at a locus controlling the phenotypes. Let jij be the probability that an allele is lli and is carried in phenotype Pj. in an arbitrary generation. Let F(P,) IX: the average fitness of P;., Let .fi
=
xhj .;
be the frequency of ai in the given generation. and let E = c PjG(Pj, j be the mean fitness of the population. Then Ihe new frequency of (1;. /I: i< given by
so that
(Al) At equilibrium Afi = 0 for each i so that, writing sj = [W(Pj)-W)/~, wt‘ obtain the set of linear homogenous equations in the m unknowns xi: II, (i= l.....n) (A?) jzlAj-yj = O If IZ < ~1, there are insufficient equations to determine the -yj. If II = HI, then standard theory tells us that eitlwr si = 0 for each ,j. or the following determinantal condition holds: ljijl == 0 (A3) Condition (A3) implies the existence of special relationships between the equilibrium frequencies (linear dependence between the rows and columns of the matrix {fij}). In the case II = ~1 = 2, condition (A3) is equivalent to ,f, a,‘fI -f2,/f2 in equation (3) of the text being equal to zero. Such relations will exist only under rather unusual conditions, so that in general we can conclude that xj = 0 for each j. This of course implies that the fitnessesof each phenotype are equal in the equilibrium population. The case IZ > m can be dealt with by pooling together alleles until the number of alleles is reduced to nz, and then proceeding as above. We thus conclude that, in equilibrium populations with the unifornl fitness model of inheritance, the fitnesses of all the phenotypes are equal
560
D.
G.
LLOYD
unless the number of phenotypes exceeds the number of alleles at the locus concerned, or special constraints are placed on the equilibrium genotype frequencies independently of the equilibrium phenotype fitnesses. Dr M. Slatkin has independently arrived at results similar to these, in the context of models of frequency and density-dependent selection.
Genetic and Phenotypic Models of Natural Selection DAVID G. LLOYD Botany Department, University of Canterbury, Christchurch, New Zealand (Received 5 July 1977) The following theorem is proposed: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness, provided that certain modes of inheritance apply; in particular, all genotypes specifying the same phenotype must have the same average fitness. If these conditions of “uniform fitness” patterns of inheritance are not met, particular genetic models of natural selection should replace an analysis of phenotypes. If the conditions are met, an analysis of the stationary conditions when the phenotypes have equal fitnesses permits quantitative statements about the outcome of selection without recourse to genetic models. Phenotypic analyses of natural selection are illustrated by models of sex ratios in plants, sexual versus asexual reproduction in plants, and parental investment by animals.
1. Introduction In genetic models of natural selection, genotype and gamete fitnesses are specified and the outcome of selection is determined in the first instance in terms of genotype and allele frequencies. Even when they are of greater interest, phenotype frequencies are obtained secondarily from the genotype frequencies. Genetic models of natural selection established the hereditary basis of evolutionary change and sponsored the modern theory of evolution. Phenotypic models of natural selection reason that fitter phenotypes (those contributing on average more genes to the next generation) inevitably increase at the expense of less fit phenotypes until the contrasted phenotypes are equal in fitness or one phenotype completely replaces the other. The verbal arguments of Charles Darwin and A. R. Wallace, although not couched in such terms, were of this kind. More recently, phenotypic models investigating adaptive strategies have commonly assumed that the outcome of selection can be predicted solely from a knowledge of phenotype fitnesses and that the particular mode of inheritance of the phenotypes is unimportant. 543
544
D.
G.
LLOYD
There appears to be some uneasiness about the assumption that models of adaptive strategies arc applicable to all patterns of inheritance of the phenotypes considered, however, as genetical models are sometimes added to strategic ones, e.g. Murphy, 1968. The present paper distinguishes modes of inheritance under which evolutionary change ceases (stationary conditions are reached) when the fitnesses of two phenotypes are equal. This property provides a simple method for obtaining fully quantitative phenotypic models of natural selection. Although phenotypic models of natural selection contain no information on genotype or allele frequencies or rates of approach to selection targets, they are nevertheless often valuable, as their recent popularity testifies. They are applicable to a whole class of genetic models, whereas genetical arguments have no validity beyond the particular mode of inheritance that is specified. And since they omit genetical considerations, phenotypic models can consider more complex ecological situations. As a result, they are more suitable than genetic models when the primary interest lies in the effects of phenotypic parameters on fitness rather than in their genetical basis. Certain circumstances such as partial self-fertilization are difficult to incorporate into genetic models of selection (Fisher, 1941; Jain & Workman, 1967). Phenotypic models avoid these difficulties. A number of phenotypic parameters can be readily handled in phenotypic models involving mixed self- and crossfertilization, as illustrated below (and in Lloyd, 1975 and in press: Charlesworth & Charlesworth, in press). 2. Genetic Conditions for Phenotypic Analyses After cross-fertilization a zygote receives one set of genes from each gamete and parent, but after self-fertilization both sets of genes are derived from the same parent. If a fitness measure is to be applicable to selffertilization (or inbreeding generally), it must therefore be defined in terms of the relative genetic contributions to the gametes or genes making up the offspring rather than as the number of offspring themselves. Fitness is defined here as the average number of genes which a genotype, phenotype or allele contributes to the next generation. Consider two phenotypes, P, and PI, which differ to any extent in any set of attributes affecting their fitnesses. The two phenotypes may occur at any The phenotypic differences are relative frequencies in one population. determined by two alleles at one or more controlling loci, i.e. by aI, a2 ; b,, b,, etc. Selection at only one locus, a, is considered initially. By definition, the frequency of the fitter allele increases from one generation to the next at the expense of the less fit allele. But phenotypes are not directly inherited.
MODELS
OF
NATIJRAL
545
SLLEC~IIOU
The combination of genes into genotypes intercedes between the phenotypes of successive generations, so it does not follow that selection of the fitter allele necessarily increases the frequency of the fitter phenotype, even in the absence of interlocus interactions. The changes in phenotype frequencies that accompany selection at controlling loci must be established for each mode of inheritance of the phenotypes. The average fitnesses of alleles a, and a2 are designated WI and W2, and the average fitnesses of P, and P, are @(PI) and W(P2). We seek to delimit a class of genetic models for which it is true that the two phenotypes have equal fitnesses at stationary conditions, i.e. fo1 which W(P,) = iT(P,) when tj, = E2. (A)
UNIFORM
FITNESS
MODES
OF INHERITANCE
In “uniform fitness” modes of inheritance, all genotypes of the same phenotype have the same average fitness. That is, for the a locus, qa,a,-P,)
= iiqa,u,-P,)
= w(u,u,-P,)
= W(P,),
(1)
W(u,u, -P2)
= w(u,u~-P,)
= i&u,-P,)
= W(PJ,
(2)
and where alal -PI, etc. are all individuals phenotype one, etc. Then,
of genotype
alal
which
have
w, = ivl)f, 1+ ~(P2lfi2 fl ’ and w
=
Wdf* __--
I
+
W2)f22
2 f2
where the frequencies of a, in individuals and fi2, and the combined frequency .f2 = f21 +f22, where fi +fi = 1.
3
of P, and P, are respectively f, , of a,, fi = fi I +fi2. Similarly
Substituting
gives w,--zw,
=
(3)
Stationary conditions at the a locus are defined as the conditions under which selective forces cause neither allele to increase in frequency from
546
I). G.
l.l.OYI~
one generation to the next. These apply when and only when i?r = wz. From equation (3), this may occur when (fi rlf --fzllfi), which we call X, is zero. If x = 0, the two alleles are represented in the same proportions in P, and P,. This can happen only in the trivial case when locus a has no effect on the phenotypes or when the substitution of a, for a, causes a switch from P, to P, in some genetic backgrounds and causes the reverse switch from P, to PI in other backgrounds. This would in turn require such unusual gene interactions that it may usually be discounted. Hence in uniform fitness genetic models, unless an allele has opposite effects in different genetic backbackgrounds, if ti;, -I, = 0, ti(P1)-ii
= 0.
(4)
The two phenotypes have equal fitness at stationary conditions. Suppose now that a higher proportion of a1 representatives than of a, representatives occur in P,. Then x is positive, and from equation 3, if WI -cj, > 0, E(P1)-5(P,) > 0. (5) That is, the phenotype with the higher proportional representation of the fitter allele is always fitter. Moreover, the fitter phenotype, P,, also increases except in the unlikely circumstance that an increase in the frequency of a, is accompanied by a decrease in the frequency of the phenotype which it specifies by itself or in combination with other genes. In general then, if not invariably, the fitter phenotype increases. By parallel arguments, when x is positive and E1 - Wz -C 0, W(P1)-G(P,)
< 0,
(6)
and the fitter phenotype (P2) again generally increases. This property also applies when x is always negative. Similarly, for other loci (b, c, etc.) which participate in specifying phenotypes 1 and 2, the phenotypes have equal fitnesses when the alleles have equal fitnesses. Hence when evolutionary change due to selection ceases at stationary conditions, the phenotypes have equal fitnesses with the genetic models stipulated, regardless of the number of loci controlling the phenotypes and the frequency of recombination between them. But where factors such as linkage disequilibrium or genetic drift interfere with the response of a locus to the selection forces operating on it, evolutionary change may not cease when the phenotypes have equal fitnesses. The analysis above is confined to loci with two alleles and to two phenotypes. Dr Brian Charlesworth (Appendix) has provided a general proof, for n alleles at a locus with m phenotypes, that in equilibrium populations the fitnesses of the phenotypes are equal unless the number of phenotypes
MODELS
OF
NATURAL
SELI:ClION
547
exceeds the number of alleles at the locus concerned, or special constraints are placed on the equilibrium genotype frequencies which are independent of the equilibrium phenotype fitnesses. (8)
VARIABLE
FITNESS
MODES
OF INHERITANCE
In many patterns of inheritance, such as those involving lethal or subvital genotypes, overdominance or epistasis, different genotypes of the same phenotype do not have equal fitnesses-equations (1) and/or (2) are not entirely valid. For such “variable fitness” modes of inheritance, w
#
~(Pl)fll
+wzM-12 __--
I fl
’
W(P,)]
[ I
and/or
and J,
-J,
#
[I(P,)
When W, - iV2 = 0, iifP,)-
-
iG(PJ # 0.
j-11 fi--
f21 j;
.
(8)
The stationary conditions when the alleles have equal fitnesses will not coincide with equal fitnesses of the phenotypes. Hence Darwin’s argument and the common assumption of strategic analyses, that characteristics associated with an advantage in reproductive contributions will increase in frequency until the fitter phenotype replaces the less fit or their fitnesses become equal, is not always correct. Consider, for example, the situation whlere dominant and recessive alleles at one locus specify contrasted phenotypic traits and the controlling locus also displays overdominance, as in severa melanic polymorphisms in Lepidoptera (Kettlewell, 1973). The three genotypes all have different fitnesses, and at equilibrium the average fitnesses of the two phenotypes are not equal. Despite the truism that natural selection operates directly on phenotypes and not alleles, phenotypic models of natural selection do not always give correct results for variable fitness patterns of inheritance. When genetic interactions that significantly violate the uniform fitness conditions are operating, phenotypic models of selection may not give even a close approximation to the outcome of selection. Genetic models of natural selection are then necessary.
54x
I>.
(C)
c;.
GENERAL
LLOYD
THEOREM
The limited independence of phenotypic models from genetic considerations can be expressed in the following statement: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness only if all genotypes specifying any one phenotype have the same average fitness and no allele controlling the phenotypes has opposite effects in different genetic backgrounds. This theorem of phenotypic fitness delimits genetical conditions under which one can define adaptive strategies in purely ecological terms without explicit reference to genetics. The theorem is generally limited to the fitnesses of zygotes, since equations (3) and (4) apply for one complete generation only to zygotes. If two phenotypes differ in their rates of survival between zygote formation and reproductive adulthood, selection will not necessarily cause the fitter adult phenotype to increase even with uniform fitness modes of inheritance. The theorem applies to a later stage of development only if there is no change in the relative fitness of the phenotypes between the zygotes and that stage. The theorem does not require any one controlling locus to be fully responsible for specifying the phenotypes. Consequently, it will still hold if the phenotypes are not entirely determined by genes but are partly elicited by environmental conditions. Selection at a locus will generally result in the fitter of two phenotypes increasing (with uniform fitness genetic models) if the locus has any influence on the phenotypes. Several aspects of the preceding analysis have already been verified in the case of gynodiecy, a sex polymorphism in plants. In gynodioecious popultions, the unisexual females contribute genes to the next sexual generation solely through ovules, while the more or less bisexual males contribute genes predominantly through pollen but also to a minor degree through ovules (Lloyd, 1976). First, for any value of the relative (d/q) seed fecundity of the sexes, the equilibrium sex ratio is the same for genetic models in which the female phenotype is determined by a dominant or recessive allele of one locus or is controlled by three different two-locus models (Lewis, 1941; Ross & Shaw, 1971; Lloyd, 1974a). All five genetic models are uniform fitness models and should therefore give the same sex ratios according to the above analysis. Second, a number of variable fitness models of gynodioecy involving either lethal sex genotypes (Lloyd, 1974~) or overdominance (Ho & Ross, 1973; Ross & Weir, 1975, 1976) give equilibrium sex ratios which differ from those of uniform fitness models and depend upon the particular genetic model operating. Third, the average fitnesses of male and female zygotes are equal at equilibrium with the uniform fitness genetic models but not with the
MODELS
OF
NATL:RAL
StLECTlUN
549
variable fitness models (Lloyd, 1974b, 1975; Ross & Weir, 1975, 1976; Charlesworth & Charlesworth, in press). Fourth, when differential survival of the sexes occurs, the average fitnesses of male and female adults are not equal at equilibrium for either class of genetic models (Lloyd, 1974b). The genetical restriction in the theorem of phenotypic selection proposed above does not operate in the process of group selection between noninterbreeding populations. Suppose that two phenotypes, P, and P, OCCUI at frequencies of 100% in different populations, and that they are determined by alleles a, and a2 respectively of a controlling locus. When inter-population selection occurs by differential extinction or multiplication, a daughter population invariably has the same allele and phenotype as its parent population. Hence the fitness of alleles coincides with the fitness of phenotypes, and the population with the fitter phenotype always increases in the mctapopulation regardless of the genetical basis of the phenotypic difference. 3. Analyses of Stationary
Conditions
The condition that contrasting phenotypes have equal fitnesses a~. stationary conditions with uniform fitness modes of inheritance provides a useful quantitative technique for analysing the outcome of selection whenever the fitnesses of the phenotypes can be expressed exactly. All that is necessary is for the difference between the fitnesses of two phenotypes to be written down. Then the direction of selection can be determined for uniform fitness genetic models under any combination of values of the parameters affecting fitness. The outcome of selection is obtained from stationary conditions when the difference in fitness of the phenotypes is zero. The contrasted phenotypes may occur in separate individuals in a population or in different populations for group selection models. If the phenotypic fitnesses in individual selection models depend on their frequencies in such a way that each phenotype has ;I relduced relative fitness whenever its frequency exceeds that at which the phenotypic fitnesses are equal, a stable polymorphism will result with equilibrium frequencies set by stationary conditions. One such polymorphism is considered below. The relative fitnesses of many other contrasted states are not frequency-dependent in this way, and the outcome of selection is that only one phenotype persists with any particular set of parameter values. The s~ond and third analyses investigate adaptive strategies of this nature. (A)
SEX
RATIOS
IN
PLANTS
In seed plants with two sexes, the equilibrium sex ratio can be obtained very simply for all uniform fitness modes of inheritance of the sex phenotypes. Plants of one or both sexes frequently produce a proportion of gametes
550
D.
G.
LLOYD
of the other sex and are more or less bisexual. In sexually dimorphic populations, males may be defined as plants transmitting their genes predominantly or exclusively through pollen and females as plants transmitting their genes predominantly or exclusively through ovules (Lloyd, 1974~). Let the ovule production of females and males be 1 and o respectively and the pollen fecundity of males and females be k and kl. The survival rates and fertilization rates are both 1 in females and respectively S and e in males. The frequency of adult females and males are p and 1 -p. The average fitnesses of seeds resulting from outcrossing and self-fertilization are 1 and i in both sexes. The proportions of seeds which are derived from random crossfertilization are ts among the seeds borne on males and t, among seeds borne on females. All parameters may vary independently of each other; this requires in particular that self- and cross-fertilization compete with each other (rather than self-fertilization preceding or following crossing, Lloyd, 1975 and in press) and that the pollen used in self-fertilization does not affect the efficiency of cross-fertilizations or vice versa. Then for the whole population : Total number of ovules cross-fertilized = pr:+ oetJ 1 -p). Total pollen production = pkl+ (1 -p)k. :, Probability of a pollen grain achieving cross-fertilization = Ho +oebU -PI k(1 -p fp2) * The fitness of male and the contribution the next generation. have equal chances female zygotes,
or female zygotes is the product of their survival rate of the adults via both pollen and ovules to zygotes of Assuming that the pollen grains of males and females of achieving cross-fertilization, the average fitness of
ZQ = I t,+2i(l-t,)+
M& + oetA3.- 141 k(l-p+pZ)
>’
and iGa = S At stationary conditions
when E3-iVp = 0, after re-arranging,
S[tsOe +2oei(l’ = (1 - I)S[t,oe +2oei(l-
td)] - [to +2i(lts)] - [tr +2i(l-
to)] +(Sto)] +(S-
&et8 l)(oets- tJ *
(9)
The ovule contributions of a plant are considered to be independent of the sex ratio, but pollen contributions increase as the frequency of females increases and more ovules are available. Pollen contributions are more
MODELS
OF
NATURAL
551
SELECTION
important for males than for females; hence the difference in fitness between the sexes (Zg-Vp) increases monotonically as the frequency of females increases. The stationary conditions therefore represent a unique and stable equilibrium when 0 < fl < 1 is specified by equation (9). The analysis allows the joint effects on the sex ratio of six parameters to be determined. Three sets of factors maintaining sexual dimorphism will be briefly considered here. If the sexes survive at equal rates and their ovules are fertilized at equal rates solely by cross-fertilization, then S = I, e = 1, rg = t,: = 1. and equation (9) reduces to : l-20+0z
P= 2(1-0)(1-Z)’
orl-fi=
l-21+01 2(1-o)(l-I)’
Sexual dimorphism can be maintained simply by the suitable advantage in ovule and pollen production rel;ative advantages are equivalent. !$econd, if the sexes survive at the same rates and their in equal numbers and fertilized at the same rates, S = only inbreeding effects and pollen production differ Equation (9) then reduces to
Sexual dimorphism
(10) two sexes having a respectively. Their ovules are produced I, e = 1, o = I, and between the sexes.
B = (fs-- t&l -2i) +(I - l)t, 2(1- I)($ - t&l - i) * is maintained at equilibrium, 0 < lj < I, if l-2i
fd
2-2i-Z
At stationary conditions, 2w, = few,. The conditions under which asexual reproduction is advantageous are not identical for individuals and populations :tince the “cost of meiosis” (Maynard Smith, 1971) does not appear in the population selection model. But in both individual and group selection models, asexual reproduction is more likely to be advantageous when the Snvule fertility and fertilization rates are low and the relative fitness of :asexually produced offspring (W./W,) is high. The known distribution of agamospermy suggests that these factors are significant in natural populations. The great majority of the sexually reproducing relatives of agamospermous plants have mechanisms promoting outcrossing (summarized in
554
n.
G.
LLOYD
Grant, 1971). These are more likely to experience a low level:of fertilization than are species which are regularly self-fertilized. Most agamospermous flowering plants are species hybrids or hybrid derivatives, and many have sterile or semisterile pollen (Grant, 1971). It has long been recognized that the principal adaptive advantage of agamospermy is to restore fertility to individuals that are sexually sterile as a result of polyploidy and hybridization (Darlington, 1939; de Wet & Stalker, 1974). Some agamospermous hybrids are sexually fertile, however. In these cases Grant suggests that agamospermy may stabilize a favourable hybrid genotype (one with high w,). Alternatively, e may be low. A number of additional factors could be added to the analysis of agamospermy to approximate natural situations more closely. The advantages and disadvantages of asexual reproduction parallel in some ways those of self-fertilization, although self-fertilization does not overcome meiotic irregularities as agamospermy does. Both ensure that a parent contributes two genomes to each offspring and both circumvent failures of crossfertilization. When agamospermy precludes sexual reproduction of the same ovules, its selective advantages are similar to those of self-fertilization which occurs prior to any opportunity for cross-fertilization. In other instances, sexual and asexual embryos compete in the same ovules. Detailed models of comparable “prior” and “competing” self-fertilization have been developed elsewhere (Lloyd, in press). They show that with certain interactions between parameters, mixtures of self- and cross-fertilization are selected under some conditions. The models elaborated for self- and cross-fertilization could be applied with some alteration to strategies for sexual and asexual reproduction. (C)
PARENTAL
INVESTMENT,
PARENT-OFFSPRING
SEPARATION
AND
SEXUAL
SELECTION
The final example examines the selective advantages of alternative behaviours on the part of an animal which has mated and produced one or more offspring. At any stage during the maturation of the offspring, a male or female parent may either (i) care for the current offspring, thereby increasing the offspring’s chances of surviving and eventually reproducing, or (ii) desert the current offspring and thereby increase the parent’s own chances of producing more progeny subsequently. In a stimulating paper on parental investment, Trivers (1972) proposed a series of semi-quantitative models exploring among other topics the circumstances under which either behaviour confers higher fitness on parents displaying it. Trivers defined parental investment as “ . . . any investment by the parent in an individual offspring that increases the offspring’s chance of
MODELS
OF
NATURAL
SELECTION
555
surviving (and hence reproductive success) at the cost of the parent’s ability to invest in other offspring”. He suggested that “At any point in time the individual whose cumulative investment is exceeded by his partner’s is theoretically tempted to desert”. Dawkins & Carlisle (1976) have objected to the latter view and conclude that “. . . a parent ‘deciding’ whether to dleselrt a child should ask the following questions. How closely related is this child to me? How likely is it to survive if I do, and if I do not, desert it? What proportion of my total future reproductive potential does this child represent? How much would it cost me to make a new child equivalent to this one? The parent should ignore its own previous investment in the particular child, except in so far as it affects the answers to these questions.” I Ipropose to show here, without entering into many ramifications of the subject, that the advantages of parental investment versus desertion can be readily quantified by an analysis of stationary conditions using parameters akin to those suggested by Dawkins and Carlisle. Two parental phenotypes are contrasted; “investors” who care for the n, current offspring over any specified period and “deserters” who separate themselves from the offspring at the beginning of this period. Parental investment over the period at issue increases the viability of each offspring, ui, by a factor, r, beyond that of deserted offspring, ZJ~.Deserters produce a number of later offspring, nld, that is greater than the number provided by investors, n,,, by a factor p. That is, I’{ = Ud(1+ ‘A), an d lZ[d = n,Jl +@. Parents are related to their offspring to a degree measured by Sewall Wright’s coefficient of relationship, r-normally I- in large biparental outbred populations (Hamilton, 1964). It is assumed for simplicity that both investors and deserters will treat later offspring in the same way as they treat their current progeny, and that the fitness functions remain the same. The fitness of the two phenotypes,
= ~cL%(l +cc)+n,,(l and
Then,
+cl)],
556 At stationary conditions,
D.
G.
LLOYD
wi - w, = 0 and a=--
Bnli
It, +?Z*i ’
(16)
or p=
41, + k> .
(17)
n/i
Equations (15) to (17) describe the extent to which investors are favoured by higher values of the investment benefit, a, by a greater number of offspring currently being invested in, by higher ratios of current: later numbers of offspring, and by a smaller cost of investment, B. The opposite factors favour deserters. The algebraic analysis points out that the cost and benefit components of fitness of the investment strategy must be formulated separately. Males and females may favour different strategies under certain conditions when any of the fitness factors show sex differences. The cost of producing a zygote is characteristically greater for females of a species than for males (Trivers, 1972). The number of subsequent offspring and the term nri/(n,+n1i) will accordingly be often lower for females, favouring an investment strategy in a wider range of circumstances than is the case for males. The cumulative investment by a parent up to any point may be important in deciding which strategy confers higher fitness, as Trivers (1972) suggested, but the effect of previous investment is indirect. Previous investment in a growing juvenile will also affect parental strategies through effects on the prospects of the juvenile if deserted. With increasing maturity of the offspring, a may be expected to diminish and the probability that desertion will be favoured increases. In most animal species which indulge in parental investment, the parent(s) ultimately enforce separation and attempt to raise another family; presumably a drops so low while the parents have prospects of raising more offspring that desertion is eventually favoured. In humans, however, investment commonly continues into the reproductive period of the children, while the parents refrain from producing more children. The advantages of investing in adult children (and in grandchildren) and in other members of nuclear and extended families compared with the gain in fitness from later children could be investigated further with the aid of the coefficient of relationship. The complex roles that investment in relatives other than children play in selecting age-specific fecundity and family structure warrant a detailed quantitative study. Many other factors could be added to the above analysis. The expected number of later offspring, for example, could be divided into a number of components-the probabilities of parental survival and of securing late1
MODELS
OF
NATURAL
SELECTION
557
mates and the fertility of later matings. The costs of later offspring or their via.bilities need not be the same as those of the earlier progeny. The effects of population structure including the mating system, interactions between parameters (as when o!is dependent on nJ, and the consequences of desertion of one mate on the strategy of the other might also be considered. It must be remembered however, that the parameters affecting parental investment and the behavioral traits necessary to carry out either strategy have complex efIects on aspects of biology other than parental investment. In being selected folr their total effects, the parameters controlling parental investment may therefore not achieve the “optimal” investment. In some conditions, the ofTspring receiving an investment may also subvert the parent’s interests (Trivers, 1974). 4. Discussion These examples demonstrate that analyses of stationary conditions can handle individual or group selection and can define boundary conditions for alternative strategies, mixed strategies or conditions in a polymorphism. The benefits of any character or activity can be treated as direct contributions to fitness, while the costs can be incorporated as reductions to the fitness contributions in other directions (cf. Williams, 1966). The effects of the behaviour of one class of individuals on the fitness of others can be assessed, as’ in the use of the theory of games to define evolutionary stable strategies in animal conflicts (Maynard Smith, 1976). For some topics at least, simple but plausible algebraic functions which exactly express the fitness of contrasting phenotypes can be readily obtained and lead to precise strategy predictions. It will not always be possible to obtain exact models incorporating several ecological factors, but ecologists and ethologists have perhaps been too willing to settle for imprecise semi-quantitative models when faced with the complexity of adaptive strategies. Analyses of stationary conditions are not directly applicable to enquiries into evolutionary processes involving forces other than selection (mutation, genetic drift, etc.) which require genetic specifications. On the other hand, analyses of stationary conditions are admirably suited to studies of interactions between conflicting selective forces, where the interest is predominantly in the effects of these forces on individual fitness rather than the genetical mechanisms providing the phenotypes. Phenotypic models of selection can treat a number of selective forces simultaneously. Phenotypic analyses are therefore particularly useful for situations which are too complex to be manipulated adequately in verbal models and for situations where genetic models introduce severe mathematical difficulties, as in consideration of the selective forces affecting self-fertilization (Lloyd, in press).
558
D.
G.
LLOYD
Topics on the reproductive biology of plants and animals seem particularly amenable to precise strategic analyses. The equal contributions of male and female gametes to the gene pool of sexually reproducing populations place a useful constraint on the fitnesses of the sexes. The offspring resulting from sexual reproduction are comparable units, even when they vary in fitness. A number of quantitative parameters, including fertility and sexuality (gender) measures, the coefficient of relationship, inclusive fitness, reproductive value. and parental investment (when split into its components and quantified) arc available for strategic analyses of reproductive behaviour and other aspects of adaptive strategies. I would like to acknowledge the comments of Dr Brian Charlesworth, which greatly helped me to refine the ideas presented here. Drs B. DommCe, V. Grant, B. I. Hayman, R. A. Littler, R. Primack, M. D. Ross and C. J. Webb also provided helpful comments.
CHARLESWORTH, DARLINGTON,
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DAWKINS, R. & CARLISLE, T. R. (1976).Nature 262, 131. DE WET, J. M. J. & STALKER, H. T. (1974).Taxon 23,689. FISHER, R. A. (1941).Ann. Eugenics 11, 53. GRANT, V. (1971).Plant Speciation. New York: ColumbiaUniversityPress. HAMILTON, W. D. (1964).J. theor. Biof. 7, 1. Ho, T. Y. & Ross,M. D. (1973).Heredity 31, 282. JAIN, S. K. & WORKMAN, P. L. (1967).Nature 214, 674. KETEEWELL, B. (1973).The Evolution of Melanism. Oxford: ClarendonPress. LEWIS, D. (1941). New Phytol. 40, 56. LLOYD, D. G. (1974a). Heredity 32, 11. LLOYD, D. G. (19746). Heredity 32, 45. LLOYD, D. G. (1975).Genetica 45, 325. LLOYD, D. G. (1976). Theor. Pop. Biol. 9, 299. LLOYD, D. G. Am. Nat. (In press). MAYNARD SMITH,J. (1971).In Group Selection (G. C. Williams,ed.j, p. 163.Chicago:
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26, 1. 35, 21. Ross,M. D. & WEIR, B. S. (1976).Evolution 30,425. TRIVERS, R. L. (1972). In Sexual Selection and the Descent of Man 1871-1971 (B. Campbell,
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G. C. (1966).
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Sex
MODELS
OF NATURAL
SELECTION
559
APPENDIX BRIAN CHARLESWORTH
Ptvof of Stationary Fitness Theoremfor n Alleles ad m Phenotypes Let there be m phenotypes P,, P, . . . P,,, and ir alleles a,, nz . . u, at a locus controlling the phenotypes. Let jij be the probability that an allele is lli and is carried in phenotype Pj. in an arbitrary generation. Let F(P,) IX: the average fitness of P;., Let .fi
=
xhj .;
be the frequency of ai in the given generation. and let E = c PjG(Pj, j be the mean fitness of the population. Then Ihe new frequency of (1;. /I: i< given by
so that
(Al) At equilibrium Afi = 0 for each i so that, writing sj = [W(Pj)-W)/~, wt‘ obtain the set of linear homogenous equations in the m unknowns xi: II, (i= l.....n) (A?) jzlAj-yj = O If IZ < ~1, there are insufficient equations to determine the -yj. If II = HI, then standard theory tells us that eitlwr si = 0 for each ,j. or the following determinantal condition holds: ljijl == 0 (A3) Condition (A3) implies the existence of special relationships between the equilibrium frequencies (linear dependence between the rows and columns of the matrix {fij}). In the case II = ~1 = 2, condition (A3) is equivalent to ,f, a,‘fI -f2,/f2 in equation (3) of the text being equal to zero. Such relations will exist only under rather unusual conditions, so that in general we can conclude that xj = 0 for each j. This of course implies that the fitnessesof each phenotype are equal in the equilibrium population. The case IZ > m can be dealt with by pooling together alleles until the number of alleles is reduced to nz, and then proceeding as above. We thus conclude that, in equilibrium populations with the unifornl fitness model of inheritance, the fitnesses of all the phenotypes are equal
560
D.
G.
LLOYD
unless the number of phenotypes exceeds the number of alleles at the locus concerned, or special constraints are placed on the equilibrium genotype frequencies independently of the equilibrium phenotype fitnesses. Dr M. Slatkin has independently arrived at results similar to these, in the context of models of frequency and density-dependent selection.
