w. schun2 and s.this paper are based on the results for a diallelic locus with the relative...

INTER-GENOTYPIC COMPETITION IN PLANT POPULATIONS 11. MAINTENANCE OF ALLELIC POLYMORPHISMS WITH

FREQUENCY-DEPENDENT SELECTION AND MIXED SELFING AND RANDOM MATING1

w. M. S C H U n 2 AND s. A. USANIS

Department of Genetics, North Carolina State University, Raleigh, North Carolina

Received June 11, 1968

EPORTS by investigators in recent years indicate that substantial amounts of allelic polymorphism are often maintained in plant populations that exhibit

various levels of outcrossing. The reason for this phenomenon is an important consideration in evolutionary

theory, particularly for those organisms that exhibit a high degree of self-pollination (where the mating system would seem to impose a severe restsiction on the amount of genetic recombination that can occur). Evidence is accumulating, however, that self-pollinators often possess more genetic flexibility than would be expected on the basis of observed levels of outcrossing, and that the distinction between outcrossers and selfers is, in this respect, not so great as previously supposed.

JAIN and ALLARD (1960) and ALLARD and JAIN (1962) analyzed an experimental population of barley and found that substantial heterozygosity remained after 18 generations of natural selection under a mating system with approxi- mately 99% self-pollination. ALLARD and WORKMAN (1963) reported similar results for three lima bean populations with about 95,% self-pollination. These authors speculated that the primary agency acting to maintain heterozygosity in the population was selection favoring heterozygotes.

HARDING, ALLARD and SMELTZER (1966) subjected one of the lima bean populations studied by ALLARD and WORKMAN to detailed analysis and concluded that selective values of the genotypes in the population were frequency-dependent. They further postulated that the pattern of frequency dependency suggested a “neighborhood effect” in which the relative fitness of heterozygotes is affected by competition from other genotypes growing in proximity.

Excess heterozygosity and frequency-dependent selection does not appear to be limited to self-pollinated plants. LEWONTIN and HUBBY (1966) found a high proportion of polymorphic loci in several populations of Drosophila pseudoobscura

1 Joint contribution from the Crops Research Division, Agricultural Research Service, U. S. Department of Agriculture, and the North Carolina Agricultural Experiment Station. Paper No. 2650 of the Journal Series of the North Carolina State University Agricultural Experiment Station, Raleigh, North Carolina, Publication No. 524 of the U. S . Regional Soybean Laboratory. The research was supported in part by Public Health Service Grant GM 11546. The computing was supported in part by NIH Grant F’Fi-00011.

Present address: Statistical Laboratory, College of Agriculture and Home Economics, University of Nebraska, Lincoln, Nebraska 68503.

Genetics .61: 875-891 April 1969.

8 76 W. M. SCHUTZ A N D S. A. USANIS

and cases of frequency-dependent selection have been reported in Drosophila melanogmter (KOJIMA and YARBROUGH 1967) and Drosophila ananmsae ( TOBARI and KOJIMA 1967).

SCHUTZ, BRIM and USANIS (1968) in the first paper in this series, presented a model which permits systematic study of population changes resulting from frequency-dependent competitive interactions among autogamous genotypes. The autogamous model is extended in this paper to allow inter-mating among the genotypes. Computer similation of evolving populations is employed to study the effect of various kinds of competitive interactions on the maintenance of heterozygosity in populations.

THEORY

SCHUTZ et al. (1968) defined a competitive system for a population of n autogamous competing genotypes in which the reproductive value in pure stand of the jth genotype, X i , is Hi and Hn is taken as unity. The reproductive value of X j

may then be written as rj = Hi + Cj, where Cj is the net effect of inter-genotypic competition between X j and the (n-1) other genotypes in the population. If each of the genotypes occurs with frequency pi, the mean reproductive value of the population is the sum of the reproductive values of the component genotypes weighted according to their frequency of occurrence, i.e.,

n n R = 2pii-i = z p i (H i 4- C , ) .

Under the assumption that inter-genotypic competition effects are additive, the reproductive value of X i due to competitive responses may be stated more ex- plicitly as

Ci =$& Pib(i/i) where b(j/ i) = b’(j/i) -Hi ,

b’(j/,) = output of X j under conditions of maximum competition with Xi (for a given plant density),

and b(j,i) $ b(i/j). If we consider a more general population in which intermating occurs among

the genotypes, the competitive system can likewise be defined in terms of pair- wise interactions among the genotypes. The population concepts examined in this paper are based on the results for a diallelic locus with the relative reproductive values of the two homozygotes (AA,aa) and the heterozygote (Aa) expressed as functions of pure-stand responses and frequency-dependent competitive effects. The total effect of each of the competitive interactions (under maximum competition) may be represented schematically as the sum of absolute values as follows:

(bfAA/aa) I + Ib(aa/AA) I I

aa Aa AA,

I I

INTER-GENOTYPIC COMPETITION 877

where b(h .k / aa ) = b’(AA/aa) -

b ( A a / A A ) = b’(Aa/AA) - H A a

V ( A A / a a ) = reproductive output of AA under conditions of maximum competition with aa

p ( A a / A A ) = reproductive output of Aa under conditions of maximum Com-

H A A , H A a , Ha, = pure stand reproductive values. petition with AA

The parameters b ( A A / a a ) , . . . , b(Aa /AA) are constant for a given plant density, but the reproductive value of each genotype is a function of the frequencies of the other genotypes and, hence, is frequency-dependent. Thus, the frequencies and reproductive values of the array of genotypes may be written as:

Genotype Frequency Reproductive Value AA f A A TAA = H A A + fAab(AA/Aa) $- f a a b ( A A / a a )

Aa f Aa r A a = 1 f fAAb(Aa /AA) + f a a b ( A a / a a )

aa f aa raa = Haa + f A A b ( a a / U ) + f A a b ( a a / A a )

where faa = 1 - f A a - fAA

Note that the pure stand reproductive value of the heterozygote ( H A a ) is taken as unity in the model. The recursion formulae relating genotypic frequencies in the t th and (t+l) generations are:

where

s and c are the proportions of selfing and random outcrossing, respectively (s+c =

I), and R” = I: r(:)p(:). The mean reproductive value of the population in the

( t + l ) generation is R’ = x r y + l ) f ? + l ) .

3

3

ADDITIVE I DOMINANCE I WERDWINANCE I

2K i I AA

K

0

-K I Qo

AA Ao

00 00

S.0, K.f.0

S=.SO, K4.O

S=.95, Kd .0

S-.QS, K - 0 . 5

0 1.0 21)

Ao

00

~. . - .._ .. 43.:. ............. -.---

c. : -..

0 1.0 2.0

PURE STAND REPROOUCTIVE VALUE (HAA)


RESULTS

Four different types of inter-genotypic competitive effects were defined by SCHUTZ et al. (1968) in terms of autogamous genotypes. The same effects can be defined for a diallelic locus in an intermating population as follows:

[b (AA/aa) + b(aa/AA), ' ' ' b(Aa/AA) + b(AA/Aa) = 01 2, overcompensatory [ b ( A A / a a ) + b ( a a / A A ) , ' ' b ( A a / A A ) + b ( A A / A a ) > 01 ; 3 ) undercompensatory [b (AA/aa) + b ( a a / A A ) , . . . , b ( A a / A A ) + b(AA/Aa) < 01;

- and 4) neutral [ b ( A a l a a ) = . . . - b(AA/Aa) = 01. Since undercompensatory competitive effects appear to be relatively rare in nature (SCHUTZ and BRIM 1967; SIMMONDS 1962), no attempt is made in this paper to study their effect upon the maintenance of heterozygosity in evolving populations. Computer simulation of evolving populations of autogamous lines ( SCHUTZ et al. 1968) indicated that overcompensatory effects are an essential ingredient in the establishment of stable equilibria. Thus, it is of interest to determine whether the same condition holds for an intermating population.

Since the formulae for equilibrium frequencies are extremely complicated, no attempt was made to obtain an algebraic solution and, in every instance, a computer was used to obtain equilibrium points. The examples considered in studying the effect of competition on the maintenance of heterozygosity are hypo- thetical infinite populations with either additive, dominant, or overdominant genetic effects assigned to the competitive relationships among the genotypes. In all cases, the competitive and pure stand conditions of interest are those which result in a higher level of heterozygosity than would be realized with no selection and no competition among the genotypes ( ~ A A

Two overcompensatory and one complementary competitive system were chosen for detailed evaluation. In all of the models studied the sum of the absolute values of the competitive effects constituting the largest interaction was 2K, where K is the magnitude of the competitive eifects expressed as a multiple of the pure- stand reproductive value of the heterozygote. Thus, if the largest interaction occurred between genotypes i and j , 2K = lb(ilj)l + lb(3/ i ) l . For the examples discussed in this paper, K was assigned values of 0.5 or 1.0, since this range of values is consistent with estimates obtained from experimental soybean populations (SCHUTZ and BRIM 1967). 0 < K < 0.5 also leads to non-trivial equilibria, but the results are not given in this paper.

In one of the overcompensatory models studied (model I) , the largest interaction was 2K = 1 3K/2/ + I-K/21 and in the other overcompensatory model (model 11) was 2K = 12KI + 101. In each case, competitive relationships [b (AA/aa) ,

etc.] for each pair of genotypes are indicated by bar graphs in Figures 1 and 2,

r A a = raa = 1 ) .

FIGURE 1 .-The model I overcompensatory competitive system with either additive, dominant, or overdominant competitive effects and three levels of selfing. Competitive relationships [b(AA/aa) , etc.] for each pair of genotypes are indicated by bar graphs with the zero response lines representing the pure stand reproductive values and are expressed as a function of a positive value, K. Contour graphs indicate pure stand reproductive values (for the homozygotes) which result in stable equilibria and more heterozygotes than expected when competition and selection do not occur. Numbers on the contour lines indicate heterozygote frequency with equal intervals between the contours.

880 W. M. SCHUTZ A N D S . A. USANIS

with the zero response lines representing the pure stand reproductive values. Thus, with additive competitive effects, the interaction parameters €or model I (shown in the first bar graph in Figure 1) are brAA/aa) = 3K/2 , b(aa/AA) = -K/2, b(Aa/aa) = 3KJ4, b(aa /Aa) = -K/4, b(AA/Aa) = 3K/4 , ~ ( A ~ / A A ) - -K /4 . The additive para-

b ( A a / A A ) = 0 for the more powerful model I1 overcompensatory system in Figure 2. The largest interaction for the complementary model studied (model 111) was 2K = / K / + I-KI. The interaction parameters for model I11 are defined by bar graphs in Figure 3 in the manner described above. Overcompensatory and complementary competitive effect similar to those defined in models I and I11 were

- - meters are b(AA/aa) = 2K7 b ( a a / A A , - 0, b ( A a / a a ) = K . b ( a a / A a ) = 0, b(AA/Aa) = K,

ADDITIVE I1

AA 2K

K

0

-Ki

DOMINANCE I1 OVEROOM II

PURE STAND REPRODUCTIVE VALUE (HAA)

FIGURE 2.-The model I1 overcompensatory competitive system with either additive, dominant, or overdominant competitive effects and s = .95. Competitive relationships [b(AA/aw)r etc.] for each pair of genotypes are indicated by bar graphs with the zero response lines representing the pure stand reproductive values and are expressed as a function of a positive value, K . Contour graphs indicate pure stand reproductive values (for the homozygotes) which result in stable equilibria and more heterozygotes than expected when competition and selection do not occur. Numbers on the contour lines indicate heterozygote frequency with equal intervals between the contours.

AA Aa Aa

S0.95, K4.O p( *. . * : . ‘.I ...... ...< ................. . 0 : -..

’-a.

**.* ....

S - 9 5 , K-0.5 k *.., .k. *..*

.05 ..-. . ! ’, .., i -05 *2 : ..... 20 :.:....

.......... 4 ................ ........................

; .... . . ; .. j .*.. *..$ - ,

0 1.0 2.0 0 1.0 2.0

INTER-GENOTYPIC COMPETITION 88 1

AA t Aa

Aa

aa

U 3

0 D S1.95, Km0.5

............... ...............

Q 0 1.0 2.0 0 1.0 2.0 0 1.0 2.0

PURE STAND REPRODUCTIVE VALUE (HAA) FIGURE 3.-The model I11 complementary competitive system with either additive, dominant,

or overdominant competitive effects and s = .95. Competitive relationships [b(AA,aa)r etc.] for each pair of genotypes are indicated by bar graphs with the zero response lines representing the pure stand reproductive values and are expressed as a function of a positive value, K . Contour graphs indicate pure stand reproductive values (for the homozygotes) which result in stable equilibria and more heterozygotes than expected when competition and selection do not occur. Numbers on the contour lines indicate heterozygote frequency with equal intervals between the contours.

observed in experimental populations (SCHTJTZ and BRIM 1967); but the powerful overcompensatory system of model I1 appears to be less likely to occur in

882 W. M. SCHUTZ A N D S. A. USANIS

nature and, in that sense, approximates an upper bound for overcompensatory effects.

Pure stand reproductive values for HA* and Ha, ( H A , = 1) which result in stable equilibria and more heterozygotes than expected (when gene frequency = s) are designated by contoured areas in adjoining graphs. Thus, the contour graphs can be interpreted in much the same manner as the phase diagrams of HAYMAN (1953). The boundaries of the areas result from graphing the equations for the equilibrium frequencies of AA, Aa, and aa subject to the restrictions that the frequencies of AA and aa must be greater than zero and less than one, and that the frequency of Aa must be less than one but greater than e, the expected value with no competition and no selection. When gene frequency = 1/, the values of e fors = 0,0.50, and 0.95, are 0.50,0.33, and .05, respectively. For other gene frequencies, e is less than these values. Thus, the areas shown are minimum regions with respect to the permissible range of H A A and Ha*. Since the equilibrium equations are high order polynomials and extremely difficult to solve and graph, the areas were approximated by generating equilibrium frequencies for HA*, Ha, 2 0 at intervals of .05 and mapping those points that satisfied the restrictions. A somewhat poorer approximation of the areas can be obtained by using linear regression techniques to characterize the equilibrium formulae from a small subset of generated equilibrium points.

The results for three levels of selfing i s = 0, 0.5, 0.95) are shown for model I, but only s = 0.95 is given for models 11 and 111. However, the effect of varying levels of selfing was much the same in all of the competitive systems.

A genetic interpretation of the lines and areas in Figures 1-3 is given in Figure 4. Broken lines are used to indicate HA, and Ha, points for which the pure stand reproductive value of the heterozygote exhibits additive ( H A A = 2HA, - Ha, = 2 - H a , ) and dominant (HA* = 1 or Ha, = 1) genetic effects. The areas between the additive and dominance lines consist of points with partially dominant pure stand relationships, and points below and to the left of the dominance lines (0 i H A , , Ha, < 1) are in the overdominance range. Underdominance effects occur above and to the right of the dominance lines ( H A A , H a , > 1 ) .

The permissible range of pure stand reproductive values for the homozygotes (ElAA, Ha, area) is larger for K = 1 than for K = s, except for the additive I and 111, dominance 111, and overdominance I11 systems. The partial dominance por- tion of the H A * , Ha, areas is much smaller for K = i /z than for K = 1 in every case.

When K = 1 in the additive 111, overdominance I11 and all dominance competitive systems, the areas of partial dominance make up the bulk of the equilibrium region; but for the additive I and I1 and overdominance I and I1 systems, partial dominance is responsible for only 30-50% of the total region. For K = i / z , overdominance makes up at least 50% of the region in every case.

The overall size of the H A , , Ha, area ias well as the partial dominance region) increases as the amount of selfing ( s ) increases for K = 1 in the additive I system, but for K = i /z ( s = 0, 0.5 not shown on graphs), the overall area decreases and the partial dominance region is virtually unchanged. The HAA, H , area (includ-


0 1.0 2.0 PURE STAND REPRODUCTIVE VALUE (HAA)

FIGURE 4.-Genetic interpretation of the areas shown in the pure stand response graphs of Figures 1-3. Areas where a is dominant to A and A is dominant to a are designated (a>A) and (A>a), respectively.

ing the partial dominance region) decreases in size as s increases for both K = 1 and K =

equilibrium region for any of the additive competitive systems analyzed in this study, but partial dominance, dominance and overdominance effects did occur. How- ever, there are additive, partial dominance, dominance and overdominance pure- stand effects for all of the dominance competitive systems except dominance 111. In the latter system, there are no dominance hor overdominance effects for the case where K = 1. The overdominance competitive systems are the only ones in which underdominance pure-stand reproductive values occur.

The manner and rate of approach to equilibrium for additive, dominant and overdominant competitive systems (models I and 111) when s = .95 and K = 1 is shown in Figure 5. Starting frequencies of fAA = f a a = .475 and fAa = .050 were chosen to illustrate the change in genotypic frequency that results when frequency-dependent competitive effects are imposed on a population where no differential selection had previously taken place. In each case H u and H a a values were selected near the middle of the partial dominance region in which aa is dominant to A A for high pure-stand reproductive output. Note that, in most cases, the populations are very close to equilibrium in less than 10 generations,

(not shown) in the dominance I and overdominance I systems. There are no additive pure stand reproductive effects in the H A A ,

884 W. M. SCHUTZ A N D S . A. U S A N I S

ADDITIVE I DOMINANCE I OVERDOMINANCE I

ADDITIVE 111 DOMINANCE 111 OVERDOMINANCE 111

C

\

0 0

NUMBER OF GENERATlONS

FIGURE 5.-Genotypic frequencies (fAA, fAa, faa) and mean reproductive values (R') in each generation for the additive, dominant, and overdominant competitive systems defined in Figures 1 and 3 when s = .95 and K = 1.

and proceed toward the equilibrium state with a minimum of oscillation. For the model I overcompensatory system, the mean reproductive value of the population declined with additive competitive effects but increased for dominant and overdominant effects. The mean reproductive value of the complementary competitive system (model 111) declined in early generations for both dominant and overdominant effects. The reproductive value of the additive complementary system was quite stable, since genotypic frequencies did not change much as the population went to equilibrium. Note that the overdominance model I system was the only one that had a higher reproductive value at equilibrium than either of the homozygotes in pure stand.

Additional information on the rate of approach to equilibrium, equilibrium frequencies, and mean reproductive values are given in Table 1 for various values of H A , and Haa. The equilibrium regions sampled are those defined in Figures 1-3


TABLE 1

Equilibrium frequencies, number of generations required to reach equilibrium; and the mean reproductiue ualues, RIe, at equilibrium in both the presence and absence of competitive

effects (s = .95 and K = I ) . In the absence of competition, f, = 1.000 and R', = Ha, unless otherwise indicated by parentheses

Pure stand Generations to' effects Equilibrium frequencies equilibrium Competition

contribution H A A Ha, fAA fAa fa. tl tz R', to R',

.IO

.IO

.IO

.25

.25

.25

.40

.40

.4 5

.10

.IO

.10

.35

.35

.35

.60

.60

.85

.IO

.35

.35

.60

.60

.60

.85

.85

1.20

.10

.IO

.IO

.30

.30

1.30 .90 .50

1.40 1 .oo .60

1.40 1.10 1.35

1.50 1.05 .60

1.65 1.15 .60

1.75 1.20 1.85

1.05 1.20 .70

1.40 1 .oo .60

1.40 .80

1.70

1.35 .80 .25

1.40 1 .oo

.284

.581

.629

.331

.672

.807

,473 .750 .570

(.010)

(.001)

.134 ,278 .I 12 .I90 .488 .406

(.003) ,305 .777 .4$8

.067 ,115 .128 .I27 .327 .358

(.425) .331 .746

.433 (.973)

.349

.575 ,650

(.047) .424 .606

.064

.I63

.347 (.184) .056 .098 .I71

(.007) ,054 .058 ,053

.083

.319

.825 ,069 .185 .556

,067 ,073 ,056

(.009)

Additive I .651 18 19 .256 13 12 .023 12 11

.613 17 19

.231 13 17

.022 14 12 (.991) (32) (38) .473 3 10 .192 17 19 .377 16 15

(306) (10) (41)

Dominance I .782 23 .402 18 .064 10 .741 23 .327 18 .038 22

.628 20

.I50 26

.436 13

(.987) (41

21 11 10 22 17 16

(41) 19 29 11

Overdominance I .385 ,548 21 16 .233 .652 14 22 .724 .I49 14 12 .118 .755 29 30 ,361 .312 19 8 ,578 .065 20 16

.145 .524 19 17

.I94 .061 11 30

.084 .483 15 12

(.151) (.425) (12) (14)

(.007) (.020) (94) (102)

Additive I1 ,062 .589 10 12 .I25 .301 11 11 .294 ,056 11 8

.055 .521 9 11

.075 .319 10 10

(.769) (.185) (12) (10)

1.15 0.67 0.55

(0.59) 1.23 0.70 0.47

(0.60) 1.16 0.74 1.06

1.41. 1.01 0.93 1.55 0.95 0.76

1.58 0.85 1.60

(.60)

1.21 1.26 1.09 1 .40 1.09 0.98

(0.66) 1.35 0.98

(0.85) 1.59

1.36 0.88 0.65

(0.82) 1.40 1.03

-.i5 -23 -.04

-.i 7 -.30 -.13

-94 -.36 -29

-.06 -.04 +.33 -.io -.20 +.is

-.17 -.35 -.25

+.is +.06 +.34

0 +.09 +.32

-.05 +.13

-.I 1

+.Ol +.08 -.17

0 +.03


TABLE I-Continued

Pure stand Generations to* effects Equilibrium frequencies equilibrium Competition

contribution HA, Ha, fAA f.4, fa, *I t 2 R', t o R',

.30

.30

.50

.50

.10

.IO

.10

.35

.35

.35

.60

.60

.85

.IO

.10

.40

.40

.70

.70

1 .oo 1 .oo 1.50

.10

.10

.IO

.20

.20

.30

.45

.10

.10

.30

.30

.50

.70

.90

.60

.20

1.20 .80

1.70 .95 .20

1.80 1.05 .30

1.80 1.20 1.80

1.10 .50

1.20 .50

1.40 .70

1.70 1.20 2.00

1.10 1 .oo .90

1.15 1.05 1.30 1.45

1.10 1.20 1.30 1.10 1.50 1.70 1.90

,757

,805 (.231) ,621 .806

(.002)

.I69 ,351 .163

(.050) .230 .503 ,521

(.272) .342 ,612 .471

.094

.I13

.167

.277 (.160) .239 .501

(.450) .260 .583 .438

(.010)

.425 ,574 .581 .715 .802 .463 .474

.117

.192

.364

.309

.4$0

.463

.473

.I10 .I33 13 (.009) (.989) (34) .171 .023 15

(.665) (.104) (14) .052 .327 11 .053 .I41 13

Dominance I1 .077 .754 17 .234 .415 15 .746 .091 11

(317) (.133) (10) .069 .701 15 .138 .359 14 .392 .087 22

.066 .592 14

.079 .310 13

.055 ,474 4

Overdominance I1 ,305 .602 13 ,644 .244 10

(.184) (.806) (10) .245 .588 13 .541 .182 12

(.269) (.571) (14) .166 .595 16 .327 .172 18

(.101) (.450) (10) .IO2 .638 23 .14!3 .268 15 .069 .492 15

(.549) (.179) (17)

Additive I11 .I51 .425 12 .297 .129 20 .373 .046 14 .133 .I53 39 .161 .037 18 .074 .463 7 .052 .474 2

Dominance I11 .765 .I17 23 .420 .388 41 .272 .364 21 .624 .066 18 .120 .440 11 .075 .463 7 .054 .473 3

14 0.72 (37) (0.60) 11 0.60

(11) (0.76) 13 1.21 14 0.85

16 1.75 14 1.15 11 0.95 (8) (0.85) 12 1.83 17 1.15 13 0.76

(12) (0.70) 10 1.82 19 1.24 16 1.81

13 1.43 8 1.19

(41) (0.59) 15 1.44 10 1.14

(24) (0.62) 14 1.54 13 1.16

(12) (0.73) 20 1.77 22 1.38 11 2.03

14 0.66 20 0.48 15 0.47 40 0.45 17 0.36 10 0.82 10 0.95

19 0.91 18 0.90 15 0.85 15 0.79 14 1.00 12 1.18 10 1.38

+.le

-.16

f.01 +.05

+.05 +.20 +.lo

+.03 +.lo +.06

+.02

+.Ol

+.33

+.04

+.60

+.24 +.52

+.14 +.43

+.07 +.18 f.03

-.44 -.52 -.43 -.70 -.69 -.48 -.50

-.19 -.30 -.45 -.31 -.50 -.52 -52

INTER-GENOTYPIC COMPETITION

TABLE l-Continued

Overdominance I11 .50 1.10 .123 .650 .227 47 36 0.96 -.14 .70 1.20 .360 .281 .360 22 14 0.96 -.24 .70 1.00 .338 .569 .093 37 33 0.90 -.lo .90 1.40 .436 .128 ,436 13 15 1.13 -.27 .90 1.30 .753 .I08 .139 57 56 0.97 -.33

1.10 1.60 .459 ,082 .459 8 12 1.32 -.28 1.30 1.80 .470 .060 .470 5 11 1.52 -.28

887

* Starting frequencies for t , and t , of fAA = f, = .475, fAa = .050 and fAA = fa, = fAa = .333, respectively, Populations were considered to be at equilibrium when frequencies were constant to three decimal places.

for s = .95 and K = 1. Systems where K =% are not shown in Table 1, but the rate of approach to equilibrium is very similar to K = 1. In most cases, equilibrium is reached in less than 30 generations, regardless of the competitive system involved. Different starting frequencies had a negligible effect, except where initial frequencies and equilibrium frequencies were similar. In some cases, the complementary (model 111) competitive systems proceeded to equilibrium more slowly, but in all cases a stable equilibrium was reached. The mean reproductive values of all of the complementary and additive I overcompensatory systems and most of the dominance I overcompensatory systems were lower at equilibrium frequencies than they would have been in the absence of competition. However, R', values for most of the overdominance I and additive I1 and all of the dominance I1 and overdominance I1 systems were enhanced by the competitive effects.

DISCUSSION

The reasons for the maintenance of considerable amounts of allelic polymorphism in natural populations are of great evolutionary importance but have not been clearly defined.

Heterosis in fitness at the locus level has been proposed as an answer to this phenomenon (ALLARD and JAIN 1962; CRUMPACKER 1966; JAIN and ALLARD 1960) , and population models have been suggested by which heterotic loci might be maintained at equilibria (KING 1967; MILKMAN 1967; SVED, REED, and BODMER 1967). Disruptive selection (SMITH 1962), differential fertility (PURSER 1966), and partial negative assortative mating (WORKMAN 1964) have been proposed as other (rather specialized) means by which heterozygosity might be maintained. However, there is now considerable experimental evidence ( ALLARD, JAIN and WORKMAN 1968; HARDING et al. 1966; KOJIMA and YARBROUGH 1967; TOBARI and KOJIMA 1967; YARBROUGH and KOJIMA 1967) that frequency-dependent selection may be an important agent in maintaining polymorphisms in both plants and animals. CLARK and O'DONALD (1 964) and CLARK (1 964) set up some simple mathematical models of balanced polymorphism under frequency-dependent selection and suggested that they might apply to mimetic and apostatic polymorphisms observed in animals.


The model utilized in this theoretical study of frequency-dependent competitive effects is based on the assumption that competitive effects are proportionally additive in nature. The reader is referred to the previous paper in this series ( SCHUTZ et al. 1968) for a discussion of the supporting evidence for this assumption as well as for references documenting the existence of overcompensatory and complementary competitive interactions among plant genotypes.

Perhaps the most striking result of the study reported herein is the wide range of conditions that can result in excess heterozygosity under a frequency-dependent competitive regime. I t is clear that overdominance of the heterozygote (heterosis) for the competitive component of fitness is not essential for the maintenance of heterozygosity, even in populations that are largely self-pollinated. Heterozygotes intermediate for competitive effects as well as pure stand reproductive ability can be maintained in the population in relatively high frequencies regardless of whether overcompensatory or complementary competitive interactions are involved. The “epistatic” effect of these frequency-dependent and frequency-independent components of fitness results in a significant reproductive advantage for the heterozygote. However, heterosis cannot be ruled out as a cause of polymorphism, since overdominant effects for both competitive ability and pure stand response can also be responsible for this phenomenon. It should be empha- sized that the contour graphs of the permissible range of pure stand values repre- sent minimum bounds, since the expected frequency of the heterozygote without competition and selection was computed for gene frequency of one-half. For other gene frequencies, the expected heterozygote frequency is less, and the permissible range of H A A and H,, is greater (i.e., the equilibrium genotypic frequencies with competition and selection remain the same for any initial gene frequency between zero and one). The magnitude of the competitive interactions (range of K values) considered in the examples is consistent with experimental estimates (SCHUTZ and BRIM 1967). However, smaller K values lead to similar results except that the equilibrium regions for H A , and H,, are somewhat smaller in size.

The equilibrium contour lines of Figures 1-3 indicate that in all of the competitive systems examined (regardless of the kind of gene action for competitive fitness), the highest level and widest range of allelic polymorphism resulted from overdominant pure stand reproductive values for the heterozygote. However, judging from the levels of heterozygosity observed in natural plant populations (ALLARD and JAIN 1962; ALLARD and WORKMAN 1963; JAIN and ALLARD 1960),

reproductive values in the additive, partial dominance or dominance regions appear to be a much more likely occurrence.

It should be noted that for the additive and dominant competitive systems the homozygote AA must always be inferior to aa for pure stand reproductive ability in order to achieve the polymorphic condition. However, this does not appear to be a serious restriction, since, in many crop plants studied, the best competitors in mixtures are not necessarily the highest yielding genotypes in pure stand ( FRANKEL 1939; HARLAN and MARTIN 1938; MONTGOMERY 1912; SUNESON 1949; SUNESON and WIEBE 1942; SCHUTZ and BRIM 1967).

The results also indicate that complementary as well as overcompensatory


competitive effects can give rise to stable equilibria in populations with mixed selfing and random mating, although the permissible range of H A A and Ha, values is much greater in the latter. This observation is in contrast to the results obtained for homozygous lines (SCHUTZ et al. 1968) where stable equilibria did not appear to exist for complementary systems.

The effect of competition on the mean reproductive value of a population at equilibrium depends on the type of competitive effect as well as the kind of gene action involved in the competitive interaction. Overcompensatory competitive effects appear to be a necessary, but not a sufficient condition, for the evolution of increased reproductive output. Unless the overcompensatory interaction is relatively large, additive, and in some cases dominance and overdominance competitive effects, lead to equilibrium populations with lower reproductive values than would occur in the absence of competition. In every case, the complementary systems studied were at a reproductive disadvantage Thus, competitive conditions which lead to excess heterozygosity do not necessarily improve the reproductive ability of a population. However, they do preserve the population’s potential for genetic recombination, a factor which may be of importance in the evolutionary stability of predominantly self-pollinated plants. Nevertheless, if competition among populations is of overriding importance, intra-population competition effects may be limited to overcompensatory cases where R’, is increased by competition. Comparisons of Figures 1 and 2 indicate that increasing the degree of overcompensation results in a larger equilibrium region for H A A and Ha. HOW- ever, an overcompensatory system as powerful as model I1 probably occurs infre- quently in plant populations.

The rate of approach to equilibrium is much more rapid in a population with mixed selfing and random mating than in a comparable three-component p o p - lation of autogamous homozygous lines (SCHUTZ et al. 1968). In the former, equilibrium results from the combined pressure of the mating system and competitive interactions whereas, in the latter, competition is the only driving force.

In the absence of experimental data, one cannot be sure that competitive interactions are involved in maintaining allelic polymorphism. However, we do know that competition is a very common occurrence in evolving populations, a factor which suggests that the theoretical mechanism demonstrated here has a good chance of being operative in natural populations. If this does occur to any significant extent, then organisms that are largely self-pollinated possess genetic versatility that has not been properly appreciated.

SUMMARY

Experimental evidence is accumulating which indicates that frequency-dependent selection may be an important agent in maintaining allelic polymorphism in evolving populations. A linear model was utilized to examine the role of frequency-dependent competitive interactions in maintaining heterozygosity at a diallelic locus. Computer simulation of evolving populations indicated that heterozygotes intermediate for competitive fitness as well as pure stand reproductive


ability may be retained in relatively high frequencies, even in populations that are largely self-pollinated. Thus, heterosis is not essential to polymorphic variation. Both overcompensatory and complementary competitive effects resulted in stable equilibria; but in the latter case, the mean reproductive value of the population invariably decreased as the result of the competitive interactions. Although overdominant pure-stand reproductive values can result in high levels of allelic polymorphism in all of the competitive systems studied, values in the additive, partial dominance, or dominance range appear to be more likely to occur in nature. Since competition is a common occurrence in evolving populations, the results suggest that predominantly self-pollinated organisms may possess un- suspected amounts of genetic plasticity due to enforced heterozygosity.

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w. schun2 and s.this paper are based on the results for a diallelic locus with the relative...

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