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THE EFFECTS OF ENVIRONMENTAL HETEROGENEITY ON THE GENETICS O F FINITE POPULATIONS

HEATHER DICKINSON* AND JANIS ANTONOVICS

Department of Biology, University of Stirling, Stirling, Scotland, U.K. and Department of Botany, Duke University, Durham, North Carolina 27706

ABSTRACT

The effects on a panmictic population of a patchy environment and stochas- tic selection were investigated by computer simulation. The model allowed for different patterns and intensities of selection and various densities of distribu- tion of the offspring. I t delineated the effects of these variables on the correla- tion between the genotype and the environment, the level of heterozygosity, the change i n gene frequency from generation to generation and the diver- gence between similar populations due to random effects.

T

is being increasingly realized that the heterogeneity of the environment can 'have not only a profound effect on evolutionary processes (BRADSHAW 1965; LEVINS 1964, 1965) but also that it can influence directly the genetic structure of populations (MAYNARD SMITH 1971). This has now been repeatedly shown in theory (for discussion and references see MAYNARD SMITH 1970; DICKINSON and ANTONOVICS 1973), in the laboratory (for discussion and references see THODAY and GIBSON 1970), and in nature (ANTONOVICS and BRADSHAW 1970 and previous papers in this series; SNAYDON 1970; and WATSON 1973). It is also evident that natural situations are often complex and cannot be adequately ap- proximated by theoretical two-niche models, by two-way disruptive selection, or by sharp boundaries between two extreme environments. These situations are likely to be exceptions and natural situations generally involve mosaic patchy environments of considerably greater complexity than two-niche environments. This complexity is the result of several features of such environments, among them the following:

(i) There may be many different intensities and directions of selection and many optima rather than two extreme selection pressures.

(ii) The heterogeneity may occur in small patches, such that each patch can support only a small population. Sampling error as regards the genotypes arriv- ing in each patch may therefore have a significant effect.

(iii) The patches, and hence the selection pressures, may fluctuate with time. Previous workers have considered the general conditions necessary for such environments to maintain a polymorphism when there are niches of different size which have different selection pressures (LEVENE 1953; and see MAYNARD SMITH

1970 for review). In this paper we investigate by computer sirnulation the gen- eral effects of a varied, patchy, fluctuating environment on the genetic structure

* Now a t computing senrice department, Unnersity of Glasgow, Glasgow, Scotland, U K

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714 H. DICKINSON A N D J. ANTONOVICS

of a population. A later paper will apply this stochastic model to the natural situation studied by WATSON (1973).

THE MODEL (see Table 1)

The population remained at a constant size, consisting of M individuals of the genotypes AA, Aa aa.

The environment was divided into

M

patches, such that each patch could sup- port one individual. The relative fitnesses of particular genotypes could be dif- ferent in all these patches (i.e. if

i

labels the patch, then the fitnesses 1-xi, 1-yi, 1-zi could vary with i)

.

So, if

xi,

y i , zi were suitably chosen, there could be any range of fitnesses.

I n all cases, the initial genotype frequencies were those which would have been reached, at equilibrium, under a deterministic selection system in which the rela- tive fitnesses had the same magnitudes as in the present model.

The genotypes were initially distributed at random over the patches.

In the cycle of mating and selection, it was assumed that the entire population mated at random and produced an infinite number of offspring. The offspring were dispersed at random over the patches, the only restriction being that a cer- tain number of offspring, N , arrived in each patch.

Selection was allowed to act and one survivor was chosen from the N offspring which arrived in each patch. It was assumed that, in a particular patch, the rela- tive probability of each genotype being the sole survivor was the product of the frequency of that genotype in the patch and the fitness of the genotype in the patch (see TABLE 1). The genotype of the survivor was decided by generating a random number, ri, in the range (0,l) and comparing this with the probabilities of survival, Ui,

Vi,

Wi, of the three genotypes.

If 0

<

ri

<

Ui the survivor was of genotype AA.

TABLE 1

The numbers, fitnesses and probnbilities of survival of the uarious genotypes in the ith patch

Total

Genotype A A Aa an

Number'

4

mi ni N

4

Frequency* u.=-

' N

mi ' N

U,=- ni

' N

w.=- 1

Fitness 1 - x i 1-Y, l--zi -

Relative probability

of survival (l--zi)ui (l-Yi) vi ( I --zi)wi T= (1-14) (1-Yi) ui

+

(1-9) Wi ( 1--zi) wi

1

Probability (I---z$) ui ( I-Yi) vi

V.= W i z

' T T

of survival U$= T

(3)

EFFECTS O F PATCHY ENVIRONMENTS 71

5

If

Ui

<

ri

<

U i

+

Vi

If

U ;

+

V l

<

i i

<

1

Hence, the selection process was subject to random fluctuations.

After selection, the degree of population adaption/differentiation was meas- ured by the genotype-environment correlation using Kendall’s 7-coefficient for tied ranks

(KENDALL

1948, Chapter 3.15). This coefficient is increased by geno- types occupying patches where they have a high fitness and, conversely, it is de- creased by genotypes surviving in less suitable patches where they have a low fitness.

This cycle of random mating, random seed dispersal, selection and the subse- quent calculation of the genotype-environment correlation was carried through for twenty generations. A regression line was fitted to the frequency of allele a in successive generations. Since the frequency of allele a follows a binomial dis- tribution, it was first transformed using an arc-sin transformation ( SNEDECOR 1956). This regression line was constrained to pass through the initial frequency of allele a which was the equilibrium value obtained from the corresponding de- terministic model (see APPENDIX). Hence the slope, b, of the regression line showed whether the gene frequency underwent any directional change away from the deterministic value. The variance, s2, of the deviations of the gene fre- quency about the regression line was also calculated.

The entire process was then repeated for nineteen more replicates. These twenty replicate runs had the same number of offspring,

N ,

arriving in each patch and the same fitnesses: they differed only in the random fluctuations in the offspring genotypes arriving in each p t c h and in the fluctuations in the selection process.

the survivor was of genotype Aa. the survivor was o i genotype aa.

This general model was restricted to a few particular cases:

(1) The number of offspring arriving in each patch was chosen to describe (a) When there was a sparse distribution of offspring only a small num-

ber

of offspring, arbitrarily chosen to be N=4, arrived in each patch. (b) The opposite extreme, of a dense distribution of offspring, was de- scribed by N = m . In this case, only the frequencies of the genotypes, rather than their numbers, were considered and these were the same in all the patches.

either a sparse o r a dense distribution of offspring.

(2) Three distinct patterns of selection were chosen (see Table 2 ) .

(a) Two-niche system: There were two types of patches. appearing in equal numbers. In half the patches (those belonging to niche X ) A A

was favored, while in the others (niche Y ) aa was favored. We con- sidered both (1) the situation with A dominant and (2) the situation with A showing no dominance effects.

(b) Three-niche system: There were three types of patch, such that either

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716 H. DICKINSON A N D J. ANTONOVICS TABLE 2

Patterns of selection

Fitness

Number of patches

Selection pattem Niche AA Aa aa

a1 2-niche X 1 1 1-s M/2

dominance Y I-s 1-s 1 M/2

a2 2-niche X 1 1-s/2 I-s M/2

no dominance Y I-s 1-s/2 1 M / 2

bl 3-niche X 1 I-s

I-s

M/3

(1-1) rectangular Y I-s 1 I-s M/3

distribution of niches 2 I-s I-s 1 M I 3

bl 3-niche X 1 I-s I-S M/4

(1-2) normal Y I-s 1 I-s M/2

distribution of niches Z I-s 1-s 1 M/4

b2 3-niche X 1 1-s/2

1-s

M/3

(2-1) rectangular Y 1-s/2 1 1-s/2 M/3

distribution of niches Z I-s 1-s/2 1 M/3

b2 3-niche X 1 1-s/2 1-s M/4

(2-2) normal Y 1-s/2 1 1-s/2 M / 2

distribution of niches Z I-s 1-s/2 1 M/4

c 6-niche 1 1 1-s/2 I-s M/G

2 1 1-S/4 1-3S/4 M / 6

3 1-S/4 1 1-s/2 M/6

4 1-s/2 1 I-S/4 M/6

5 1-3S/4 1-S/4 1 M / 6

6 I-s 1-s/2 1 M/6

equal numbers [Table 2, b l ( 1-1 ) and b2 (2-1 )

1,

or there were twice as many patches favoring the heterozygotes as there were favoring each homozygote [Table2, b l (1-2) and b2 (2-2)].

(c) Six-niche system: The maic features of multiple-niche selection were demonstrated by a six-niche system. The six types of patch appeared in equal numbers and the fitnesses were chosen to mimic an environ- ment where the patches changed gradually from one extreme, where AA was favored, through an intermediate range favoring Aa, to the other extreme favoring aa.

W e also considered the situation where there was no selection whatsoever. Although on a deterministic model the parameter S (see Table 2) would be referred to as the selection pressure, the introduction of random fluctuations in the selection process means that S corresponds neither to the selection pressure on a particular generation nor to the average selection pressure averaged over several generations.

There were three sources of stochastic variation in the general model: (i) The finite size of the population.

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EFFECTS O F PATCHY ENVIRONMENTS 71 7 (iii) The fluctuations intrinsic to the selection process used.

Since the fluctuations introduced both by the selection process and by the random sampling of offspring genotypes arriving in each patch would be aver- aged out if there were a large number of patches (i.e., a large population), these three sources of stochastic variation were interdependent. Our selection schemes, in which the stochastic element is introduced through consideration of selection at the level of the individual organism, are in contrast to those of WRIGHT (1969), in which the selection pressure on the entire population fluctuates from genera- tion to generation.

The consequences of stochastic selection for genotypes of particular fitnesses were compared with the consequences of deterministic selection on genotypes of the same relative fitness. The deterministic models assumed that there was a dense distribution

OF

offspring over the patches and that the number of adults supported by each niche was very large; hence the values of S (see Table 2) were in this case the actual selection pressures. The consequences of such deterministic models can be deduced algebraically and details are given in the APPENDIX.

The consequences of the two-niche models were also compared with those of a similar deterministic two-niche model in which there was migration of male gametes only (DICKINSON and ANTONOVICS 1973).

RESULTS

a) Two-niche model: Figure 1 shows the results for a typical case (population size 60; N=4: relative fitnesses as in Table 2 (a2), given by S=0.8 and no domi- nance. The overall gene frequency in niche

X ,

the gene frequency in niche Y , the genotype-environment correlation and the departure of the genotype fre- quencies from Hardy-Weinberg expectations are shown for twenty generations and for eight replicates. The horizontal dashed lines indicate the results obtained with the same regime of migration and with the same relative fitnesses on a de- terministic model; the dash and dotted lines correspond to a deterministic model with the same relative fitnesses but with migration of male gametes only.

In time, the gene frequencies resulting from the stochastic model tended to diverge from the deterministic results (Figure l a ) . I n one run, shown by a dotted line, there was a tendency towards fixation of one allele. The gene frequencies in niches

X

and

Y

showed marked fluctuations about the deterministic values and although, on an average, they diverged less than predicted deterministically, they could diverge more (Figure Ib)

.

This trend was also reflected by the genotype- environment correlation, which was usually lower than on the deterministic model (Figure I C ) . The run which drifted towards fixation is again dotted. Any marked deviation of the gene frequencies from their deterministic values was accompanied by a lowering of the genotype-environment correlation, although the converse was not necessarily true. The heterozygote frequency was, on average, very close to that predicted by the Hardy-Weinberg law, in agreement with the deterministic model (Figure I d ) .

(6)

FIGURE 1.-Results of 8 replicate runs for a typical 2-niche cause (see text).

A

I I I 1

5

1

0

.I

5

m

(7)

EFFECTS O F PATCHY ENVIRONMENTS 719

(c) Genotype-environment correlation.

cI1

LLI

6.20

z g

10

LLJLLJ

l-

1

a g

-

-I

0

‘ - 2 0

-3

0

5

IO

15

x)

GE

N%ATON

S

-3

(8)

720 H. DICKINSON A N D J. ANTONOVICS

(Figure 1 b)

.

The divergence of the gene frequencies was reflected in the geno- type-environment correlation which was appreciably higher when there was migration by males only (Figure IC). Habitat selection by females also resulted in an excess of heterozygotes over Hardy-Weinberg expectations (Figure Id). The main interest of the results lies not in viewing each situation separately, but in deducing general trends by comparison of the results for different intensi- ties of selection and for the particular cases outlined earlier. To facilitate such a

comparison, the following parameters were calculated for the twenty replicates over the first twenty generations.

TABLE 3

The variance of the regression coefficients, var(b). All values of var(b) have been multiplied by IOJ

Population size N=4 N = m

( M ) S 0.0 0.2 0.5 0.8 0.2 0.5 0.8

a1 2-niche dominance

a2 2-niche- no dominance

bl 3-nich- (1-1) rectangular

distribution of niches

bl 3-niche-normal (1-2) distribution

of niches

b2 3-niche- (2-1) rectangular

distribution of niches

b2 3-niche-noma1 (%2) distribution

of niches

c 6-niche

36 60 120 240 36 60 120 24Q 36 60 120 240 36 60 120 2443 36 60 120 240 36 60 120 240 36 60 120 240

0.23 0.39 0.20 0.03 0.11 0.13 0.12 0.02 0.06 0.07 0.04 0.01 0.03 0.02 0.01 0.00

. . . 0.18 0.09 0.03 . . . 0.15 0.08 0.02 . . . 0.07 0.04. 0.01

, . , 0.02 0.01 0.00

. . . 0.23 0.15 0.08 . . . 0.11 0.09 0.06 . . . 0.07 0.05 0.03 . . . 0.02 0.02 0.01

. . . 0.14 0.07 0.02 . . . 0.08 0.05 0.01

. . . 0.04 0.03 0.01 . . . 0.02 0.01 0.00

. . . 0.16 0.10 0.03 . . . 0.07 0.07 0.02 . . . 0.05 0.04 0.01 . . . 0.02 0.01 0.00

. . . 0.14 0.08 0.04. . . . 0.07 0.05 0.02 . . 0.04. 0.03 0.01 . . . 0.02 0.01 0.00

. . . 0.17 0.09 0.03 . . . 0.08 0.06 0.03

. . . 0.05 0.03 0.01 . . . 0.02 0.01 0.01

0.19 0.08 0.01 0.16 0.08 0.00 0.06 0.03 0.00 0.03 0.01 0.00

0.19 0.09 0.02 0.21 0.07 0.01 0.07 0.03 0.01 0.03 0.01 0.00

0.23 0.16 0.03 0.19 0.11 0.02 0.06 0.05 0.01 0.03 0.02 0.00

0.19 0.09 0.01 0.16 0.05 0.01

0.M 0.01 0.00 0.02 0.01 0.00

0.17 0.10 0.02 0.17 0.06 0.01 0.05 0.02 0.00 0.02 0.01 0.00

0.15 0.08 0.02 0.17 0.05 0.01

0.04 0.02 0.00 0.02 0.01 0.00

(9)

EFFECTS O F PATCHY ENVIRONMENTS 721 (i) The mean, b, over the twenty replicates, of the regression coefficients, b, relating gene frequency and number of generations.

(ii) The variance of the regression coefficients, var ( b )

,

among the twenty replicates (the regression coefficients having been first transformed using Fisher’s z-transformation ( SNEDECOR 1956) )

.

This gave a measure of the divergence be- tween replicate lines (Table 3 ) .

(iii) The mean change in gene frequency from generation to generation (Table

4).

(iv) The mean genotype-environment correlation (Table

5).

TABLE 4

The m a n change in gene frequency from generation to generation. All ualues of the change in gene frequency h u e been multiplied b y 10

Population size N = 4 N = DO

(MI S 0.0 0.2 0.5 0.8 0.2 0.5 0.8

a1 2-niche dominance

a2 2-niche no dominance

bl 3-niche- (1-1) rectangular

distribution of niches

bl 3 niches- (1-2) normal

distribution of niches

b2 3-niche- (2-1) rectangular

distribution of niches

b2 3-niche- (2-2) normal

distribution of niches

c 6-niche

36 60 120 240 36 60 120 240 36 60 120 240 36 60 120 240 36 60 120 240 36 60 120 240 36 60 120 240

0.41 0.35 0.38 0.37 0.36 0.29 0.30 0.30 0.24 0.24 0.22 0.21 0.17 0.18 0.16 0.15

. . . 0.41 0.41 0.42 . . . 0.35 0.35 0.32 . . , 0.24 0.24 0.23

. . . 0.18 0.18 0.16

. . . 0.41 0.42 0.41 . . . 0.35 0.37 0.34 . . . 0.25 0.25 0.24 . . . 0.18 0.18 0.17

. , . 0.41 0.41 0.39

.

. . 0.35 0.35 0.33 . . . 0.25 0.25 0.23 . . , 0.19 0.17 0.17

, . . 0.42 0.44 0.42

, . . 0.35 0.35 0.34

. . . 0.24 0.25 0.23

, . . 0.18 0.18 0.16

. . . 0.42 0.43 0.40

. . . 0.35 0.35 0.34 . . . 0.24 0.24 0.23 . . . 0.18 0.18 0.17

, . . 0.43 0.43 0.41

. . . 0.35 0.35 0.34 . . . 0 . N 0.24 0.23 . . . 0.18 0.18 0.17

0.42 0.43 0.39 0.33 0.33 0.31 0.22 0.23 0.20 0.16 0.16 0.15

0.46 0.46 0.42 0.35 0.38 0.35 0.24 0.24 0.23 0.18 0.17 0.17

0.43 0.43 0.45 0.36 0.38 0.35 0.25 0.24 0.22 0.18 0.17 0.16

0.44 0.44 0.43 0.37 0.36 (3.35 0.25 0.24 0.21 0.17 0.17 0.16

0.44. 0.45 0.44 0.36 0.38 0.36 0.25 0.24 0.22 0.17 0.17 0.16

0.46 0.M 0.44 0.35 0.38 0.37 0.25 0.25 0.23 0.17 0.17 0.16

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722 H. D I C K I N S O N A N D J. A N T O N O V I C S

TABLE 5

The mean genotype-enuironmt correlation

Population

SlZB N = 4 N = m Deterministic

( M ) 5' 0.0 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8

~ ~ ~

a1 2-niche 36 0.00 0.06 0.21 0.48 dominance 60 0.01 0.07 0.22 0.49

240 0.00 0.07 0.23 0.44 120 -0.01 0.08 0.23 0.50

a2 2-niche-no 36 . . 0.05 0.16 0.36

dominance 60 . . . 0.05 0.16 0.35

120 . . . 0.06 0.16 0.35 240 . . . 0.05 0.16 0.35

bl 3-niche- 36 004 0.14 0.32

(1-1) rectangular 60 0.04 0.14 0.33 distribution 120 0.05 0.14 0.33 ofniches 240 0.04 0.14 0.33

bl 3-niche- 36 0.05 0.13 0.31

(1-2) normal 60 0.04 0.13 0.31

distribution 120 0.04 0.13 0.31 ofniches 240 0.04 0.13 0.31

b2 3-niche- 36 0.04 0.13 0.28

(2-1) rectangular 60 0.04 0.13 0.28 distribution 120 0.05 0.13 0.28 ofniches 241) 0.04 0.13 0.28

b2 3-niche- 36 . . . 0.04 0.12 0.25 (2-2) normal 60 . . . 0.04 0.11 0.25

distribution 120 . . 0.04q 0.12 0.25

ofniches 241) . . . 0.09 0.11 0.25

c 6-niche 36 0.03 0.11 0.22

60 0.04 0.10 0.22

120 0.04 0.11 0.22

240 0.03 0.10 0.21

0.10 0.31 0.62 O.u)+ 0.51+ 0.72+ 0.09 0.29 0.62

0.10 0.31 0.62

0.07 0.22 0.45 0.08 0.22 0.45 0.06 0.22 0.45 0.15+ 0.40+ 0.59f 0.07 0.22 0.45

0.08 0.23 0.45 J

1

0.06 0.21 0.52 0.06 0.19 0.50

0.05 0.20 0.51 0.06 0.20 0.52

0.05 0.17 0.49

0.05 0.18 0.49

1

0.05 0.19 0.50 0.05 0.19 0.50

0.05 0.19 0.50 J

0.06 0.18 0.38 0.06 0.18 0.37 0.06 0.17 0.36

0.05 0.17 0.37 0.06 0.18 0.36

1

0 . S 0.16 0.33 0.05 0.16 0.33 0.05 0.15 0.32

0.04 0.15 0.33 0.05 0.15 0.33

1

J

0.05 0.14 0.28 0.05 0.14 0.28

0.04 0.14 0.28 0.05 0.14 0.28

(v) The mean percentage excess (or deficit) of heterozygotes over Hardy- Weinberg expectations (Table 6 ) .

Parameters (iv) and (v) are compared with the corresponding deterministic results and, for the two-niche model, these parameters are also presented for the deterministic model with migration of male gametes only.

The mean regression coefficient, b, was never significantly different from zero, (e.g. the highest value, for a population of size 36, was 0.0064), showing that there was no general tendency for either allele to increase.

(11)

EFFECTS OF PATCHY ENVIRONMENTS 723

TABLE 6

Mean percentage excess (or deficit) of heterozygotes over Hardy-Weinberg expectations

Population

SlZB N=4 N= m Deterministic

(MI S 0.0 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8

a1 %niche- 36 dominance 60 120 240

a2 %niche-no 36 dominance 60 120 240

bl 3-niche- 36 (1-1) rectangular 60

distribution 120 ofniches 240

bl 3-niche- 36 (1-2) normal 60 distribution 120 ofniches 240

b2 3-niche- 36 (2-1) rectangular 60 distribution 120 ofniches 2u)

b2 3 - n i c h e 36 (2-2) normal 60 distribution 120 of niches 240

c 6-niches 36 60 120 2443

0.9 1.0 1.1 1.1 0.2 -0.2 -0.5 0.0 0.0 0.0 0.6 1.2 0.5 0.6 0.8 0.2 1.2

-0.5 0.3 0.4 0.5 0.4 0.4 0.6 0.5+-3.6+-9.9+ -0.4 0.2 0.3 0.3 0.2 0.0 0.2

. . 0.7 1.5 5.1 0.0 0.4 0.9

. . 1.1 2.0 5.3 -0.2 0.1 0.8

1

0.0 0.0 0.0 . . 0.3 2.2 6.1 -0.5 -0.1 -0.4 0.6f 4.8+ 14.3+ . . 0.4 1.7 5.5 -0.3 -0.6 -0.6 J

. . 1.2 0.6 -2.3 -0.6 -1.9 -9.4 -0.2 . . -1.2 0.6 -1.4 -0.2 -1.9 -8.8

. . 0.1 -0.5 -3.4 -0.8 -2.7 -10.4 . . 0.2 -0.2 -3.3 -0.6 -2.7 -11.0 J

1

. . 2.7 6.6 11.2 1.9 6.2 9.4 2.6 . 3.2 6.4 11.8 2.4 7.8 10.3

. . 0.2 6.1 10.2 1.9 5.9 8.8 . . 2.2 6.0 10.6 2.3 6.0 8.4

. . 1.9 5.0 10.4 1.0 4.2 8.8 1.8 . . 2.8 5.1 10.9 1.5 5.4 9.2

. . 1.3 4.3 9.4 1.2 4.2 8.2 . . 1.7 4.5 10.0 1.6 4.2 8.1

1

i

. . 2.4 6.4 12.3 1.6 6.3 12.6 2.6 . . 2.8 6.6 12.0 2.9 7.3 12.4

. . 1.8 5.6 11.8 2.2 6.8 12.7 . . 2.3 6.0 12.1 2.3 6.8 12.1

i

. . 0.1 4.3 8.7 0.5 3.9 8.6 . . 2.3 5.1 9.8 1.5 5.5 8.4

. . 1.4 3.9 8.5 1.3 4.4 8.5

. . 1.6 4.1 8.7 1.6 4.2 7.4

-2.2 -11.1

6.7 8.3

4.8 8.3

7.1 12.5

4.7 8.1

+ These results correspond to migration of male gametes only-i.e., the model of DICKINSON and ANTONOVICS (1973).

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724 H. DICKINSON A N D J. ANTONOVICS

The average change in gene frequency from generation to generation decreased significantly as the population size increased, but was not affected by other factors

(Table

4).

When there was a dense distribution of offspring ( N = ")

,

the mean genotype- environment correlation was close to the deterministic value; but when there was a sparse distribution of offspring ( N =

4)

,

the correlation was appreciably lower on the stochastic model than on the deterministic model (Table

5).

On both models, the correlation increased when there was intense selection and the when gene undergoing selection was dominant.

If

there is dominance, the parameter measured in a field situation is likely to be the correlation between the phenotype and the environment. The latter could be calculated using the same rank-correla-

tion methods of KENDALL (1948) and was generally of the same magnitude as the genotype-environment correlation although, under intense selection, it could be slightly greater.

The results (Table 6) show that on the deterministic model, the genotype fre- quencies were as predicted by the Hardy-Weinberg law. As may be deduced from MAYNARD SMITH'S (1966) model, this result holds for any selection pressures even if the niches support populations of different sizes, provided there is domi- nance and the two populations form a single panmictic unit. However, if there is not dominance, this result holds only in the particular case considered here, when the selection pressures in the two niches are equal and opposite and the popula- tions are of the same size. On the stochastic model, the genetic structure of the population remained, o n average, close to Hardy-Weinberg predictions, although there were marked oscillations about the mean irequencies. When there was no dominance and intense selection (S = 0.8), the random sampling effect due to a sparse distribution of offspring

( N

=

4)

tended to produce more heterozygotes than predicted deterministically.

b) Three-niche model: I n many respects, the behavior of parameters (i)-(v) is the same on this model as on the stochastic two-niche model, so only the points where the results differ are noted below.

The variance between replicates, u a r ( b ) , showed the same trends as before except that a slightly greater differential in both fitnesses and population sizes was required to produce a significant change in the variance. Under the more severe selection pattern bl, with intense selection ( S = 0.8), there was in general a significantly lower variance when there was a dense distribution of offspring

( N = ") and when more patches favored the heterozygotes [bl(l-2)]. The same trends were apparent for selection pattern b2 but were not significant. The variances resulting from selection patterns b l and b2 were not significantly dif- ferent.

The genotype-environment correlation was lower i f more patches favored the heterozygotes. The correlation resulting from the less severe selection pattern b2

was considerably lower than that resulting from selection pattern bl; it was also considerably lower than in the two-niche case.

(13)

EFFECTS OF PATCHY E N V I R O N M E N T S 725 (2-1 ) and (2-2), which gave overall heterozygous advantage, produced an excess of heterozygotes over Hardy-Weinberg expectations. The trend to either an excess or a deficit was magnified by intense selection. Under selection patterns bl there was a similar effect to that which arose with two niches and no domi- nance, in that with intense selection and a sparse distributtion of offspring there were more heterozygotes than predicted deterministically.

c) Six-niche model: As before, only the points where this model differs from

the two-niche model are noted.

There was not, in general, any significant differences between the variance of the regression coefficients, var(b), as the density of distribution of the offspring changed. Both this parameter and the mean change in gene frequency from generation to generation were of the same magnitude as in the two-niche and three-niche models. The genotype-environment correlation was appreciably lower in the six-niche model. The heterozygote frequency was close to that pre- dicted deterministically but it increased as the selection became more intense and was greater than that predicted by the Hardy-Weinberg law.

DISCUSSION

The generality of the model presented here makes it applicable to a wide variety of contexts. The term ‘patch’ does not necessarily correspond to an area of fixed size but may also refer to environmental resources which are capable of supporting one individual. Similarly, the stochastic variations in the selection process may be interpreted in several ways. Fluctuations in the selection pres- sures obviously may arise from variations in the biotic or physical environments: predation is likely to be a selective process with a high stochastic element; or the force of selection on a species may depend on its time of arrival and its state of maturity; or where plants successively replace each other in a habitat, the new arrival may occupy a habitat which is still partially occupied by its predecessor and selection on it may therefore not be identical, but o n l y similar to that on earlier generations established under perhaps slightly different circumstances. Alternatively, it may be assumed that the environment remains constant yet the expression of the alleles at a given locus is affected by the remainder of the genome.

(14)

726 H. DICKINSON A N D J. ANTONOVICS

different from the fate of a large population with the same genetic resources in an environment where there are two large, adjacent patches of contrasting types. Since it is inevitable, both in the field and in the laboratory, that selection will be influenced by random fluctuations and since the results show that this can cause a wide variation from generation to generation in the genotype frequencies, any deduction of relative fitnesses from gene o r genotype frequencies may be misleading unless observations have been made for several generations. D. L.

HARTL

(personal communication) has suggested that the best measure of aver- age fitness when fitness is variable over space is the arithmetic average and the appropriate measure when fitness is variable over time is the geometric mean of the fitnesses.

The results show that the mean genotype-environment Correlation increases as the intensity of selection increases but decreases if there is a sparse distribution of offspring. This emphasizes that in a field situation even the mean of the genotype-environment correlatoin over several generations may be a poor indi- cator of the intensity of selection, unless the number of offspring arriving in an area which can support one adult is also measured.

In spite of this it is clear from the results that in certain circumstances the genotype-environment correlation provides evidence about the action of selec- tion.

WATSON

(1973) showed that in Plantago Zanceolata in a heterogeneous sward there was a high genotype-environment correlation for certain genetically controlled characters (e.g. growth habit, leaf index)

,

while for others (e.g. length of scape and length of spike) there was no significar,t correlation. The present model shows that this could result if there were significant selection on the former characters but low selection on the latter. In this field situation the correlation was probably increased by localized pollen and seed dispersal, and by flowering time differences (assortative mating). The present theoretical results show that if the number of individuals arriving on a patch is large (i.e., there is a dense distribution of offspring) a significant genotype-environment correlation can readily result from selection for populations of 60 or more; such a correlation could be reinforced by localized gene dispersal o r by any degree of assortative mating. In the situation studied by WATSON (1372) only 2% of the seed popula- tion survived to the seedling stage, indicating that there was in this instance a relatively dense distribution of off spring ( N

>

50)

.

The effects of small population size in increasing the variance of gene fre- quency have been documented for isolated populations both in the absence of selection and with fluctuating selection pressures, for partially isolated popula- tions in which selection is balanced by migration and f o r clusters of populations which are not subject to selection but which experience migration between adja- cent groups (stepping-stone model) (see WRIGHT 1969, for review; also MARU-

YAMA 1971). The present model extends this work to include the situation where

the effects of multiple-niche selection are balanced by migration. When there was no selection our results agreed with the formula for the variance in gene frequen- cies

(WRIGHT

1969),

(15)

EFFECTS O F PATCHY ENVIRONMENTS 727 where q is the mean gene frequency,

n is the number of generations,

M

is the population size.

When the heterogeneous pattern of selection was acting, the variance was lower, showing that in such instances gene frequencies remained closer to their deter- ministic values and random fixation was less probable. This is in harmony with the work of WRIGHT (1931) on migration into a single population, which showed that there was a lower variance for higher levels of migration. CAVALLI-SFORZA (1969) has shown that, in certain human populations in isolated villages there is a greater divergence between small populations than between large popula- tions. He has argued from this that the genes are selectively neutral. The present results show that such phenomena could result if the genes involved were not neutral but subject to selective heterogeneity within each population.

Stochastic fluctuations can also have profound effects on laboratory experi- ments with small populations. Such fluctuations may affect the repeatability of an experiment (THODAY and GIBSON 1970)

,

may result in completely different outcomes on replicate populations (WRIGHT and KERR 1954; PARK 1954) and will produce a larger genetic variance between replicates the smaller the popu- lations (DOBZHANSKY and PAVLOVSKY 1957). In laboratory experiments on dis- ruptive selection in Drosophila, usually less than 40 individuals are selected from the offspring to produce the next generation. We have shown that for finite popu- lations subject to stochastic selection the divergence obtained between two select- ed lines may be appreciably lower than when selection is deterministic (Figure Ib,

IC,

and Table 5 ) . This is in agreement with the suggestion of ROBERTSON (1 971 ) that the degree of divergence which can be obtained between two repli- cate lines is limited by random sampling effects. It should be noted that in labora- tory experiments on Drosophila the phenotypic parameters are usually measured in a large sample of the offspring produced by the selected individuals, whereas the results presented in Figure 1 refer to the gene frequencies of the selected individuals themselves. In order to compare with the results of experimental designs such as those discussed by THODAY and GIBSON (1970) and ROBERTSON

(1917), the gene frequencies predicted by the model after selection and subse- quent random mating should be considered; these are approximately half-way between the values shown and 0.5, and so show even less divergence.

The other area explored in the present investigation is the extension of environ- mental heterogeneity to more than two niches c)r to situations more complex than clinal variation between two habitats. The results of the selection schemes for several niches emphasize both the limitations of the present model and the lack of data on selection in natural situations.

For the same fitnesses and the same number of offspring arriving in each patch, the genotype-environment correlation was generally lower if there were more niches since each genotype had more chance of being found in a niche which was less than ideal for it.

(16)

728 H. DICKINSON A N D J. ANTONOVICS

niches may vary considerably and this may have an important effect. For ex- ample, in the three-niche case there is no data to indicate whether the selection patterns b l (1-1) and b2 (2-I), in which the resources or environmental factors responsible for the selection pressures follow a rectangular distribution, are more realistic than selection patterns b l (1-2) and b2 (2-2) in which the environ- mental parameters are, approximately, normally distributed. WATSON (1 973) has shown that selection pressures on Plantago lanceolata are related to vege- tation height and that the vegetation height is, very approximately, normally distributed. However, the precise relationship between selection pressure and vegetation height is not known, so any precise conclusion about the distribution

of selection pressures is not possible. Indeed there are few estimates even of the magnitude of selection pressures in natural situations (see ANTONOVICS 1971 for a brief review) and present work is directed at refining earlier estimates and deducing ecological factors which account for selection (e.g., MCNEILLY, personal communication) ; the estimation of the spatial variation of selection pressures is obviously a more arduous task but is an area deserving of considerably more attention.

The second difficulty arises from our lack of knowledge of the genetic basis for adaptation. Are heterozygotes generally adapted to intermediate environments or, as argued by PARSONS (1971), are they more often adapted to extreme envir- onments? If the former is the case and the environments are normally distributed, there will invariably be overall heterozygous advantage [e.g., Table 2, b l (1-2)

and b2 (2-2)

3,

and polymorphism will be readily maintained. Indeed this can be true even i f the environments have a rectangular distribution (e.g., Table 2, c). Heterogeneous environments are therefore likely to be a cause of overall heterozygous advantage, even though in any given niche there may be no such advantage. The application of the models here to any particular situation would require a n estimation both of how the selection pressures vary with the environ- ment and of the size of population which can be supported by each environmental category.

(17)

EFFECTS O F PATCHY ENVIRONMENTS 729

distribution of the off spring. Many of the studies of stochastic processes in evolu- tion have been on genes maintained in populations because of their neutrality; the present study shows how stochastic processes may affect genes maintained in a multiple-niche environment by selective forces and this clearly would warrant further theoretical treatment. However, detailed comparison of evolution under two-, three- and multiple-niche situations is hampered by the multitude of possi- ble combinations of relative fitnesses, and by the multitude of possible assump- tions regarding both niche size and the genetic basis for adaptation. The applica- tion of the model to a particular example will be attempted in a later paper (DICKINSON and

WATSON,

in preparalion) but in general our lack of knowledge of the above factors precludes generalizations about natural situations.

This work was supported by a grant from the Science Research Council, U.K.

APPENDIX

Deterministic models: In order to delineate the effects of stochastic selection and finite popu- lation size the various stochastic models were compared with the corresponding deterministic mgdels. The deterministic models assumed that the population was infinite, there was a dense distribution of offspring over the patches, and selection pressures remained constant. Otherwise the assumptions were as in the stochastic models: the population size was regulated independently in the separate niches, the entire population mated at random, the offspring dispersed at random over the niches, and then selection acted.

(a) Two niches

a1 Dominnnce: The gene frequencies at equilibrium may be derived from MAYNARD SMITH’S (1966) model of a two-niche environment with no habitat selectiqn by females. Noting that we are concerned only with the case where the selection pressures are equal and opposite, MAYNARD

SMITH’S equation (4) yields the average gene frequency,

where p8, pv are the frequencies of allele A in niches X and Y respectively.

MAYNARD SMITH’S equation (2) (with H = 0 i n his notation) then yields

S(1- 42/91

% ( P , - PJ = -

where S is as in Table 2.

From equations ( A . l ) and (A.2),

Using the fitnesses given in Table 2, the genotype frequencies in each niche after selection are as i n Table A l . Hence the rank correlation coefficient (KENDALL 1948) for the genotype- environment cwrelation is

where

I

.={

-}{----

1

(18)

TABLE AI

x

Equilibrium genotype frequencies in each nick after selection (selection pattern ai) Genotype Niche AA Aa aa Total

g

5

5

- 2P (1-P) (1-P) 2 (IS) 13 ( 1-P) 2 14 ( I-P) 2 IS(l-P)2 1

--

P2 X U Y PZ(1-S) 1s (2P-PT 2P(l-P) (IS) 14 (2P-P ) (1-P) 2 I--s (2P-P )

-~

4 I

5

Y

[ 2--2S+S2 ( 1 -P)

'1

2P ( 1-9) [I-S(l-P)2] [ 1-S(2P--P2) 1 [%-2S+S~P(LP)] (1-P)Z [IS( 1-P) 21 [ IS(2P-P2)] 2

s

6

(19)

EFFECTS OF PATCHY ENVIRONMENTS

x {P2[2P(l - - ) ( I - S )

+

(1 --P)*] + 2 P ( 1 --P) [-+"I - S )

+

(1 --P)*]

- (1

-

P ) Z ( l - S ) [PZ(l- S )

+

2 P ( 1 - P) (1

-

S ) ] ) D, = 4{l/e(22- 1 2 - 12)} = 1

73 1

and

where

T I = [ 2 - 2 S + S ~ ( I - P ) 2 ] P 2

T , = [2-2S +S2(1 -P)2]2P(I --P) T , = [2

-

2s

+

S2P (2

-

P) ] (1

-

P) 2

After some algebra and the use of equation (A.l) this reduces to

S S

7 =

___-

= 0.93571

-

(A.4)

(2

-

S) p' (- 13

+

10 4 2 ) 2 - s .

The Hardy-Weinberg law predicts the frequency of heterozygotes in the entire population to be

2P(1 -P).

[2 - 2s

+

S2( 1 - P)2]2P(l- P)

2[1

-

s

( 1

-

P)2] [ 1

-

s

(2P - Pa)]

2 P ( l - P ) From Table A.l, the actual frequency of heterozygotes is

and, using equation ( A. l ) , this reduces to

showing that the percentage excess of heterozygotes is zero.

a2 N o dominance: This model and all subsequent deterministic models are simplified by noting that, from consideration of symmetry, the equilibrium gene frequency in the population as a whole is 0.5. Since all the organisms mate at random and the relative Gtnesses are as in Table 2 (a2), the genotype frequencies after selection are as in Table A2. Hence the equilibrium gene frequencies in the separate niches are

4 - 3 s

''=

4(2-S) and

Using the same methods as above, the genotype-environment correlation may be deduced from the contingency Table A2, and hence

3s S

7 = = 0.67082 --

2 p'5(2-S) 2 - s

TABLE A2

Equilibrium genotype frequencies i n each niche after selection (selection pattern a2)

Niche

X

Y

Total

1

(20)
(21)

EFFECTS O F PATCHY ENVIRONMENTS 733

Since the overall gene frequency is 0.5, the Hardy-Weinberg law predicts the overall fre- quency of heterozygotes as 0.5. From Table A2, the actual frequency of heterozygotes is 0.5, and so the percentage excess of heterozygotes is zero.

(b) Three niches

For the sake of generality It was assumed that the sizes of the sub-populations supported by the three niches were in the ratio 1: z: 1 although when comparing with the stochastic models only the values z = 1 [c.f. Table 2, b l (1.1) and b2 (2.1)] and x = 2 [c.f. Table 2, bl (1.2) and b2 (2.2)] were used.

Since the genotype-environment correlation and the percentage excess heterozygosity are deduced for the remaining models in the same way as above (see section a2), only the contin- gency tables and the final results are presented.

( b l ) Contingency table-See Table A3. Genotype-environment correlation:

__

W[2(3 - S) (4 - 3 s )

+

(6

-

55') (2

-

S ]

r =

(4-33s)

4

{ [ 2 ( 2 - S ) 2

+

2(1 - S ) (4-33S)i

x [2(2

-

S ) (10

-

9s) f ~ ( 4 - 3s) (5- S)] [I 2x1) 1 ooS[ (4 - 35') z -2 (2

-

S )

1

(2

-

S) (4

-

3 s ) (2

+

5)

Percentage excess heterozygosity =

(b2) Contingency table-See Table A4. Genotype environment correlation:

2-5'2(6--S) +35'(4-5') T =

2(2

-

S)

4

{ [5 (4 - S)

+

z( 101

-

S)] [4 - S

+

2 ( 2 - S )

1

Cl

,+

221 I

loos2

(4 --SI (2

+

5)

Percentage excess heterozygosity =

(c) Si2 niches

Contingency table-see Table A5. Genotype-environment correlation:

S(1474560 - 1515520s

+

560768P - 88176S3

+

4971S41 r =

2(2 - S ) (16

-

5 s ) (16

-

3s)

x ~{15[768-44&3+ 63Sz][3840- 172883- 171S21J

200s (32

-

9s)

3( 16

-

5s) (16

-

3s) Percentage excess heterozygosity =

-

TABLE A4

Equilibrium genotype frequencies in each niche after selection (selection pattern b2)

Genotype

Niche A A An aa Total

X 1 2-s I-s 1

2 (2-9

--

Z(2-S) 2(-)

Y

z

X

(22)

734 H. DICKINSON A N D J. ANTONOVICS

TABLE A5

Equilibrium genotype frequencies in each niche after selection (selection pattern c )

Genotype

Niche AA Aa aa Total

4

1 6 5 s

4-S 16-3s

4-3s 1 6 5 s

1 4

6

W2-S)

768-448S+63S2 Total

2(16-38) ( 1 6 5 s )

1

2 -

1 6 5 s

8

16-3s

8

1 6 3 s

1 2

-

768-32OS+27P (1 6-3s) ( 1 6-5s)

1 4-3s

16-5s

--

16-3s

4

16-5s

--

1

1

76&-448s+63s2

( 16-3s) ( 16-5s) 6

LITERATURE CITED

ANTONOVICS, J., 1971 The effects of a heterogeneous environment on the genetics of natural populations. American Scientist. 59 : 593-599.

VIII. Clinal patterns at a mine boundary. Heredity 25: 349-362.

BRADSHAW, A. D., 1965 Genet. 1 3: 115-155.

CAVALLI-SFORZA, L. L., 1969

DICKINSON, H. and J. ANTONOVICS, 1973

DOBZHANSKY, T. and 0. PAVLOVSKY, 1957

KENDALL, M. G., 1938

LEVENE, H., 1953

LEVINS, R., 1964

ANTONOVICS, J. and A. D. BRADSHAW, 1970 Evolution in closely adjacent plant populations.

Evolutionary significance of phenotypic plasticity in plants. Advan.

‘Genetic drift’ in an Italian population. Scientific American 21 :

Theoretical considerations of sympatric divergence.

An experimental study of interaction between genetic 30-37.

Am. Naturalist 107: 256274.

drift and natural selection. Evolution 11: 311-319.

Rank mrrelation methods. Third edition. Griffin & Co., London. Genetic equilibrium when more than one ecological niche is available. Am. Naturalist 87: 331-333.

(23)

EFFECTS O F PATCHY ENVIRONMENTS 735

MARUYAMA, T., 1971 The rate of decrease of heterozygosity in a population occupying a circu-

MAYNARD SMITH, J., 1966 Sympatric speciation. Am. Naturalist 100: 637-650. -, 1970

---,

PARK, T., 1954. Experimental studies of interspecies competition 11. Temperature, humidity,

PARSONS, P. A., 1971 Extreme-environment heterosis and genetic loads. Heredity 26 : 479-482. ROBERTSON, A., 1971 A note on disruptive selection experiments in Drosophila. Am. Naturalist

104: 561-569.

SNAYDON, R. W., 1970 Rapid population differentiation in a mosaic environment. I. The re- sponse of Anthoxanthum odoratum populations to soils. Evolution 24: 257-269.

SNEDECOR, G. W., 1956 Stastical methods. Fifth edition. Iowa State University Press.

THODAY, J. M. and J. B. GIBSON, 1970 The probability of isolation by disruptive selection. Am.

WATSON, J., 1973 Evolution in a heterogeneous environment. Ph.D. thesis, Stirling University. WRIGHT, S., 1931 Evolution in Mendelian populations. Genetics 16: 97-159. -

,

1969

Evolution and the genetics of populations. Volume 11. The theory of gene frequencies. Uni- versity of Chicago Press, Chicago and London.

Experimental studies of distribution of gene frequencies in very small populations of Drosophila melanogaster 11. Bar. Evolution 8: 225-240.

lar or linear habitat. Genetics 67: 437-4454.

Genetic polymorphism in a varied environment. Am. Naturalist 1M: 487-490. 1971What use is sex? J. Theoret. Biol. 30: 319-335.

and competition in two species of Tribolium. Physiol. Zool. 27: 177-238.

Naturalist 104: 219-230.

Figure

TABLE 1
FIGURE 1.-Results
TABLE 3
TABLE 4
+6

References

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