Assortative mating and gene flow in the Lesser Snow Goose: A modelling approach

THEORETICAL

POPULATION

BIOLOGY

22, 177-203 (1982)

Assortative Mating and Gene Flow in the Lesser Snow Goose: A Modelling Approach JOAN M. GERAMITA Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

FRED COOKE Deparfment of Biology, Queen’s University,

Kingston, Ontario K7L 3N6. Canada

AND ROBERT F. ROCKWELL Department of Biology, City College of New York, New York, New York

Received February 12, 1980; revised February 26. 1982

Nonrandom mating with respect to color is well documented in the dimorphic Lesser Snow Goose. A set of mathematical models and data from the long-term study at La Perouse Bay, Manitoba, Canada are used to examine various hypotheses advanced to explain the mechanisms behind the assortative mating. Among the factors considered are mate choice based on familial color, accidental formation of genetically unrelated families, and nonuniform distribution of colors in the region where mate selection occurs.

INTRODUCTION

The two colour phases, blue and white, of the dimorphic Lesser Snow Goose are nonrandomly distributed both in space and time. Dzubin et al. (1975) and Cooke et al. (1975) show that they are distributed clinally in both breeding area and wintering grounds. Coach (1961) has reported changes with time in the distribution of the phases. Various explanations have been advanced concerning this nonrandom distribution, in space and time. Coach (1961, 1963) felt that the distribution of the morphs could be explained in terms of differential selection pressures with blue being favoured by warmer temperatures, white by cooler. Hunting pressure was also thought to favour blue-phase birds. However, explanations based on differential selection pressures seemed less plausible when gene flow among the various 177 0040.5809/82/050177-27$02.00/O Copyright c 1982 by Academic Press. Inc All rights of reproductmn m any form reserved.

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parts of the population was shown to be much greater than hitherto thought (Cooke et al., 1975). Given the massive gene flow, it seems at first sight surprising that any differences in phase ratio in different parts of the range should still remain, Cooke et al. (1976), however, have shown that geese generally choose a mate of the same colour phase as that of the family in which they are raised. Rockwell and Cooke (1977) point out that this assortative mating based on familial colour would retard any equilibration of the phase ratios; and so different ratios in different parts of the range could still be expected despite the high level of gene flow. In this paper we develop a model which demonstrates the compatibility of the mating rule proposed by Cooke et a/. (1976) and the field data on changes in phase ratio and other observable phenomena. We use the model to assess the relative influences of various factors in the breeding biology of the geese on these changes, and show that an obvious less complicated rule is not compatible with field observations. The first attempts to simulate the changes in phase ratios in various populations of snow geese were by Seiger and Dixon (1970). They attempted to predict the evolutionary progression of the population on the basis of different levels of imprinting and dispersion. Since many facets of the breeding biology of snow geese were unknown at that time, their parameters were chosen arbitrarily. The conclusion that varying the parameters of coefficient of imprinting, dispersion, and initial population size and phase ratio could produce qualitatively different results emphasized the necessity of acquiring more information about the life history of the geese and more realistic estimates for such population parameters as average clutch size. This has been done and it is now possible to construct models which reflect the real world more faithfully. This paper is organized in six sections. Section I consists of a history of the changing phase distribution in space and time. It also includes a natural history description of the breeding and brood-rearing behaviour of the geese which forms the basis for the model. Section II begins with a discussion of the simplifying assumptions which were made when the model was constructed. It proceeds with a short and totally nontechnical description of the model. A more detailed description follows. Specific functions appear in the Appendix. The second description should provide a clear view of (1) the population information needed to construct the model, (2) the type of data needed to make the model reflect a specific breeding colony, and (3) the reasons for some of the simplifying assumptions. The assignment of values to the constants in the model, the selection of initial state vectors, and the robustness of the model are described in Section III. Section IV gives the results of the simulations and a discussion of some of the immediate implications of these results. Section V is the discussion and Section VI a

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summary. The Appendix contains mentioned in the main body.

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specific

functions

179 and derivations

I Phase Distribution

in Space and Time

The Snow Goose (Anser caerulescens) consists of two races, the Greater Snow Goose (A. caerulescens atlantica) and the Lesser Snow Goose (A. c. caerulescens). The Greater Snow Goose breeds in the high Eastern Arctic, primarily Northern Baffin, Bathurst, and Ellesmere Islands and winters on the Atlantic Coast of the United States. Since pair formation occurs on the wintering grounds or during spring migration, there appears to be little gene flow between the Greater Snow Goose and the populations of the Lesser Snow Goose. Blue-phase birds are for the most part unknown in the Greater Snow Goose population, though examples do exist. To the east and south of the breeding range of the Greater Snow Goose, the Foxe Basin and Hudson Bay populations of the Lesser Snow Goose (hereafter referred to as the Hudson Bay population) can be found. These populations are the ones considered in this paper and they winter predominantly near the Gulf coast in Texas and Louisiana. Both colour phases occur, with the western breeding colonies having around 25% blue phase and the eastern and more northerly colonies having up to 90% blue phase. A summary of the phase ratios in the colonies of this population for the years 1968-1972 is given by Dzubin et al. (1975). In the wintering grounds too, white-phase birds predominate in the western part of the range, blue-phase birds in the east. Within individual nesting colonies, the blue-phase birds have a more easterly migration route and wintering distribution than white-phase birds (Lemieux and Heyland, 1967). Birds from all breeding colonies overlap on these wintering grounds (Cooke et al., 1975). Further west are the colonies of Wrangel Island in the USSR, Banks Island in Northern Canada, and several small colonies on the shore of the Beaufort sea. These birds are almost all white phase and winter primarily in California and British Columbia. They overlap with the Hudson Bay population at no stage in their life history and therefore there is little opportunity for gene flow between the two populations. The status of the colonies in the central Arctic (around Perry River) is more problematical. Five to ten percent of these birds are blue phase and some winter along the Gulf Coast. There is presumably some gene exchange between this population and the Hudson Bay population, but as the latter population is so much greater, the effect of this exchange on the Hudson Bay population is presumed to be minimal. The temporal change in phase distribution is considered only for the Hudson Bay population. Early records suggest that blue-phase birds were

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once more separated from white-phase birds than at present. Graham (1769) states At Prince of Wales, York and Severn settlements are killed yearly above ten thousand of the geese 1white phase] only. The blue geese are the size, shape and make of the white geese and are as numerous at Albany, Moose and East-main settlements as the white geese are at Prince of Wales, York and Severn; but they have very few white geese at the southern settlements and as few blue at the northern, there not being above 20 or 30 blue geese killed at Prince of Wales Fort etc. a season. At Albany Fort only, are killed above ten thousand blue geese in a season.

Hearne (1795), however, records occasional mixed flocks, with white birds predominating, flying over Churchill, Manitoba. In the Gulf Coast winter grounds too, Lynch (personal communication) notes that the blue phase at the turn of the century used to be more isolated from the white-phase birds, the former wintering in the Mississippi Delta, the latter along the Texas coast. Human disturbance, marsh management, and the development of rice fields which provide an attractive alternative food source for the geese have all had the effect of increasing the wintering ground contact of the two morphs. Coach (1961) provided evidence that blue-phase birds were increasing at all the Hudson Bay colonies where data were available with the exception of the huge Bowman Bay colony which is predominantly blue phase. He interpreted this increase in blue as being due to selection pressures favouring the blue phase and he predicted that “if present climatic conditions and hunting pressure continue, then most Hudson Bay populations can be expected to exceed 75% blue phase birds by 1980.” The raw data with which Coach worked were in some cases less than ideal since they were based on visual estimates from a fixed-wing airplane. It seems clear from Coach’s data that the predominantly white-phase colonies were increasing in the frequency of blue birds, but whether the predominantly blue-phase colonies were also doing so is less certain. Since 1961 there have been strong indications of increase in blue phase among nesting pairs at the McConnell River Colony (Kerbes, 1975) and at the La Perouse Bay colony (Cooke, 1978), both colonies of Western Hudson Bay. The rate of increase is not as great as predicted by Coach. Hanson et al. (1972) claimed no change in phase ratios at the McConnell River colony and at the La Perouse Bay colony and an increase in the blue phase of the predominantly blue-phase Cape Henrietta Maria colony. Because of biases inherent in the method of data collection (Boag, 1974) their conclusions must be considered suspect, and recent reports (Lunsden, personal communication) show that blue-phase birds are decreasing in frequency at Cape Henrietta Maria. At the three colonies on Baffin Island, Kerbes (1975) found from aerial photos of nesting pairs that the blue phase had decreased from Coach’s (1961) counts in 1955. It appears then rather than a transient polymorphism

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with blue slowly increasing, the post-1961 data suggest an increase in white phase in the predominantly blue eastern colonies and an increase in blue at the predominantly white western colonies. A slow equilibration of phase ratios with time is occurring. This might suggest that gene flow rather than selection pressure differences account for the changes in phase ratio at the various Hudson Bay Snow Goose colonies. We present a model which shows that a fairly simple rule for mate selection produces results which are consistent with the known behaviour of certain phenotypic variables of the goose population at the colony at La Perouse Bay; i.e., percentage of white birds among the breeders, percentage of mixed pairs among the breeding pairs, and percentage of mixed pairs with a blue female. The model also shows that the gene flow caused by following this rule is sufficient to explain the change in phase ratios in the various colonies without the additional hypothesis of differential selection on the morphs. The assumption of no selection pressure differences between the colour phases is consistent with 10 years’ data collected at La Perouse Bay colony (Smith and Cooke, in preparation) though not necessarily true of other Hudson Bay colonies where no data are available. Data on the breeding biology of the geese have also been collected principally at La Perouse Bay and should be extrapolated to other colonies with caution. Breeding Biology of Snow Geese For a complete understanding of the model some basic parameters of snow goose biology need to be emphasised. Geese arrive at the breeding colony already paired, and nesting begins as soon as nest sites become available. Pairs establish nest sites and lay clutches of 3 to 6 eggs. Occasionally birds lay eggs in the nests of other members of their own species and thus a gosling hatched from such an event will not be genetically related to the parents of that nest (Finney, 1975). This intraspecific nest parasitism (ISNP) is often referred to as dumping. It rarely exceeds 10% of all eggs laid. Another phenomenon which influences the genetic composition of the brood is extra-pair bond copulation, where a male from an adjacent nest copulates with a sitting female (Mineau, 1978). If this occurs prior to completion of the clutch, the male of the pair bond may not be the genetical father of all the eggs. Incubation lasts on average 23 days and on hatching goslings are nidifugous, remaining in the nest no more than 24 hr. They then accompany their parents to the salt marshes nearby where they feed and grow rapidly. There is the possibility of some goslings becoming attached to other families during the first days of hatch, but most families remain intact as evidenced by recapture of marked families 4-5 weeks after hatch. Families which have acquired goslings who are not the offspring of both members of the attending pair, no matter how it happened, will be called modified families.

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Soon after the goslings have fledged, the families leave the breeding grounds and, as family groups, migrate south, often as far as Louisiana and Texas. The families remain intact and survivors usually return to the same colony the following spring. If the female loses her mate she usually remates bringing her new mate to her natal colony. A male losing his mate does not usually return to his natal colony but follows his new mate to her breeding colony. The young birds do not nest in their yearling summer, but feed at the periphery of the nesting area. They usually pair in their second winter or during the migration when birds from different nesting colonies are associating. Thus birds forming pair bonds are likely to be from different nesting colonies. The pair thus formed returns to the natal colony of the female. (Cooke et al., 1975). Most 2-year-old birds returning to the natal colony are already paired. Not all birds breed as 2-year-olds; some do not nest until their fourth summer. Annual adult mortality is 15-25 % and first-year mortality is in excess of 50%. The commonest age classes in a typical snow goose colony are 4- and S-year-olds (Cooke, unpublished). II Several assumptions have been made to simplify the construction of this model. One is that colour is controlled by one gene locus with blue completely dominant over white so that homozygous and heterozygous blues are indistinguisable. In view of the considerable variation observed among blue birds, this may be an oversimplification. Cooke and Mirsky (1972), however, indicate that it is a good approximation to the truth and Rattray (198 1) supports this assumption over several other likely choices. We have constructed a model which does not overlap the generations as breeders. This means that trends which show up in the model would develop more slowly in an actual goose colony. We have assumed that there is no female immigration since this seems to be the rule at La Perouse Bay in spite of the exception in 1978 documented by Geramita and Cooke (1982). We have assumed no change over several generations in the proportions of the genotypes present on the wintering ground taken as a whole. Since this model is meant to consider short-range effects, this assumption is the correct one to make. Any change in the proportions on the wintering grounds would be very slow due to the large proportion of birds from prediminantly onecolour colonies. Short Description of the Model The input (initial state vector) needed for the model is a description of the parental generation in terms of the proportions of the various pairs of

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genotypes. Using straight Mendelian genetics (blue completely dominant over white) each pair type produces a family whose colour is known. Some of these are altered as described in Section I. These families will be referred to as modified families. This information allows the goslings to be classified jointly by genotype and the colour composition of the family they were raised in. The mating rule being modelled follows the findings of Cooke (1978) and says: a gosling generally chooses a mate the colour of the family it was raised in; if raised in a two-colour family it mates at random. Specifying what is meant by “generally” is at the heart of the model. In Section IV this is made precise and the implications of various specifications are examined. For the population on the wintering ground it is necessary to specify how the colours and preferences of potential mates are distributed and also how colony members are distributed. After mates have been selected (a matter of calculating probabilities) the phenotypic variables are extracted and a new state vector constructed to allow the process to be repeated for the next generation. Description of the Model Mated pairs of geese can be divided into six disjoint (i.e., no pair goes into more than one class) classes of mated pair types. We denote the dominant homozygous blue bird by BB, the heterozygous blue by Bb, and the homozygous white by bb. The six pair types are given in Table I. We shall habitually denote the distribution of such classes in a colony by the vector x = (x, , x2, xj, x4, x5, x6) where xi is the proportion of parent pairs of type i and the xi sum to one. The vector x is the initial state vector mentioned before. In Section III, we discuss methods of estimating x for a specific colony. The distribution of mated pairs for the total population, everyone on the wintering ground, is denoted by y = (v, , y2, 1~~)y,, JJ~,y,). The offspring of the pairs with distribution x can be divided into the three genotypes with the proportions predicted by Mendelian inheritance given in TABLE

I

Classes of Mated Pairs Class number

Genotypes of pair

1 2 3 4 5 6

BB-BB BB-Bb BB-bb Bb-Bb Bb-bb b&bb

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Table II. We denote the distribution of genotypes given in Table II by z=(z,,z*,z& According to the mating rule we are modelling, birds can have one of three preferences. A bird can insist on a blue mate, insist on a white mate, or be indifferent to mate colour. Since a bird can be one of three genotypes, there are nine possible preference-genotype combinations. We shall see that some of these are so unlikely that it is a reasonable simplification to eliminate them. Let us consider how the various combinations may arise. There are two kinds of families: those which contain only the offspring of the attendant pair and those which we refer to as modified. The families of pair types 1 or 2 are all blue and so produce goslings that insist on blue mates (blue insisters). The families of pair types 3 and 5 are mixed and so the young are indifferent to colour. Families of pair type 6 are all white and so produce birds that insist on white mates (white insisters). We notice that so far all blue insisters have been blue themselves and all white insisters have been white. Now consider a family with parents of pair type 4, two heterozygous blues. Some of these will be all blue (producing blue insisters) and some will be mixed (producing indifferents). The proportion of each type of family can be calculated if the distribution of clutch sizes for the colony is known. Now consider those mixed families with a single white gosling. If the gosling has siblings they will be indifferent, but the white gosling itself ought to be a blue insister since it only sees blue parents and siblings. Similar situations can arise when we consider families which have been modilied. Because these goslings form such a small proportion of the young of the year, we simplify the model and do not consider these two preferencegenotype possibilities: a white insister which is itself of genotype BB or Bb (blue); or a blue insister which is itself white. All the goslings in a mixed family are considered to be indifferent. We are left with six disjoint classes of preference-genotype classes (from now on we shah call them preference classes; see Table III). To specify the word “generally” in the mating rule we use the parameter t which is a number between 0 and 1 and represents the proportion of insisters TABLE Genotypic

II

Classes of Offspring

Class number

Genotype

Proportion

1 2 3

BB Bb bb

xl + 0.5x, + 0.25x, 0.5x, + X) + 0.5x, + 0.5x, 0.25~~ + 0.5x, + x6

ASSORTATIVE

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TABLE

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III

Preference Classes Class number

Preference

Genotype

Colour

I

Blue insister Blue insister Indifferent Indifferent Indifferent White insister

BB Bb BB Bb bb bb

Blue Blue Blue Blue White White

2 3 4 5 6

who choose the other colour. As is explained in Section IV there is an alternate parameter r which can be used to define generally. It too is between 0 and 1 and represents the proportion of insisters who mate at random. Next we must establish where on the wintering grounds these goslings go and what colour and preference type potential mates they find when they get there. The model assumes three regions in the area where pair formation occurs; one where blue birds predominate, one where white are much more frequent, and a region with a more even mixture of colours. These areas are to some extent artificial constructs designed to reflect the existence of three fairly distinct wintering area step clines (Cooke et al., 1975). The specific functions used to calculate the distribution of goslings among the three regions and the composition of the regions are given in the Appendix (Part 1). In Section III the data needed for these functions are discussed. We point out here that it is necessary to calculate preference types for the whole population which uses the wintering area and this is done in a way consistent with the analogous calculation for a colony. It remains to use the pieces we have described to calculate the probability that a female gosling of a certain genotype from the summer colony will mate with a male of a certain genotype. From this distribution we can extract probabilities of the three phenotypic variables about which there are data: The percent of white birds among the breeding pairs, the percent of mixed pairs among the breeding pairs, and the percent of mixed pairs with a blue female.

III Before the model described in Section II can be used we have to specify values for the various constants which appear in the several functions. We give these values in the Appendix (Part 2). Here, we give a description of the data on which we based the choice of those constants. The information

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encoded in the constants represents another set of assumptions concerning the model. Since the model is not sensitive to small changes in these constants, however, the “assumptions” made in choosing the specific values we did are unlikely to introduce enough error to lead us to wrong con elusions. With respect to the total population of geese breeding in the Hudson Bay area, we have assumed that approximately 40% of these are blue phase based on breeding grounds counts by Kerbes (1975) and an approximate 5 to 3 homozygous blue to heterozygous blue ratio based on extrapolations from the assortative mating patterns at La Perouse Bay. The genotypic distribution of the total population of geese was set at 0.25 RB, 0.15 Bb, and 0.60 bb. The distribution of colours on the wintering ground was determined using Table 2 in Cooke et al. (1975) which was updated with information from the Bowman Bay colony. White, mixed, and blue regions were designated by longitude. These same regions were then used with distribution data from La Pirouse Bay to specify the dispersal of birds from a given colony among the regions. Table IV indicates the relative composition of each region. It was necessary to compute the proportion of type-4 families (two heterozygous parents) with at least one white gosling. This was done using a clutch-size distribution recorded at La Perouse Bay in 1975 (Cooke, unpublished) and the probability of a white gosling from these parents as 0.25. The proportion calculated was rounded to 0.7. The dump rate (proportion of nests affected by the inclusion of a gosling not the offspring of the attending pair) was set at 0.23. This value was based on the experience at La Perouse Bay. The effect of varying this rate is discussed in Section IV. We look now at the input vector x. Since we expect our model to mirror the trends in the real world, it is important that the entries in x be computable from field data. The sum x, + x2 + x, is the proportion of blue pairs in the colony; x3 + xg the proportion of mixed pairs; and x6 the proportion of white pairs. Data on these proportions are usually available. If one knows the proportion of mixed pairs with white offspring then it is TABLE Wintering

IV Regions

Region

Longitude

All birds wintering there W)

I Blue 2 Mixed 3 White

85-92 93-94 95-99

20 40 40

Birds there that are blue (o/o) 80 40 20

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possible to estimate x3 and x5 separately (Appendix Part 3). Similarly, if the proportion of blue pairs with white offspring is known then it is possible to estimate xq (Appendix, Part 3). This leaves x, + x, to be assigned. An opinion about the relative frequency of homozygous blue to heterozygous blue would allow one to assign values to X, and x2. In fact, the functions in the model usually combine these terms into expressions such as x, +x2 or x, + 0.5x, + 0.25x, so that some of the errors made in assigning values will be cancelled. In addition, since small changes in the entries of x cause only small changes in the output of the model, the trends indicated by the model will be unaffected by small errors in the estimates. We have avoided using vectors x where x, +x2 + x, = 0 (no blue-blue pairs) or xg = 0 (no white pairs). As the model is constructed it cannot accommodate these cases. Since they are not situations which occur, we were not interested in them. We have considered three typical colonies. The predominantly blue one corresponds roughly to Bowman Bay, Baffin Island; the mixed one to Cape Henrietta Maria, Ontario; and the white one as closely as possible to La Perouse Bay, Manitoba. Blue, mixed, and white values were specified for all colony-related constants. These are given in the Appendix. The three colonies were specified by the initial state vectors blue

x = (0.45, 0.22, 0.04, 0.08, 0.06, 0.15)

mixed

x = (0.20,0.10,0.04,0.08,0.20,

0.38)

white

x = (0.04, 0.06, 0.03, 0.06, 0.11, 0.70).

In each case we have set 2x, =x,. This is not accidental. This equation arises early in the construction of the model and offers some insight into possible mechanisms controlling the colour ratio we see in the tield. From an initial vector x representing the genotypes of mated pairs we can calculate the proportions of the three genotypes among the parents and among the offspring (assuming of course our one-gene Mendelian model; see Table V). The equation 2x, = xq represents a state of equilibrium in the proportions TABLE Proportions

Parents Offspring Change from parents to offspring

V

of Each Genotype

BB

Bb

bb

x, + 0.5x, + 0.5x, x, +0.5x, + 0.25x,

0.5x, + x, + 0.5x, 0.5x, + xj + 0.5x, + 0.5x,

0.5x, + 0.5x, + x, 0.25x, + 0.5x, + x,

0.25x, - 0.5x,

x3 - 0.5x,

0.25x, - 0.5x,

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of genotypes from generation to generation. (We note here that we are not discussing gene frequency which in the absence of selective pressure we assume constant.) This situation always occurs when there is random mating and is the essence of the Hardy-Weinberg law. In the context of our vector x this equation says that there is no change in the percentage of white birds from one generation to the next when the proportion of pairs composed of two heterozygous blues is exactly twice the proportion of pairs with one white and one homozygous blue. If, however, due to nonrandom mating and nonrandom return to the breeding areas, x, # 2x, then the colour composition of the colony can change from year to year without any other factors being involved. For example, if xq is greater than 2x, then there will be more homozygous goslings than there were parents. Reference to Table V shows that this means more white goslings than parents. At the predominantly white La Perouse Bay colony the evidence is that type-3 pairs (B&-M) are relatively rare; certainly they are less than half the number of type-4 pairs. This indicates that the observed increase in blue birds at that colony is due to blue male immigration rather than blue gosling increase. This observation explains why we look at the proportion of mixed pairs with a blue female. If this is less than 0.5 we know there are more blue males in that cohort of pairs than there are blue females. Since we assume a l-l sex ratio among goslings this means more blue males returned as mates than were generated as goslings.

IV We constructed the model described in Section II to determine whether the behaviour of the geese in their mate selection (as verbalized in the mating rule) is sufficient to explain the levels of assortative mating and the changes in colour composition observed within the Hudson Bay population. Our original plan was to show that the rule “geese always choose mates according to the colour of their family” was sufficient to maintain the levels of mixed pairs observed at La Perouse Bay when the nonuniform distribution of colours in the wintering ground and the nongenetic modification of families were incorporated into the model. This is equivalent to setting t = 0 and r = 0 in the present model. Table VI gives the data collected at La Perouse Bay from 1969 to 1977. If we assume roughly that one generation takes about 4 years we can use these data to evaluate the results of the simulations. Table VI shows a steady decrease in the percentage of white birds, a slightly increasing percentage of mixed pairs, and an increase in the percentage of mixed pairs with a blue female with that percentage remaining less than 50%. This last reflects the fact that at La Perouse Bay it appears

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La Pkrouse Bay Data

mixed pairs (‘Xl)

Mixed pairs with blue female (?,I )

Generation

Year

White (‘“I)

0

1969 1970 1971 1972

71 75 76 74

15 16 15 17

33 38 33 41

I

1973 1974 1975 1976

73 72 72 72

16 16 16 18

38 38 44 44

2

1977

71

18

44

that blue increase is accomplished through male immigration rather than increase in blue among goslings. The graphs (with t = 0) in Fig. 1 show that our original plan could not be successful. We did discover that the inclusion of the nonuniform distribution on the wintering ground and the nongenetic family modification in the model yielded a system which always had some mixed pairs (l-3%). However, even allowing the proportion of families affected by various forms of nongenetic modification to approach 1 was insufftcient to raise the mixed pair level to more than 10%. We had been aware of cases of goslings from one-colour families who chose mates of the other colour and had been inclined to dismiss these as “noise” in the system. It now seemed likely, however, that the noise was, in fact, the phenomenon which should be examined. The mating rule was rephrased to “geese generally choose mates....” The parameter t was used to make precise the word “generally;” i.e., a proportion t of the goslings from one-colour families choose mates of the other colour. This guarantees a minimum level (t times the proportion of one-colour families) of mixed pairs without saying how or why such choices are made. For example, using the initial state vector for the white colony t = 0.05 guarantees 4% mixed pairs; t = 0.10 guarantees 8% mixed pairs. The other graphs in Fig. 1 report the values of the phenotypic variables obtained using t = 0.05, t = 0.10, and t = 0.15. These results show that determining a value for t (i.e., ensuring a certain level of mixed pairs with females from one-colour families) also causes a decrease in white birds and this is at least partially caused by blue male immigration. These are the trends observed at La Perouse Bay as shown in Table VI. 653/22/2

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GERAMITA, COOKE, AND ROCKWELL

NO. OF GENERATIONS

NO. OF GENERATIONS

NO. OF GENERATIONS

FIG. 1. White colony simulation. Several values of t.

With the results of the model before us, a search for data on goslings of known family history was made. These data were presented in Cooke (1978) and can be summarized by saying that roughly 11% of goslings of onecolour parents chose other colour mates. Some of this can be explained by nongenetic family modification and some by the presence of white siblings in a family with two heterozygous blue parents. These two phenonena, however, explain only about 5 % of the blue goslings and 2% of the white goslings choosing the other colour. This leaves 6-10% unexplined. The parameter t has precisely that role: to insert unexplained nonfamilial colour choice into the model. The results of the simulations depicted in Fig. 1 show that it is this “mistake making” which controls the level of assortative mating and male immigration. The simulations also underline the importance of determining the biological mechanism by which certain birds from one-colour families choose familial colour and others do not. Simply setting t equal to some

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number is not a final answer. The simulated white colony produced by letting t = 0.15 mimicked La Perouse Bay but differed in two interesting ways. First X, =x3 in the simulation. This means that more blue than white goslings are being produced. (The equality x,, = 2x, giving equal numbers of blue and white goslings is discussed in Section III.) At the real colony xq = 4x, which means that more white goslings are being produced. In both “colonies” the blue phase is increasing but in the real world the source of the increase appears to be solely blue male immigration. A second difference sheds some light (and raises some question). At La Perouse Bay, the type-3 pairs (homozygous blue-white) seem to be rare. xj = 0.02, and are about 10% of all mixed pairs. In the simulated colony this type is less rare, x3 = 0.05, and is about 30% of the mixed pairs. The model causes more homozygous blue birds to be mated to white birds than seems so in the real world. This raises at least two questions: All things being equal (i.e., all blue family, etc.) are heterozygous blue birds more likely to make “mistakes” than homozygous blues? and are mistake-making (or

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2.

NO.

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GENERATIONS

4 GENERATIONS

White colony simulation.

Several values of r.

t =.15 t=.10

t =.o

t=o

c

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GENERATIONS

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NO.

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w 3 al

NO. OF

FIG. 3.

GENERATIONS

Blue colony simulation.

Several values of

t. r=.6

24

t

OI NO.

OF

GENERATIONS

NO.

I2 NO. OF

FIG. 4.

2 OF

34

GENERATIONS

34 GENERATIONS

Blue colony simulation.

Several values of r.

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white between birds distinguishing perhaps non-mistake-making) heterozygous and homozygous blues? An obvious choice mechanism is that a certain proportion of the birds mate at random while the rest choose family colour. This may be interpreted as a gosling from any family has a probability r of choosing at random or as a family having a probability r of having all its young choose at random. Figure 2 gives the results of simulations with several values of the parameter r. These are in fact slightly less satisfactory than using the parameter t. However, the values of r equal to 0.4 and 0.6 are consistent with the data and the r parameter has the advantage of being more readily interpretable in a biological sense. The type-3 families still compose approximately 30% of the mixed pairs and the ratio of xX to X, is still greater than Data from colonies other than La Ptrouse Bay are not detailed. In Section I, however, we have set forth a summary which indicates that at blue colonies the trend is to increasing white. Figures 3 and 4 show the results of simulations using an initial state vector corresponding to a predominately blue colony. These show a white increase even with f = 0 but the level of mixed pairs is quite low (6%). Figures 5 and 6 record simulations using an

t=o t=.o t=.I0

5 z2 P j51&

14

E

w t

t=.o5

t=o

3 z

241e 1=.IO

t=.15

50

t =.I5

4

‘t&---

St

%YFTTTNO.

OF

4

0

GENERATIONS

NO

_

I

OF

2

3

GENERATIONS

t =.05 >=.I0

-----xx

I:! NO. OF

FIG. 5.

34 GENERATIONS

Mixed colony simulation.

Several values of I

4

194

GERAMITA,COOKE,ANDROCKWELL

24

cl NO.

12 OF

34

0

GENERATIONS

NO. OF

I2 NO. OF

FIG. 6.

I2

34 GENERATIONS

34 GENERATIONS

Mixed colony simulation.

Several values of r.

initial vector from a mixed colony (50% white). In this case, increasing t retards the white increase. Additional data from other Hudson Bay colonies will allow us to better assess the model as a description of a species wide characteristic.

V. GENERAL CONSIDERATIONS Predictions concerning temporal change in the genetic composition of natural populations are always hazardous. This is particularly true for subdivided populations in which assortative mating, gene flow, nongenetic modification of family structure, and natural sefection may be among the factors affecting gene frequencies. Early attempts to predict genetic changes in the Hudson Bay colonies of the Lesser Snow Goose (e.g., Coach, 1963; Seiger and Dixon, 1970) were unsuccessful in part because they did not consider the effects of several factors now known to operate in these colonies. It is clear that the inclusion of such factors is necessary to predict realistically frequency changes in terms of plumage colour genes. It is only

ASSORTATIVE

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195

because so much is presently known about the population structure and breeding biology of these birds that we felt an attempt at prediction should be made at this time. It is not difficult to predict analytically the effects of single factors such as gene flow or assortative mating on the dynamics of gene frequencies in the colonies. To predict the effects resulting from the simultaneous actions (and interaction) of the factors now known to operate in these colonies, however, one must rely on computer simulations of sophisticated models. The model used here has focused on three phenotypic variables (percent of colony which is white, percent of mixed pairs, and percent of mixed pairs with a blue female) for which we have considerable baseline information from the La Perouse Bay colony. The simulation studies of the model presented in the preceding sections have led to predictions of future trends and levels for these variables in the colonies which should be more realistic than previous ones and can be tested in the field. The simulation studies have also led to clarification of the potential interactions among the factors affecting these variables in nature. These predictions and clarifications are synopsized in the following. Given the amount of gene exchange between colonies, there would be rapid equilibration of gene frequencies at all breeding colonies if mating were at random. In that case, the frequency of mixed pairs would increase rapidly to approximately 48%. If, on the other hand, mate selection were always according to the preference hypothesis of Cooke et al. (1976), the frequency of blue birds would decrease at the La Perouse Bay colony and the frequency of mixed pairs would decrease even more rapidly (Fig. 1, t = 0). It is only when lo-15% mistake making in terms of the mate selection hypothesis is allowed that the predicted changes in the frequencies of blue individuals, of mixed matings, and of mixed matings with blue females approximate the values obtained in field data at La Perouse Bay. Intensive field studies are presently being conducted to ascertain the actual level of mistake making in mate selection as well as its behavioural basis. Preliminary results are in close agreement with the level predicted by our model. The changes in the relative frequency of blue individuals in the Hudson Bay colonies, may, thus, not reflect transient polymorphisms based on natural selection as suggested by Coach (1963) but, rather, gradual equilibration among the colonies due to gene flow tempered by assortative mating based on familial colour. The assortative mating is itself modulated by “mistake making” such that the level of mixed matings is determined by the degree to which birds use familial colour in choosing a mate. We feel that as long as 10-15 % of the birds of one-colour families choose a mate of the “wrong” colour, the level of mixed mating presently found in the field will remain constant.

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It is important to note that it is not necessary to invoke fitness differentials between the colour phases to explain the present colour distribution among the Hudson Bay colonies nor changes in that distribution. While this is not in itself sufficient proof that selection does not affect the gene frequencies, we would again stress that in 12 years of research at La Perouse Bay we have found no evidence of overall fitness differences between the phases. The colour composition of Lesser Snow Goose families can be modified on the breeding grounds by such factors as intraspecific nest parasitism (egg dumping) and fostering. Our simulation studies indicate that while such nongenetic modification of family structure could by itself alter the genetic dynamics of the colonies, the effect of this factor is swamped by the observed “mistake making.” For example, increasing the proportion of nests affected by nongenetic family modification by 0.10 results in an increase of only 0.0 1 in the proportion of mixed pairs. In a similar fashion we have shown that the nonrandom distribution of colour phases when pair formation occurs would have little effect on the predictions of the model unless there were considerably more segregation of the colour phases than presently exists. As with mistake making, we are presently studying the actual levels of both nongenetic modifications of families and nonrandom distribution of colour phases in the field. The biological factors incorporated into this model have been identified through intensive study of the La Perouse Bay colony of the Lesser Snow Goose. The simulation study of the model using La Perouse Bay data for baseline conditions has led us to a particular view of the simultaneous actions and interaction of these factors as determinants of the genetic structure and dynamics of this colony of snow geese. We feel that this general view applies to other colonies of this species. To ascertain whether the model and its biological bases are in fact more universal, we suggest the collection of data on the genetic structure and dynamics of other colonies of snow geese and to compare that data to predictions from our model. Differences between observation and prediction may lead us to find new biological factors affecting genetic structure in this species. Agreement would support the general validity of our views. The preliminary data collected thus far is in full agreement with the predictions from our model.

VI. SUMMARY

The nonrandom spatial and temporal distribution of the colour phases of the Lesser Snow Goose has been thought to reflect the interaction of many factors. Lack of knowledge as to the identity and extent of impact of many of these factors limited early attempts to explain or model the colour dimorphism in colonies of this species. This paper presents a model which

ASSORTATIVE

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197

incorporates those factors which have been shown to exist and to affect the colour dynamics of colonies of Lesser Snow Geese. The model is examined with computer simulation using initial conditions empirically determined from 12 years of research on the La Perouse Bay snow goose colony in northern Manitoba. These simulations have led to predictions about the levels of action of several other factors only recently uncovered at La Perouse Bay. The simulations have also been used to predict the patterns and temporal trends of colour phase distribution in colonies which have not yet been fully studied. Evaluation of these colonies will afford a test of the universality of the model. Several conclusions concerning the interaction of factors affecting the dynamics of the colour phases in the species have been motivated by the model. These are presented and discussed.

APPENDIX

Part I Let x be the vector (x, ,..., x6) defined in Table I. The aim of this section of the Appendix is to define the 3 x 3 matrix M(x) where mi,i is the probability that a female offspring of the colony defined by x of genotype i mates a male of genotype j (Table II). From M(x) we can compute the values of the phenotypic variables in which we are interested. The proportion of white birds is 0.5m,, + 0.5~ + 0.5m,, + 0.5m,, + mj3. The proportion of mixed pairs is m,3 + m23+ m,, + mj2. The proportion of mixed pairs with a blue female is

ml3 + m23 + m3, + m32 Let p = (p, ,..., p6) be the distribution of preference types for the total population using the wintering ground, where the types are defined in Table III. We define a set of matrices which will relate genotype wintering region and preference. Let Tk, k = 1, 2, 3, be the 6 x 6 matrix with ij entry the probability that a bird of preference type i is in region k, behaving as if it were of preference typej. Only the diagonal entries of Tk will be nonzero. With the preference hypothesis we are using, it is possible to divide the geese into classes based on their willingness to mate a certain colour. These “acceptance” classes are the pool of potential mates a choosing bird has available. Since our view is that of the breeding colony where the females are the returning birds, we think of females as choosing and the males as the “pool’‘-the fact that not every male is in the pool reflects the male preferences. Table VII gives six such classes. Classes 1 and 2 are identical as are Classes 3 and 4.

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GERAMITA,

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TABLE VII Acceptance Classes Class number 1 and 2

3 and 4

6

Description

Blue-acceptors that are blue

Blue-acceptors

White-acceptors White-acceptors that are white

Proportion of birds in class

1 - P, - P,

I-P,

1 -P, - P,

5

p, + p,

The repeated classes are a convenience in the definition of the matrices C, and A below. The C, could be 4 x 3 matrices but then the formation of the matrix A which gives the probabilities that a female of preference i will mate a male of genotype j would be more complicated. In addition to repeated classes, it should be noted that the classes overlap. For example, class 3 (which is also class 4) of birds willing to mate a blue bird includes class 1 (which is also class 2) of blue birds willing to mate a blue bird. The proportions (or probabilities) given in Table VII tell what part of all the birds fall into a certain class. To express the proportion of birds willing to accept blue mates who are blue themselves we use 1 -Ps

-Pb

l-P,

.

We must relate the three characteristics of male genotype, wintering region, and acceptance class. We do this with a set of matrices C,, k = 1, 2, 3. C, is the 6 x 3 matrix with ij entry the probability that a bird in wintering region k which is of acceptor type i has genotype j. It is formed from T, and p. Let t , ,..., t, be the diagonal entries of Tk. t, PI + t, P3

t2 P2 + t4 P4

C4= I ti Pi

C4= I ti Pi

0

Same as row 1 t1

PI + Cf=l

c, =

t3 P3

tiPi

t2 P2 + t4 P4

Ci=I

tiPi

ts Ps X:=1 tiPi

Same as row 3 t3 P3 CF=3 ti Pi

tS PS X:=3

+

t6 P6

ti Pi

ASSORTATIVE

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FLOW

Let R,, k = 1, 2, 3, be the 6 X 6 diagonal matrix analogous to Tk but describing the situation for a specific colony rather than the whole universe. Then A = R, C, + R,C, + R, C, is the 6 x 3 matrix with ij entry the probability that a female from the specified colony of preference type i will mate a male of genotype j. The vector q = (q i ,..., q6) is the distribution of preference types for a specified colony corresponding to p for the whole population. Compute q using x and Q, where p is the proportion of type-4 families with at least one white gosling, by q = xQ with

Q=

1

0

0

0

0

0.5

0.5

0

0

00

o7

0

0

0

100

0.25(1 - p)

OS(1 - p)

0.25~

0.5~

0.25

0 .

0

0

0

0.5

0.5

0

0

0

0

0

01

The vector q represents the preferences which would hold if there were no nongenetic family modification and if the word “generally” meant “always.” Recall that d, t, and r are real numbers between 0 and 1. The vector z = (z, , z2, z3) gives the distribution of genotypes of the offspring as given in Table II. Let q’ = (q;, q; ,..., q:) be given by

q; = q2 - dz, q2 - rq2 - tq2 4; = q3 .+ dz, q1 + rq,

s;= q4+ dz3q2+ rq2 4; = 45 + d(z, + Z&6 +

rq6

4; = q6

-

d(z,

f

z2)q6

-

rq6

-

‘q6’

Form Q:

&

q;

0

q;

0

0

0

0

q;

0

q;

0

0

0

0

0

0

4;

4;

200

GERAMITA,

COOKE,

AND

ROCKWELL

We let B = OA and write

b,,+tq1 b,,+tq,

b I2 b 22 fq,

b,, +d+-P3 fP4

M(x) is the matrix we set out to define.

‘q6

b 33

I

Part 2 In this part of the appendix, we specify all the constants used in the functions in Part 1 and give the rationale for choosing the values we did. There are two types of constants-those assigned to a specific colour class of breeding colony, and those describing the total population using the wintering ground. For each type of colony, white, blue, and mixed, we needed a set of diagonal matrices R, relating preference type and wintering area. Naturally there were not data relating these concepts since the preference type is our invention not that of the geese. We used data (Cooke, unpublished) which gave a distribution of mated pairs from La Pirouse Bay along the wintering ground. The pairs were of four types: white-white, blue female-white male, white female-blue male, and blue-blue. The distribution of white-white pairs was taken to represent the distribution of white insisters from a white colony. Similarly, the white female mixed pairs were used to represent white indifferents, the blue female mixed pairs to represent blue indifferents, and blue pairs to represent blue insisters. Our reasoning is this: if there were such things as white insisters, they would be in white-white pairs and the proportion of females from La Perouse Bay in region k who were white insisters would be less than the proportion of white-white pairs from La Perouse Bay present in region k. On the other hand, the proportion of white female mixed pairs from La Perouse Bay in region k would be less than the proportion of white indifferent females from La Perouse Bay since some of the latter would be in the white-white pairs. For the blue birds the problem is compounded by the genotype question. However, the data on pair distribution and the estimates of genotype frequency were combined (and rounded) to produce the diagonal elements of R,, for the three wintering regions given in Table IV (all the other entries in R, are zero). For the white colony R, - (0.2, 0.2, 0.2, 0.2, 0.1, 0.15) R, - (0.4, 0.4, 0.5, 0.5, 0.6, 0.25) and R,- (0.4, 0.4, 0.3, 0.3, 0.3, 0.6). For the blue version of the white values obtaining colony we used a “reciprocal” R, - (0.6, 0.6, 0.3, 0.3, 0.3, 0.4), R,- (0.25, 0.25, 0.6, 0.6, 0.5, 0.4), and R, - (0.15, 0.15, 0.1, 0.1, 0.2, 0.2). We considered that the mixed colony should distribute itself in much the same way that the universe of birds does

ASSORTATIVE

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201

and so set R, = T, where the T, were the corresponding matrices used to describe the whole population. For these matrices, we used the data given in Table IV. An example of our reasoning: Region 1 is 80% blue and includes about 20% of the total population so about 40% of the blue birds would be in region 1. In a heavily blue region, one might expect more blue families than the average, and therefore more blue insisters than the average. So the proportion of blue insisters would be more than 0.4 and the proportion of blue indifferents less than 0.4. We also decided-in the absense of any “universal” pair distribution data such as we had for La Plrouse Bay-that the white indifferents and blue indifferents would go to each region in equal proportions. We set the diagonal elements of T, = (0.5, 0.5, 0.3, 0.3, 0.3, O.l), T, = (0.3, 0.3, 0.5, 0.5, 0.5, 0.4), and T, = (0.2, 0.2, 0.2, 0.2, 0.2, 0.5). These ligures give about 15 % of white birds in region 1 while Table IV suggests a figure closer to 7%. This means that blue indifferent birds in the simulation have a greater probability of mating white birds in region 1 than the Table IV data suggest. Correcting this “error” would strengthen our conclusions rather weaken them. For the total population on the wintering ground we specified a preference vector p= (0.20, 0.05, 0.05, 0.10, 0.20, 0.40). In this case we made the entries p, + p3, pz + p4, ps + p6 correspond to the assumed genotype frequency (0.25,O. 15,0.60) as discussed in Section III. Part 3

Suppose the proportion of mixed pairs x3 + xs = X is known. Let W be the proportion of mixed pairs with white goslongs. X and Ware collectable data. Let t,~ be the probability that a type-5 family will have at least one white offspring. Let d proportion of nests affected by intraspecilic nest parasitism and z3 the proportion of white offspring in the colony. Then (dz3 - w>x, + (dz,( 1 - lu) + v/ - W)x, = 0 x1 + x, = x can be solved simultaneously to obtain

x5=

- (dz, - W)X y/(1 -dzJ ’

Similarly, if we use p in place of I,Y,x, + x2 in place of x3, and xq in place of x5 then we can separate the sum x, + x2 + xq into x, + x2 and x,.

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GERAMITA,COOKE,AND

ROCKWELL

ACKNOWLEDGMENTS These studies formed part of an investigation being carried out under contract with the Canadian Wildlife Service. Financial assistance was also provided by the Natural Sciences and Engineering Research Council Canada and the Manitoba Department of Renewable Resources. We also thank J. T. Smith and J. Davis for comments and suggestions about several versions of these models, and J. Emlin for comments on their presentation.

REFERENCES BOAG. P. T. 1974. “A descriptive and Functional analysis of Post-hatch Flocking in the Lesser Snow Goose.” Unpublished B.Sc. Thesis. Queen’s Univ., Kingston, Ontario, Canada. COOCH.F. G. 1961. Ecological aspects of the Blue-Snow Goose complex, Auk 78, 72-89. COOCH,F. G. 1963. Recent changes in distribution of colour phases of Chen c. caerulescens, Proc. Int. Ornithol. Congr. 13, I 182-l 194. COOKE. F. 1978. Early learning and its effect on population structure. Studies of a wild population of Snow Geese, Z. Tierpsychol 44, 344-358. COOKE,F.. AND MIRSKY, P. J. 1972. A genetic analysis of Lesser Snow Goose families, Auk 89, 863-87 I. COOKE, F., MACINNES, C. D., AND PREVETT,J. P. 1975. Gene flow between breeding populations of Lesser Snow Geese, Auk 93, 493-5 10. COOKE, F.. FINNEY G. H., AND ROCKWELLR. F. 1976. Assortative Mating in Lesser Snow Geese, Behav. Genet. 6 127-140. DZUEXN,A., BOYD, H., AND STEPHEN,W. J. D. 1975. Blue and Snow Goose distribution in the Mississippi and Central Flyways 195l-1971, Prog. Notes 54, Canadian Wildlife Service, Ottawa. FINNEY, G. H. 1975. Reproductive strategies of the Lesser Snow Goose, Anser caerulescens caerulescens. Unpublished Ph. D. Thesis, Queen’s Univ., Kingston, Ontario, Canada. GERAMITA.J. M., AND COOKE,F. 1982. Evidence that fidelity to natal breeding colony is not absolute in female snow geese, Canad. J. Zool., to appear. GRAHAM A. 1769. Andrew Graham’s Observations on Hudson Bay 1767-1791. Hudson’s Bay Archives. HANSON, H. C.. LUMSDEN H. G.. LYNCH, J. J., AND NORTON, H. W. 1972. Population characteristics of three mainland colonies of Blue and Lesser Snow Geese nesting in the southern Hudson Bay Region, Ontario Fish and Wildlife Res. Branch Res. Rep. (Wild.) No. 92, 38 pp. HEARNE, S. 1795. A journey from Prince of Wales’ Fort in Hudson’s Bay to the Northern Ocean, 1769-1772. Hudson’s Bay Archives. KERBES,R. H. 1975. Lesser Snow Geese in the Eastern Canadian Arctic, Canadian Wildlife Service, Rep. Ser. 35, 49 pp. LEMIEUX, L.. AND HEYLAND,J. M. 1967. Fall migration of blue geese Chen caerulescens and lesser snow geese, Chen hyperborea hyperborea from the Koukdjuak River, Batlin Island, Northwest Territories, Natur. Canad. 94, 677-694. MINEAU, P. 1978. The breeding strategy of a male snow goose, Anser caerulescens caerulescens. Unpublished M. SC. Thesis, Queen’s Univ., Kingston, Ontario, Canada.

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B. 1981. “Genetics of the Colour Polymorphism of the Lesser Snow Goose.” Unpublished B.Sc. Thesis, Queen’s Univ., Kingdom, Ontario, Canada. ROCKWELL, R. F., AND COOKE, F. 1977. Gene flow and local adaptation in a colonially nesting dimorphic bird, Amer. Nutur. I1 1, 91-97. SEIGER. M. B., AND DIXON, R. D. 1970. A Computer simulation of the effects of two behavioral traits on the genetic structure of semi-isolated populations, ELdution 24, 90-97. RATTRAY,