Article,
Classifications of selection migration structures and conditions for a protected polymorphism
Evolutionary biology, (1982)
Abstract
The idea that levels and forms of genetic variability can be related
to temporal and spatial patterns of environmental heterogeneity is a
widely promulgated theme in evolutionary biology. Recent reviews per-
taining to genetic polymorphisms under conditions of variable selection
and migration are given by Hedrick et at (1976) and Felsenstein (1976) ,
which include numerous references to experimental. field, and theoretical
studies.
ies.
Three classes of migration patterns have been predominantly studied
in theoretical population genetics:
a. The island model (Wright, 1943) consists of an array of islands
exchanging genes uniformly. Equivalently, the island model involves N
"equidistant" islands thai share a common migrant pool drawn equally
from all demes.
The Levene population subdivision structure (1953) is a direct gen·
eralization of Wright's model that allows for variable deme (island) sizes.
Deakin (1966) introduced a homing or sessile tendency so that some
individuals would remain in their deme of birth rather than enter the
migrant pool. He assumed that a fixed proportion of the population is
sessile while the remainder of the population acts according to the Levene
model.
b. The stepping-stone model and. more generaJly. isolation-by-dis-
lance migration patterns assume that the rates of migration between
demes depend on the distances between them. Isolation by distance based
on a o ne, two, or even higher dimensional layout (strata defined by phys-
ical position, social or behavioral characteristics) of the population has
been investigated by Maleeo! (1948, 195 1. 1959, 1967), Jain and Bradshaw
(1966), Kimura and Weiss (1964) , Maruyama (1970), among others. This
class of models generall y involves no differential selection within or be-
tween locaJities. In the context of environmental selection gradients. some
cline stepping-stone models have been extensively investigated (e.g., Slat-
kin , 1973; Nagylaki, 1974. 1975, 1976b, 1978, 1979; Nagylaki and Lucier,
1980; F leming, 1975; Karlin and Richter-Dyn, 1976).
c. Migration matrix models are designed to deal with general migra-
tion patterns (e.g., Malecot, 195 1, 1959; Bodmer and Cavalli-Sforza. 1968;
Carmelli and Cavalli-Sforza. 1976; Smith , 1969). Most authors have lim-
ited their attention to linear pressures. i.e ., mutation andlor migration
from an external fixed population, but having no mating or natural se-
lection differences operating.
In both classes of migration structures (b) and (c), computations have
been mainly directed to evaluating the correlation of gene freque ncies
over space and the changes of these with time.
T his work is part of a continuing series of theoretical studies that
seek to understand the effects of different types of spatially and temporally
varying selection regimes coupled with migration patterns on the exist-
ence and nature of polymorphism (Karlin, 1976. 1977a, b ; Karlin and
Richter-Dyn, 1976; Karlin and Campbell . 1978. 1979, 1980). In this chapter
we establish the conditions for the existence of a protected polymorphism
(protection means that none of the alleles become extinct even when
initiall y rare) for a hierarchy of migration patterns. These results will
permit qualitative comparisons of the influence of different structures of
migration exc hange in contributing to the maintenance of polymorph isms.
In this review we describe a series of hybrid migration structures
composed from canonical, e.g .• Levene. Deakin . stepping-stone. circu-
lant migration patterns that entail one or several clusters of demes. Deme
clusters can reflect the background terrain. geographical relationships ,
geological. climatic. or other ecological and environmental factors. and
also behavioral, social, or exogenous genetic-environmental character-
istics . Moreover. by introducing suitable fict itious deme arrays we can simulate the effects of seasonal variation in selection by a spatial selection
gradient. T he seasons often induce cyclic variation.
Some of the models are partly motivated by observations from insect
populations that divide into demes or groups of demes , depending on the
spacing of the plants on which they feed. Natural groupings of demes can
also be associated with arrays of islands, an archipelago. tributaries of
a river, inlets along a coast. a range of hills, spacing of flora. or moun-
tain-valley-canyon topography.
The partitioning of demes into clusters often distinguishes local pop-
ulation interactions against far movements. For example, with respect to
plant dispersal we can contrast t he nearby seed droppings with the long
migrations mediated by vectors (insects, mammals).
A three-tiered clustering of some human populations may arise from
the family-tribe, nation, and race structures. Other criteria for groupings
may relate to social economic status, life-styles, religion and customs.
and educational levels .
Conditions for clustering may be based on aspects ofthe environment
such as degree and kind of salinity. food availability . moistness, exposure,
etc. T he modeling of migration should reflect various levels of clustering
and associated with the clustering is a related pattern of selection.
We will investigate typically the following questions: To what extent
is the maintenance of polymorphism facilitated by the nature of hierar-
chical determinations of deme clustering? What are the consequences
associated with asymmetries in population exchanges? What are the rel-
ative influences of migration rates between and within clusters of demes?
Also, how do we compare spatial versus temporal variations in selection
and migration parameters?
The text is arranged as follow s. The migration patterns on which we
focu s are described in Sections 2-4. In Section 2 we review the formu-
lation of the Levene and Deakin migration models and their extensions
that allow variable (habitat-dependent) rates of homing, several stages of
migration in each life cycle, and different characteristic deme sizes. The
nature of c1inal flow , i.e., migration exchanges per generation limited to
neighboring demes whose rates can vary with respect to positions andlor
differ in reciprocal directions. is described. A number of relevant circulant
migration patterns are set forth. A final class of basic migration forms
involving d irectional migration via a distinguished (major) deme dispers-
ing to or receiving from "subordinate" demes is detailed. The process
of delayed germination in plant species with seed pools can be modeled
by such a migration scheme.
Sections 3 and 4 delineate more-complex migration structures encompassing clusters of several demes each. A population is assumed to
naturally divide into clusters with each cluster consisting of a number of
demes where the migration contrasts refer to intra- and interdeme cluster
movements. For example. a "slar" migration form is developed where
demes along each ray communicate only through a central deme. The
combined effects accruing from temporal and spatial variation can be
studied expeditiously by regarding the aggregate population as consisting
of a multicluster deme formation. Other relevant hybrid migration struc-
tures, such as Kronecker products and generalized circulant migration
systems, are also elaborated. For all these cases we ascertain the con-
ditions for a protected polymorphism and discern their dependence on
the model parameters.
The analytic apparatus used in ascertaining protection for the general
multideme migration-selection model is reviewed at the close of Section
1. Sections 7-10 present conditions for a protected polymorphism ap-
propriate for the models of Sections 2-4.
It is of interest to contrast migration structures as to their degrees
of mixing and isolation. Two such notions were introduced in Karlin
(1976). Several additional concepts and their analyses are set forth in
Section 5. A number of means of comparing selection heterogeneity are
introduced and some robust results interpreted in Section 6. Specifically,
we address the issue of the relationship between spatial or temporal se.
lection " heterogeneity " and the existence of a protected polymorphism.
In Section II we compare the opportunities for protection with mi.
gration once per season versus once per generation for a multideme pop.
ulation subject to seasonal selection variation. The conditions for pro·
tection in a multideme seed pool process are delineated in Section 13. A
number of relations of deme size distributions and allele protection are
de veloped in Section 14. Some models of mUltiple migration stages per
generation are investigated in Section 15. We set forth in Section 16 a
number of result s pertaining to the existence of protection attendant to
the addition or deletion of demes. In this vein. we examine the effects
with respect to protection of the unification or the separation of different
parts ofa population range. It is also of interest to ascertain the similarities
and contrasts in the equilibrium gene frequency patterns that accrue from
an enlarged neutral zone where in other respects the selection migration
structure is unchanged.
The discussion of Section 17 summarizes in qualitative terms some
of the implications and contrasts of the quantitative results of the previous
sections. Mathematical proofs and analyses are relegated to Appendices
A-F. The reader not concerned with the technical developments hence-
forth may best concentrate on Sections 1-4 to understand the spirit of the formulations and motivations on the various migration structures, and
then skip to Section 17 for the qualitative summary and discussion of the
results.
Tags
- eigenvectors
- maintenance_of_variation
- matrix_models
- migration
- population_dynamics
- population_genetics
- spatial_structure
