Abstract
I formulate and study a series of simple onelocus twoallele models for maternal (parental) selection. I show that maternal (parental) selection can result in simultaneous stability of equilibria of different types. Thus, in the presence of maternal (parental) selection the outcome of population evolution can significantly depend on initial conditions. With maternal selection, genetic variability can be maintained in the population even if none of the offspring of heterozygous mothers survive. I demonstrate that interactions of maternal and paternal selection can result in stable oscillations of genotype frequencies. A necessary condition for cycling is strong selection.
MATERNAL effects refer to situations in which an individual's phenotype depends (besides other factors) on the phenotype of its mother. Well known to biologists for decades, maternal effects have been considered as a nuisance that makes interpretation of biological data more difficult and that should be removed from experiments if possible (Falconer 1989). Recent years, however, have been characterized by a growing realization that maternal effects may actually be very important in evolution (e.g., Cheverud and Moore 1994; Wade and Beeman 1994; Fox and Mousseau 1996; Roosenburg 1996; Rossiter 1995, 1996; Sinervo and Doughty 1996; Wade 1996; Wolfet al. 1997). This realization is coming partially from theoretical models that have shown that maternal effects can result in very interesting and sometimes counterintuitive evolutionary dynamics (Cheverud 1984; Kirkpatrick and Lande 1989; Lande and Kirkpatrick 1990; Orr 1991; Ginzburg and Taneyhill 1994; Wade and Beeman 1994). The concept of maternal effects can be generalized to the concept of “kin effects” (Cheverud 1984; Cheverud and Moore 1994) incorporating the effects of any relatives on an invidivual's phenotype. “Maternal selection” is one of various maternal effects. This notion describes situations in which an individual's fitness depends (besides other factors) on the phenotype of its mother (Kirkpatrick and Lande 1989). Most of the theoretical work on the evolutionary consequences of maternal selection has been done within a quantitative genetics framework (e.g., Dickerson 1947; Villham 1963, 1972; Falconer 1965; Cheverud 1984; Kirkpatrick and Lande 1989; Lande and Price 1989; Lande and Kirkpatrick 1990). There have been only a handful of theoretical studies considering major locus effects. An extreme case of fitness that depends entirely on the mother's genotype at a single diallelic locus was considered by Wright (1969, pp. 57–59) and Nagylaki (1992). Wright (1969, pp. 148–149) and Wade and Beeman (1994) studied singlelocus twoallele models in which offspring of a specific genotype with a parent of a specific genotype has a reduced fitness.
Here I consider the dynamics of a series of simple onelocus twoallele models for maternal (and paternal) selection. These models are closely related to those introduced and discussed by Wade (1996) who, however, did not consider the dynamic behavior. Although the models studied in this article probably oversimplify real situations, they may be useful in identifying different dynamic possibilities and important parameters and in training our intuition about more complex (and realistic) systems.
A MODEL FOR MATERNAL SELECTION
I consider a single randomly mating diploid population with nonoverlapping generations. I assume that fitness (viability) of an individual depends on its genotype at a single diallelic locus as well as on the genotype of its mother at this locus. Let i = 1, 2, 3 correspond to genotypes AA, Aa and aa, respectively, and let w_{i,j} be the fitness of an individual with genotype i raised by a mother with genotype j. Genotype frequencies are equal in both sexes after one generation of selection and segregation. Let x, y and z be the frequencies of adults with genotypes AA, Aa and aa, respectively. The nine mating types and the frequencies of the corresponding matings and offspring are given in Table 1. Using this table, the adult frequencies x′, y′ and z′ in the next generation are defined by
The dynamic system (1) has two monomorphic equilibria (1, 0, 0) and (0, 0, 1) corresponding to the fixation of genotype AA and aa, respectively. Conditions for stability of these equilibria, which can be found in a straightforward manner, are given in
Result 1 (stability of monomorphic equilibria): Monomorphic equilibrium (1, 0, 0) is locally stable if and is unstable if
Thus, a monomorphic equilibrium is stable if fitness of the homozygote having a homozygous mother is larger than the average of the fitnesses of heterozygotes having a heterozygous mother and a homozygous mother. Reversing both inequalities results in the conditions for protecting polymorphism in this model. Analysis of attractors of (1) other than the monomorphic equilibria requires additional simplifying assumptions about fitnesses w_{ij}. Some special simplifying cases are examined below.
Symmetric model for maternal selection
A standard approach for simplifying analysis of population genetic models is to introduce some symmetry in the model. In this section, I will assume that fitnesses w_{i,j} are symmetric in the sense that w_{1,1} = w_{3,3}, w_{1,2} = w_{3,2}, w_{2,1} = w_{2,3}. This symmetric case can be described by the following fitness matrix:
where α, β, γ and δ are nonnegative. Note that neither genotype AA can have a mother with genotype aa, nor can genotype aa have a mother with genotype AA. This symmetric model implies that fitnesses of homozygotes raised by homozygous mothers are identical, fitnesses of homozygotes raised by heterozygous mothers are identical and fitnesses of heterozygotes raised by homozygous mothers are identical. The dynamic equations (1) become
Result 1a (stability of monomorphic equilibria): Monomorphic equilibria (1, 0, 0) and (0, 0, 1) are (simultaneously) locally stable if
Let us consider polymorphic equilibria of Equation 3.
Results 2 (existence of a symmetric polymorphic equilibrium): There always exists a symmetric polymorphic equilibrium with x_{sym} = z_{sym}. The genotype frequencies at this equilibrium are x_{sym} = u/(2u + 2), y_{sym} = 1/(u + 1), z_{sym} = u/(2u + 2), where
Results 3 (stability of the symmetric polymorphic equilibrium): The symmetric polymorphic equilibrium is locally stable if
Results 4 (existence of a pair of unsymmetric polymorphic equilibria): A pair of unsymmetric polymorphic equilibria with genotype frequencies (x*, y*, z *) and (z*, y*, x*) exists if
If α > γ,δ or α < γ,δ, the unsymmetric polymorphic equilibria do not exist. If α is closer to δ than to γ (that is, Δ_{1} ≡ α − δ < Δ_{2} ≡ α − γ), then the unsymmetric polymorphic  sufficiently small β/α equilibria exist for (for β/α < Δ_{1}/Δ_{2}). If α is closer to δ than to γ (i.e., Δ_{1} > Δ_{2}), then the unsymmetric polymorphic equilibria exist for sufficiently large β/α (for β/α > Δ_{1}/Δ_{2}).
Can these equilibria be stable? Conditions for stability of these equilibria can be found in a straightforward manner but are rather cumbersome. These conditions simplify if β = δ = 0, that is, if offspring of heterozygous mothers are inviable. In this case, the genotype frequencies at unsymmetric equilibria are
Result 5: If β = δ = 0, the necessary and sufficient condition for both existence and simultaneously stability of a pair of asymmetric polymorphic equilibria is γ > 2α.
This somewhat conterintuitive result means that genetic variability can be maintained in the population even if none of the offspring of heterozygous mothers survive. This is a general feature of models where there is competition between sibs within a family and a feature of the maternaleffect selfish gene model of Wade and Beeman (1994).
Multiplicative model for maternal selection
Let an individual's fitness, w, be a product of two parameters, one of which, v, depends on an individual's own genotype, whereas another one, m, depends on its mother's genotype
Symmetric model for parental selection
In this section, I assume that both parents contribute to an offspring's fitness. Let w_{i,jk} be the fitness of an individual with genotype i raised by a mother with genotype j and a father with genotype k (i, j, k = 1, 2, 3). Using Table 1, the adult frequencies x′, y′ and z ′ in the next generation are defined by
To illustrate this, let us assume that only offspring of heterozygous parents and of different homozygous parents are viable, and offspring of all other pairs of parents are inviable. In this case the following is true.
Result 6: Let α = β = 0, δ, γ > 0. Then if 4γ > δ, the only stable equilibrium of (12) is a symmetric polymorphic equilibrium (with x* = z *). If 4γ < δ, the system does not have any stable equilibria.
In the latter case, the system cycles with period two (see Figure 1). In general, a dynamic system (12) cycles if the fitness δ of individuals with both parents heterozygous is sufficiently large relative to the fitness γ of individuals with different homozygous parents, which in turn is sufficiently larger than fitnesses α and β of other individuals.
CONCLUSIONS
Here I have studied a series of simple onelocus twoallele models for maternal (parental) selection. Srb et al. (1965, Chapter 11) give several examples for maternal effects that can be attributed to a single diallelic locus; see Wade (1996) for more discussion of the relevance of maternal effects controlled by a small number of loci with large effects. My results indicate similarity between dynamic behaviors under maternal selection and fertility selection. The latter is wellknown to be much more complicated than the dynamics resulting from viability selection (e.g., Owen 1953; Bodmer 1965; Hadeler and Liberman 1975). I have shown that maternal selection can result in a simultaneous stability of equilibria of different types. Thus, in the presence of maternal (parental) selection, the outcome of population evolution can significantly depend on initial conditions. With maternal selection, genetic variability can be maintained in a population even if none of the offspring of heterozygous mothers survive. I have demonstrated that interactions of maternal and paternal selection can result in stable oscillations of genotype frequencies. A possibility for cycling and even chaos in theoretical models incorporating maternal effects has been already demonstrated by Ginzburg and Taneyhill (1994). In Ginzburg and Taneyhill's model, cycling resulted from ecological factors. In contrast, in the model considered here cycling is brought about by genetic factors and maternal selection.
The counterintuitive results about the maintenance of variability and cycling require selection to be strong. The following two examples of very strong maternal selection are interesting in this respect. The first example (Srbet al. 1965, pp. 319–320) concerns Drosophila where the sexlinked recessive gene fused causes partial sterility as well as fusion of two longitudinal wings' veins. At least one wildtype allele of the fused locus must be present in either the mother or in the progeny in order for embryogenesis to proceed normally. As a result, offspring with identical genotypes may die or develop to maturity, depending on the genotype of their mother. The second example is a class of dominant lethal genetic factors called Medea that is widespread in natural populations of the flour beetle, Tribolium castaneum (Beemanet al. 1992). These factors cause maternaleffect lethality of all offpring not inheriting a copy of the factor [see Wade and Beeman (1994) for a theoretical study of the evolutionary dynamics of these factors]. These examples suggest that conditions necessary for complex dynamics might be satisfied at least for some biological systems.
Results and findings reported here are complementary to those obtained within the quantitative genetic framework (e.g., Dickerson 1947; Villham 1963, 1972; Falconer 1965; Cheverud 1984; Kirkpatrick and Lande 1989; Lande and Price 1989; Lande and Kirkpatrick 1990). The latter typically considers an additive quantitative trait that is controlled by both direct genetic and maternal factors and that is subject to phenotypic selection. The main focus is usually on the dynamics of the mean value of the trait, whereas (constant) genetic variability is tacitly assumed to be maintained by some factors that are not considered explicitly. The major locus framework utilized here and in Wright (1969), Nagylaki (1992), and Wade and Beeman (1994) focuses on maternal effects contributing directly to fitness. This framework allows for analyzing conditions for the maintenance and dynamics of genetic variability and the existence of multiple stable equilibria. Generalization of this approach for the case of multiple alleles and loci would be very desirable.
Acknowledgments
I am grateful to MaryCarol Rossiter and Michael Wade for sharing manuscripts before publication and to reviewers for valuable comments and suggestions. This work was partially supported by National Institutes of Health grant GM56693.
APPENDIX
In analyzing the dynamics of Equation 3 it is convenient to use new variables u = 2x/y and v = 2z/y, which are defined for y ≠ 0. The dynamic equations for u and v are
The symmetric equilibrium is stable with respect to symmetric perturbations (such that perturbed values of u and v are equal), if at this equilibrium
Combining Equation A3 with Equation A2 at equilibrium, one finds that at asymmetric equilibria
If β = δ = 0, the values of u and v at the unsymmetric equilibria are (0,α/(γ/2 − α)) and (α/(γ/2 − α), 0). Thus, these equilibria are feasible if α < γ/2. The eigenvalues of the stability matrix at these equilibria are 2α/γ and 1 − 2α/γ. For the unsymmetric equilibria to be stable these eigenvalues should lie between −1 and 1, which is obviously the case given that the equilibria exist. This completes the proof of Result 5.
Using Equations 4 and 6 from Hadeler and Liberman (1975), one finds that if α = β = 0, the only equilibrium of Equation 12 satisfies to the cubic
Footnotes

Communicating editor: W. H. Li
 Received March 17, 1997.
 Accepted February 23, 1998.
 Copyright © 1998 by the Genetics Society of America