Season-specific dominance broadly stabilizes polymorphism under symmetric and asymmetric multivoltinism

Abstract

Seasonality causes intraannual fitness changes in multivoltine populations (defined as having multiple generations per year). While it is well known that seasonally balanced polymorphism can be established by overdominance in geometric mean fitness, an unsettled aspect of the deterministic theory is the relative contribution of various season-specific dominance mechanisms to the potential for polymorphism. In particular, the relative importance of seasonal reversals in allelic dominance, where the alleles at a locus alternate in recessivity of their deleterious effects, merits clarification. Here, I analyze the parameter space for the discrete generation two-season multivoltine model and find that biallelic polymorphism is easily maintained owing to an abundance of stabilizing dominance schemes, and moreover, a substantial fraction of these schemes are nonreversing with the season (∼25–50%). In addition, I derive the approximate equilibrium allele frequency cycle under bivoltinism and find that the amplitude of allelic oscillation is maximized by nonreversing dominance if the homozygous fitnesses (per annum) are roughly symmetric. Lastly, I derive conditions for the intralocus evolution of dominance. These predict a long-term trend toward maximally beneficial reversal. Overall, the results counter the disproportionate emphasis placed on dominance reversal as a stabilizing mechanism and clarify that nonreversing dominance is expected to frequently characterize seasonally fluctuating alleles under both weak and strong selection, especially in their early history.

Introduction

The discovery of allelic oscillations in temperate Drosophila melanogaster, where hundreds of biallelic sites spread across the genome display seasonal changes in allele frequency of $\sim$4–20% (Bergland et al. 2014; Machado et al. 2021; see also Rudman et al. 2022; Bitter et al. 2024; Nunez et al. 2024; reviewed in Johnson et al. 2023) has renewed interest in cyclic regimes of natural selection. The simplest model of cyclical selection is the deterministic bivoltine model with nonoverlapping generations and a two-season selection regime. Levins (1968) pointed out the perpetual mismatch of selected traits and seasonal environments in such populations (e.g. the winter generation are the offspring of parents selected for traits favorable in the summer and vice versa) and argued that developmental parameters evolve to mitigate this burden. Abrupt seasonal switches in populations with >2 generations per year also incur a deleterious load, but at least this situation permits within-season adaptive evolution for those generations experiencing a phase of constant selection. The amount of genetic variation in either kind of multivoltine population (defined as >1 gens/year) is determined in part by the prevalence of seasonally antagonistic alleles that are characterized by overdominance with respect to geometric mean fitness. Classic theory on fluctuating selection in diploids established this criterion as a sufficient condition for protected polymorphism (Haldane and Jayakar 1963; Hoekstra 1975). Since cycles of fitness changes are a special class of fluctuating selection (a broader term that includes models with randomly varying fitnesses, e.g. Gillespie 1973), the criterion of geometric mean overdominance carries over as a sufficient condition for protected polymorphism under cyclical selection. This criterion conveys that even in the absence of overdominance within a generation, a time-averaged heterozygote advantage can emerge over longer time spans. For multivoltinism, the focal comparison is the rank order of geometric mean fitnesses over a single year.

More recent work on two-season models (Wittmann et al. 2017; Bertram and Masel 2019) decomposes fitness into season-specific components: selection coefficients (s₁, s₂) and dominance coefficients (h₁, h₂); in particular, this work investigates the contribution of dominance to the stabilization of polymorphism. Wittmann et al. (2017) conclude from their multilocus modeling that beneficial reversal of dominance, where the seasonally maladapted allele is always partially or completely recessive, is a powerful stabilizing mechanism for genome-wide polymorphism under either multiplicative nonepistasis or diminishing returns epistasis. In their single-locus treatment; Bertram and Masel (2019) similarly conclude that dominance reversal stabilizes alleles across a broad range of selection intensity, in contrast to certain cases of nonreversal (e.g. additivity) which were shown to be preferentially stabilizers of strong-effect alleles (with fitness effects on the order of s = 0.1 or greater).

These analyses tended to focus on contrasting special relations for the dominance parameters. Wittmann et al. (2017) assumed symmetric dominance reversal (h₁ = h₂) for the analytical results and much of the simulations (except for their Figure 10). In their analysis of protected polymorphism; Bertram and Masel (2019), except for their equation 4 compared symmetric dominance reversal to the complement relation (h₁ = 1 − h₂, “cumulative overdominance,” Dempster 1955), which includes additivity as a special case. A more general treatment of two-season selection ought to clarify the following: (1) What fraction of season-specific dominance schemes stabilizes allelic variation? (2) What is the relative contribution of dominance reversal and nonreversal to this overall fraction?, and (3) What is the effect of dominance on the magnitude of allele frequency change between seasons? In addition, I investigate stability on a longer time scale, and ask: (4) Subsequent to the establishment of polymorphism, is dominance expected to evolve in a particular direction?

To investigate these questions, I assume a single locus, two allele model with a seasonal fitness cycle in which selection occurs for m generations of season 1, followed by n generations of season 2 (Table 1). I obtain conditions on season-specific dominance for establishing a persistent allelic oscillation and demonstrate that a considerable potential for polymorphism exists across parameter space and moreover, that a sizeable input owes to nonreversing allelic dominance. Additionally, I derive an approximation for the equilibrium allele frequency cycle when m = n = 1 (bivoltinism), as well as conditions for the spread of intralocus dominance modifiers into a resident bivoltine equilibrium.

Table 1.

Fitnesses of diploid genotypes under two-season selection.

	AA	Aa	aa
Season 1 (m gens)	1	$1-{\mathrm{h}}_1{\mathrm{s}}_1$	$1-{\mathrm{s}}_1$
Season 2 (n gens)	$1-{\mathrm{s}}_2$	$1-{\mathrm{h}}_2{\mathrm{s}}_2$	1

Materials and methods

Consider a panmictic multivoltine population of infinite size with discrete nonoverlapping generations subject to the viabilities in Table 1, where the dominance coefficients (h₁, h₂) at a biallelic locus are each bounded by [0,1] and the selection coefficients (s₁, s₂) are each bounded by (0,1). It is known that a sufficient condition for the establishment of protected polymorphism under temporally varying selection is geometric mean overdominance (Haldane and Jayakar 1963). If v($i,j$) represents the viability of diploid genotype i in generation j of season 1 and v(i,k) represents the corresponding fitness in generation k of season 2, the condition for an annual heterozygote advantage under a binary seasonal fitness cycle is equivalent to:

$$\sqrt[m+n]{\left({\displaystyle \prod _{j=1}^m}v(\mathrm{Aa},j)\right)\left({\displaystyle \prod _{k=1}^n}v(\mathrm{Aa},k)\right)}>\sqrt[m+n]{\left({\displaystyle \prod _{j=1}^m}v(\mathrm{AA},j)\right)\left({\displaystyle \prod _{k=1}^n}v(\mathrm{AA},k)\right)},\sqrt[m+n]{\left({\displaystyle \prod _{j=1}^m}v(\mathrm{aa},j)\right)\left({\displaystyle \prod _{k=1}^n}v(\mathrm{aa},k)\right)}$$ (1)

where the rank order of homozygous geometric mean fitnesses is left unspecified.

Reformulating Equation 1 with the season-specific parameters of Table 1 yields

$$(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^m(1-{\mathrm{h}}_2{\mathrm{s}}_2{)}^n>(1-{\mathrm{s}}_1{)}^m,(1-{\mathrm{s}}_2{)}^n.$$ (2)

Note that Table 1, together with the parameter bounds already mentioned, excludes any case of underdominance or within-generation heterozygote advantage. These are presumed to be uncommon contributors to cyclically balanced polymorphism.

Results

Potential for polymorphism

The condition for geometric mean overdominance in Equation 2 can be reformulated by incorporating the ratio of generations per season (r = m/n), which reduces the number of parameters from six to five:

$$(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^r(1-{\mathrm{h}}_2{\mathrm{s}}_2)>(1-{\mathrm{s}}_1{)}^r,1-{\mathrm{s}}_2$$ (3)

and which could also be stated as

$$(1-{\mathrm{h}}_1{\mathrm{s}}_1)(1-{\mathrm{h}}_2{\mathrm{s}}_2{)}^{\frac{1}{r}}>1-{\mathrm{s}}_1,(1-{\mathrm{s}}_2{)}^{\frac{1}{r}}$$ (4)

With the ratio-adjusted geometric mean fitness terms in Equations (3–4), we can define the ratio-adjusted selection coefficients ($\mathrm{\sigma }$)

$${\mathrm{\sigma }}_1=1-(1-{\mathrm{s}}_1{)}^r,{\mathrm{\sigma }}_2=1-(1-{\mathrm{s}}_2{)}^{\frac{1}{r}}.$$ (5)

Given any (r, s₁, s₂), Equation 3 reduces to the stabilizing region $P_r$ on the unit square of dominance coefficients (Appendix A, Fig. 1):

Fig. 1.

Regions of the dominance unit square. The four major patterns of season-specific dominance correspond to a unique quadrant on the unit square of (h₁, h₂) values (legend). The shaded portion conveys the geometrically overdominant region $P_r$, where bold lines and filled points connote inclusion. $P_r$ always intersects 2, 3, or 4 quadrants. a–c) When ${\sigma }_1>{\mathrm{s}}_2$, the upper bound on season 1 dominance is a function of ${\mathrm{h}}_2$ and is given by ${\mathrm{h}}_{1,\max }$ (dashed); the ${\mathrm{h}}_1$-intercept is the ratio of selection coefficients $\frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}$. d–f) When ${\sigma }_1\le {\mathrm{s}}_2$, the upper bound on season 2 dominance is a function of ${\mathrm{h}}_1$ and is given by ${\mathrm{h}}_{2,\max }$ (dashed); the ${\mathrm{h}}_2$ -intercept is $\frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}$. [Examples: a–c) $\mathrm{r}=\frac{13}{2}$. a) ${\mathrm{s}}_1=0.021$, ${\mathrm{s}}_2=0.048$, b) ${\mathrm{s}}_1=0.01$, ${\mathrm{s}}_2=0.048$, c) ${\mathrm{s}}_1=0.122$, ${\mathrm{s}}_2=0.55$. d–f) $r=1$. d) ${\mathrm{s}}_1=0.017$, ${\mathrm{s}}_2=0.048$, e) ${\mathrm{s}}_1=0.035$, ${\mathrm{s}}_2=0.048$, f) ${\mathrm{s}}_1=0.52$, ${\mathrm{s}}_2=0.55$].

$$P_r:=\{\begin{array}{c}\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_2<1\wedge 0\le {\mathrm{h}}_1<{\mathrm{h}}_{1,\max }\}\mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\\ \\ \{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_1<1\wedge 0\le {\mathrm{h}}_2<{\mathrm{h}}_{2,\max }\}\mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\end{array},$$ (6a)

where ${\mathrm{h}}_{1,\max }=\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-{\mathrm{h}}_2{\mathrm{s}}_2}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}$ and ${\mathrm{h}}_{2,\max }=\frac{1-{\left(\frac{1-{\mathrm{s}}_1}{1-{\mathrm{h}}_1{\mathrm{s}}_1}\right)}^r}{{\mathrm{s}}_2}$ are the upper boundaries.

$P_r$ is equivalently represented as

$$P_r:=\{\begin{array}{cc}\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_1<{\displaystyle \frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}\wedge 0\le {\mathrm{h}}_2<{\mathrm{\eta }}_{2,\max }}\right\}& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\\ \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_2<{\displaystyle \frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}\wedge 0\le {\mathrm{h}}_1<{\mathrm{\eta }}_{1,\max }}\right\}& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\end{array},$$ (6b)

where ${\mathrm{\eta }}_{2,\max }=\frac{1-\frac{1-{\mathrm{s}}_2}{{(1-{\mathrm{h}}_1{\mathrm{s}}_1)}^r}}{{\mathrm{s}}_2}$ and ${\mathrm{\eta }}_{1,\max }=\frac{1-\frac{1-{\mathrm{s}}_1}{{(1-{\mathrm{h}}_2{\mathrm{s}}_2)}^{\frac{1}{r}}}}{{\mathrm{s}}_1}$.

The area of the stabilizing region corresponds to the probability that random uniform sampling of (h₁, h₂) from [0, 1]² results in a protected polymorphism. This is the “potential for polymorphism” (Asmussen and Basnayake 1990; Trotter and Spencer 2007), conditional on biallelism, and is given by the piecewise formula,

$$\mathrm{Area}(P_r)=\{\begin{array}{cc}{\displaystyle \underset{0}{\overset{1}{\int }}}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\\ {\displaystyle \underset{0}{\overset{1}{\int }}}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\end{array}$$ (7a)

$$=\{\begin{array}{cc}\{\begin{array}{c}{\displaystyle \frac{(1-{\mathrm{s}}_2)\ln (1-{\mathrm{s}}_2)+{\mathrm{s}}_2}{{\mathrm{s}}_1{\mathrm{s}}_2}\mathrm{if}r=1}\\ {\displaystyle \frac{r\left(1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}\right)-{\mathrm{s}}_2}{(r-1){\mathrm{s}}_1{\mathrm{s}}_2}\mathrm{if}r\ne 1}\end{array}& \mathrm{and}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\\ \{\begin{array}{c}{\displaystyle \frac{(1-{\mathrm{s}}_1)\ln (1-{\mathrm{s}}_1)+{\mathrm{s}}_1}{{\mathrm{s}}_1{\mathrm{s}}_2}\mathrm{if}r=1}\\ {\displaystyle \frac{{(1-{\mathrm{s}}_1)}^r+r{\mathrm{s}}_1-1}{(r-1){\mathrm{s}}_1{\mathrm{s}}_2}\mathrm{if}r\ne 1}\end{array}& \mathrm{and}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\end{array}.$$ (7b)

The main factors determining the size of the stabilizing region are the ratios $\frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}$ and $\frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}$. When ${\mathrm{\sigma }}_1>{\mathrm{s}}_2$, the h₂-intercept of ${\mathrm{h}}_{1,\max }$ is $\frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}$; and if ${\mathrm{\sigma }}_1\le {\mathrm{s}}_2$, then the h₁-intercept of ${\mathrm{h}}_{2,\max }$ is $\frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}$. These boundaries enclose $P_r$ on the unit square of dominance coefficients and so an increase in the intercept expands the $P_r$ region; when the geometric mean fitnesses of the homozygotes are equal, the intercepts are maximized at 1. An additional factor that influences the size of $P_r$ is the concavity of these boundaries; an intensification of the concavity for larger s₁, s₂ expands $P_r$, and appreciably so under strong selection.

In Fig. 2a–c, the size of the stabilizing region is plotted for: (a) $P_{1/3}$, (b) $P_1$, (c) $P_{13/2}$. With r = 1 (symmetric multivoltinism), it is only under a severe asymmetry of the selection coefficients that $\mathrm{Area}(P_1)$ falls to negligible values (Fig. 2b). For ∼80% of $({\mathrm{s}}_1,{\mathrm{s}}_2)\in [0,1{]}^2$, the potential for polymorphism is 0.1 or greater; this is also true when restricting focus to regimes of weak selection (e.g. $0<{\mathrm{s}}_1,{\mathrm{s}}_2<{10}^{-3}$) and moderate selection (${10}^{-3}<{\mathrm{s}}_1,{\mathrm{s}}_2<{10}^{-1}$) (Fig. 2d). $\mathrm{Area}(P_1)$ is maximized when the selection coefficients are symmetric, which results in $\mathrm{Area}(P_1)\approx 0.5$ under weak-to-moderate selection. Only under rather large selection coefficients is $\mathrm{Area}(P_1)$ ever much greater than one-half, e.g. if ${\mathrm{s}}_1={\mathrm{s}}_2=s$, $\mathrm{Area}(P_1)\ge 0.6$ only if $s>0.45$.

Fig. 2.

The potential for seasonally balanced polymorphism. a–c) Contours plot the size of the region P_r. Each point corresponds to the fraction of all possible (h₁, h₂) schemes that bring about geometric mean overdominance, i.e. Area(P_r). Bold lines correspond to fitness equality of the homozygotes: $(1-{\mathrm{s}}_1{)}^r=1-{\mathrm{s}}_2$. a–c, bottom row) For small selection coefficients, the range of areas is restricted to the approximate interval (0, 0.5). a) r = 1/3, b) r = 1, c) r = 13/2. d) Fraction of selection schemes for which the potential for polymorphism is >0.1 under weak selection (s₁,s₂ < 0.001, dashed) and general selection (solid). Generation ratios (r) are chosen from the interval [1/3, 50].

With asymmetric multivoltinism (Fig. 2a and c), there is an overall reduction in the size of the stabilizing region as compared to r = 1. Nevertheless, a considerable potential for polymorphism exists over a substantial fraction of selection coefficient pairs. For example, $\mathrm{Area}(P_{1/3})>0.1$ is true for 72% of all $({\mathrm{s}}_1,{\mathrm{s}}_2)\in [0,1{]}^2$, and for 67–68% of all weak and moderate regimes. With r = 13/2, the corresponding proportions for which $\mathrm{Area}(P_{13/2})>0.1$ are 56% (general selection) and 37–39% (weak, moderate). Even for a biologically extreme value such as r = 50, $\mathrm{Area}(P_{50})>0.1$ is true for 12% of general $({\mathrm{s}}_1,{\mathrm{s}}_2)$ and 4–5% of weak and moderate regimes (Fig. 2d). Like the symmetric case, $\mathrm{Area}(P_r)$ is maximized when the geometric mean fitnesses of the homozygotes are equal, for which $\mathrm{Area}(P_r)\approx 0.5$ in the weak-to-moderate regime.

The contribution of dominance categories to the stabilizing potential

$\mathrm{Area}(P_r)$ is a measure of the overall abundance of parameter sets that are characterized by an annual heterozygote advantage. The relative contributions ($R_C$) of the four major categories of season-specific dominance are proportional to the subregion areas resulting from the intersection of $P_r$ with each of the quadrants on the unit square (Fig. 1). In particular, the formulas for $R_C$ are determined by whether the stabilizing region intersects 2, 3, or 4 quadrants (Fig. 3, Appendix A), where $R_C=\mathrm{Area}(P_r\cap C)/\mathrm{Area}(P_r)$ for dominance category C, and

Fig. 3.

Qualitative distribution of stabilizing dominance. Region plots indicate which dominance categories (beneficial reversal, b; A-allele dominance, A; a-allele dominance, a; deleterious reversal, d) contribute to the potential for seasonally balanced polymorphism, evaluated for selection scheme (s₁, s₂) and the generation ratio r (plot label). (Top row) General selection coefficients. (Bottom row) Small selection coefficients. The space of selection schemes for which all four categories (b, A, a, and d) contribute to stable polymorphism tends to the line ${\mathrm{s}}_2=r{\mathrm{s}}_1$ in the limit of weak selection.

$$\begin{array}{rl}C\in & \{\left\{({\mathrm{h}}_1,{\mathrm{h}}_2):0\le {\mathrm{h}}_1<{\displaystyle \frac{1}{2}\mathrm{and}{\displaystyle \frac{1}{2}<{\mathrm{h}}_2\le 1}}\right\},\left\{({\mathrm{h}}_1,{\mathrm{h}}_2):{\displaystyle \frac{1}{2}<{\mathrm{h}}_1\le 1\mathrm{and}0\le {\mathrm{h}}_2<{\displaystyle \frac{1}{2}}}\right\},\left\{({\mathrm{h}}_1,{\mathrm{h}}_2):0\le {\mathrm{h}}_1<{\displaystyle \frac{1}{2}\mathrm{and}0\le {\mathrm{h}}_2<{\displaystyle \frac{1}{2}}}\right\},\\ & \left\{({\mathrm{h}}_1,{\mathrm{h}}_2):{\displaystyle \frac{1}{2}<{\mathrm{h}}_1\le 1\mathrm{and}{\displaystyle \frac{1}{2}<{\mathrm{h}}_2\le 1}}\right\}\}\end{array}$$ (8)

These entries are, respectively, A-allele dominance (A), a-allele dominance (a), beneficial reversal of dominance (b), and deleterious reversal of dominance (d). The formulas for the intersection areas, $\mathrm{Area}(P_r\cap C)$, follow straightforwardly from the geometric picture of the stabilizing region (Fig. 1) and are given below:

Case i: $P_r$ intersects only two quadrants: b and A, or b and a

$$\mathrm{Area}(P_r\cap A)=\{\begin{array}{cc}{\displaystyle \underset{\frac{1}{2}}{\overset{1}{\int }}}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}\le {\displaystyle \frac{1}{2}}}\\ 0& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}\le {\displaystyle \frac{1}{2}}}\end{array}$$ (9a)

$$\mathrm{Area}(P_r\cap a)=\{\begin{array}{cc}0& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}\le {\displaystyle \frac{1}{2}}}\\ {\displaystyle \underset{\frac{1}{2}}{\overset{1}{\int }}}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}\le {\displaystyle \frac{1}{2}}}\end{array}$$ (9b)

$$\mathrm{Area}(P_r\cap b)=\{\begin{array}{cc}\mathrm{Area}(P_r)-\mathrm{Area}(P_r\cap A)& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}\le {\displaystyle \frac{1}{2}}}\\ \mathrm{Area}(P_r)-\mathrm{Area}(P_r\cap a)& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}\le {\displaystyle \frac{1}{2}}}\end{array}$$ (9c)

$$\mathrm{Area}(P_r\cap d)=0$$ (9d)

Case ii: $P_r$ intersects only three quadrants: b, A, and a

$$\mathrm{Area}(P_r\cap A)=\{\begin{array}{cc}{\displaystyle \underset{\frac{1}{2}}{\overset{1}{\int }}}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}>{\displaystyle \frac{1}{2}\mathrm{and}{\mathrm{h}}_{1,\max }\left({\displaystyle \frac{1}{2}}\right)\le {\displaystyle \frac{1}{2}}}}\\ {\displaystyle \underset{0}{\overset{{\mathrm{\eta }}_{1,\max }\left(\frac{1}{2}\right)}{\int }}}({\mathrm{h}}_{2,\max }-1/2)\mathrm{\partial }{\mathrm{h}}_1& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}>{\displaystyle \frac{1}{2}\mathrm{and}{\mathrm{h}}_{2,\max }\left({\displaystyle \frac{1}{2}}\right)\le {\displaystyle \frac{1}{2}}}}\end{array}$$ (10a)

$$\begin{array}{rl}& \mathrm{Area}(P_r\cap a)=\{\begin{array}{cc}{\displaystyle \underset{0}{\overset{{\mathrm{\eta }}_{2,\max }\left(\frac{1}{2}\right)}{\int }}}({\mathrm{h}}_{1,\max }-1/2)\mathrm{\partial }{\mathrm{h}}_2& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}>{\displaystyle \frac{1}{2}\mathrm{and}{\mathrm{h}}_{1,\max }\left({\displaystyle \frac{1}{2}}\right)\le {\displaystyle \frac{1}{2}}}}\\ {\displaystyle \underset{\frac{1}{2}}{\overset{1}{\int }}}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\displaystyle \frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}>{\displaystyle \frac{1}{2}\mathrm{and}{\mathrm{h}}_{2,\max }\left({\displaystyle \frac{1}{2}}\right)\le {\displaystyle \frac{1}{2}}}}\end{array}\\ & \end{array}$$ (10b)

$$\mathrm{Area}(P_r\cap b)=\mathrm{Area}(P_r)-\mathrm{Area}(P_r\cap A)-\mathrm{Area}(P_r\cap a)$$ (10c)

$$\mathrm{Area}(P_r\cap d)=0$$ (10d)

Case iii: $P_r$ intersects all four quadrants

$$\mathrm{Area}(P_r\cap A)=\{\begin{array}{cc}{\displaystyle \underset{0}{\overset{\frac{1}{2}}{\int }}}{\mathrm{\eta }}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1-{\displaystyle \frac{1}{4}}& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\mathrm{h}}_{1,\max }\left({\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{1}{2}}\\ {\displaystyle \underset{0}{\overset{\frac{1}{2}}{\int }}}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1-{\displaystyle \frac{1}{4}}& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\mathrm{h}}_{2,\max }\left({\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{1}{2}}\end{array}$$ (11a)

$$\mathrm{Area}(P_r\cap a)=\{\begin{array}{cc}{\displaystyle \underset{0}{\overset{\frac{1}{2}}{\int }}}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2-{\displaystyle \frac{1}{4}}& \mathrm{if}{\mathrm{\sigma }}_1>{\mathrm{s}}_2\mathrm{and}{\mathrm{h}}_{1,\max }\left({\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{1}{2}}\\ {\displaystyle \underset{0}{\overset{\frac{1}{2}}{\int }}}{\mathrm{\eta }}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2-{\displaystyle \frac{1}{4}}& \mathrm{if}{\mathrm{\sigma }}_1\le {\mathrm{s}}_2\mathrm{and}{\mathrm{h}}_{2,\max }\left({\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{1}{2}}\end{array}$$ (11b)

$$\mathrm{Area}(P_r\cap b)={\displaystyle \frac{1}{4}}$$ (11c)

$$\begin{array}{rl}\mathrm{Area}(P_r\cap d)& =\mathrm{Area}(P_r)-\mathrm{Area}(P_r\cap A)-\mathrm{Area}(P_r\cap a)\\ & -\mathrm{Area}(P_r\cap b),\end{array}$$ (11d)

where ${\mathrm{h}}_{1,\max }\left(\frac{1}{2}\right)=\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}$, ${\mathrm{\eta }}_{1,\max }\left(\frac{1}{2}\right)=\frac{1-\frac{1-{\mathrm{s}}_1}{{\left(1-\frac{{\mathrm{s}}_2}{2}\right)}^{\frac{1}{r}}}}{{\mathrm{s}}_1},{\mathrm{h}}_{2,\max }\left(\frac{1}{2}\right)=$  $\frac{1-{\left(\frac{1-{\mathrm{s}}_1}{1-\frac{{\mathrm{s}}_1}{2}}\right)}^r}{{\mathrm{s}}_2}$, and ${\mathrm{\eta }}_{2,\max }\left(\frac{1}{2}\right)=\frac{1-\frac{1-{\mathrm{s}}_2}{{\left(1-\frac{{\mathrm{s}}_1}{2}\right)}^r}}{{\mathrm{s}}_2}$. Cases i–iii are exhaustive (Appendix A). See Tables 2 and 3 for the evaluated form of the integrals for $r=1$ and $r\ne 1$.

Table 2.

Areas of P₁ subregions (symmetric multivoltinism, $r=1$).

Integral	Evaluated expression
$\underset{\frac{1}{2}}{\overset{1}{\int }}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2$	$\frac{{\mathrm{s}}_2+2{\mathrm{s}}_2\ln \left(\frac{2-{\mathrm{s}}_2}{2(1-{\mathrm{s}}_2)}\right)-4\cot {\mathrm{h}}^{-1}(3-2{\mathrm{s}}_2)+\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{\frac{1}{2}}{\overset{1}{\int }}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1$	$\frac{{\mathrm{s}}_1+2{\mathrm{s}}_1\ln \left(\frac{2-{\mathrm{s}}_1}{2(1-{\mathrm{s}}_1)}\right)-4\cot {\mathrm{h}}^{-1}(3-2{\mathrm{s}}_1)+\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{{\mathrm{\eta }}_{1,\max }\left(\frac{1}{2}\right)}{\int }}\left({\mathrm{h}}_{2,\max }-\frac{1}{2}\right)\mathrm{\partial }{\mathrm{h}}_1$	$\frac{2{\mathrm{s}}_1-{\mathrm{s}}_2+2\ln \left(\frac{1-{\mathrm{s}}_1}{2-{\mathrm{s}}_2}\right)+2{\mathrm{s}}_1\ln \left(\frac{2-{\mathrm{s}}_2}{2(1-{\mathrm{s}}_1)}\right)+\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{{\mathrm{\eta }}_{2,\max }\left(\frac{1}{2}\right)}{\int }}\left({\mathrm{h}}_{1,\max }-\frac{1}{2}\right)\mathrm{\partial }{\mathrm{h}}_2$	$\frac{2{\mathrm{s}}_2-{\mathrm{s}}_1+2\ln \left(\frac{1-{\mathrm{s}}_2}{2-{\mathrm{s}}_1}\right)+2{\mathrm{s}}_2\ln \left(\frac{2-{\mathrm{s}}_1}{2(1-{\mathrm{s}}_2)}\right)+\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{\eta }}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1$	$\frac{{\mathrm{s}}_1+2\ln (2-{\mathrm{s}}_1)+2{\mathrm{s}}_2\ln \left(\frac{2}{2-{\mathrm{s}}_1}\right)-\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1$	$\frac{{\mathrm{s}}_1+2\ln (2-{\mathrm{s}}_1)+2{\mathrm{s}}_1\ln \left(\frac{2}{2-{\mathrm{s}}_1}\right)-\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2$	$\frac{{\mathrm{s}}_2+2\ln (2-{\mathrm{s}}_2)+2{\mathrm{s}}_2\ln \left(\frac{2}{2-{\mathrm{s}}_2}\right)-\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{\eta }}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2$	$\frac{{\mathrm{s}}_2+2\ln (2-{\mathrm{s}}_2)+2{\mathrm{s}}_1\ln \left(\frac{2}{2-{\mathrm{s}}_2}\right)-\ln (4)}{2{\mathrm{s}}_1{\mathrm{s}}_2}$

Table 3.

Areas of P_r subregions under asymmetric multivoltinism $(r\ne 1).$

Integral	Evaluated expression
$\underset{\frac{1}{2}}{\overset{1}{\int }}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2$	$\frac{r({\mathrm{s}}_2-2)\left({\left(\frac{2}{{\mathrm{s}}_2-2}+2\right)}^{\frac{1}{r}}-1\right)-{\mathrm{s}}_2}{2(r-1){\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{\frac{1}{2}}{\overset{1}{\int }}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1$	$\frac{r{\mathrm{s}}_1+{\mathrm{s}}_1-2-\left(({\mathrm{s}}_1-2){\left(\frac{2}{{\mathrm{s}}_1-2}+2\right)}^r\right)}{2(r-1){\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{{\mathrm{\eta }}_{1,\max }\left(\frac{1}{2}\right)}{\int }}\left({\mathrm{h}}_{2,\max }-\frac{1}{2}\right)\mathrm{\partial }{\mathrm{h}}_1$	$\frac{2{(1-{\mathrm{s}}_1)}^r+r(2-{\mathrm{s}}_2)\left(({\mathrm{s}}_1-1){\left(1-\frac{{\mathrm{s}}_2}{2}\right)}^{-1/r}+1\right)+{\mathrm{s}}_2-2}{2(r-1){\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{{\mathrm{\eta }}_{2,\max }\left(\frac{1}{2}\right)}{\int }}\left({\mathrm{h}}_{1,\max }-\frac{1}{2}\right)\mathrm{\partial }{\mathrm{h}}_2$	$\frac{(2-{\mathrm{s}}_1)(1-{\mathrm{s}}_2){\left(1-\frac{{\mathrm{s}}_1}{2}\right)}^{-r}-r\left(2{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}+{\mathrm{s}}_1-2\right)+{\mathrm{s}}_1-2}{2(r-1){\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{\eta }}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1$	$\frac{{\left(1-\frac{{\mathrm{s}}_1}{2}\right)}^{-r}\left({\left(1-\frac{{\mathrm{s}}_1}{2}\right)}^r((r-1){\mathrm{s}}_1-2{\mathrm{s}}_2+2)-(2-{\mathrm{s}}_1)(1-{\mathrm{s}}_2)\right)}{2(r-1){\mathrm{s}}_1{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{h}}_{2,\max }\mathrm{\partial }{\mathrm{h}}_1$	$\frac{1-\frac{(2^r{(2-{\mathrm{s}}_1)}^{1-r}-2){(1-{\mathrm{s}}_1)}^r}{(r-1){\mathrm{s}}_1}}{2{\mathrm{s}}_2}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{h}}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2$	$\frac{1-\frac{\left(2r{(2-{\mathrm{s}}_2)}^{\frac{1}{r}}+2^{\frac{1}{r}}r({\mathrm{s}}_2-2)\right){\left(\frac{{\mathrm{s}}_2-2}{{\mathrm{s}}_2-1}\right)}^{-1/r}}{(r-1){\mathrm{s}}_2}}{2{\mathrm{s}}_1}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}{\mathrm{\eta }}_{1,\max }\mathrm{\partial }{\mathrm{h}}_2$	$\frac{{\left(1-\frac{{\mathrm{s}}_2}{2}\right)}^{-1/r}\left((r(2{\mathrm{s}}_1+{\mathrm{s}}_2-2)-{\mathrm{s}}_2){\left(1-\frac{{\mathrm{s}}_2}{2}\right)}^{\frac{1}{r}}+r({\mathrm{s}}_1-1)({\mathrm{s}}_2-2)\right)}{2(r-1){\mathrm{s}}_1{\mathrm{s}}_2}$

As a general principle, the relative contribution of nonreversing dominance, i.e. $\frac{\mathrm{Area}(P_r\cap (A\cup a))}{\mathrm{Area}(P_r)}=\frac{\mathrm{Area}(P_r\cap A)}{\mathrm{Area}(P_r)}+\frac{\mathrm{Area}(P_r\cap a)}{\mathrm{Area}(P_r)}=R_A+R_a$, is at least one-quarter of the overall potential for polymorphism; symmetry of the homozygous annual fitnesses maximizes the nonreversing contribution at ∼50% (Fig. 4a). The nonreversing input to the polymorphic potential owes completely to the more-fit allele whenever the ratio of selection coefficients, $\frac{{\mathrm{\sigma }}_2}{{\mathrm{s}}_1}$ (if ${\mathrm{\sigma }}_1>{\mathrm{s}}_2$) or $\frac{{\mathrm{\sigma }}_1}{{\mathrm{s}}_2}$ (if ${\mathrm{\sigma }}_1\le {\mathrm{s}}_2$), falls below one-half (Fig. 4b). The reversing contribution is overwhelmingly comprised of the beneficial reversal class (Fig. 4c).

Fig. 4.

Quantitative distribution of stabilizing dominance. a) Fraction of P_r that lies in nonreversing quadrants; the minimum is 0.25. (below) For small selection coefficients, values fall within the approximate interval (0.25, 0.5). b) Fraction of P_r in the a-dominance quadrant, relative to the fraction of P_r lying in nonreversing quadrants. c) Fraction of P_r in the beneficial reversal quadrant, relative to the fraction of P_r lying in reversing quadrants.

Allele frequency oscillation under bivoltinism

To investigate whether seasonal dominance broadly exhibits allelic oscillation, let us now consider the special case of bivoltinism, m = n = 1. Assuming discrete nonoverlapping generations, the deterministic evolution of allele frequencies in a bivoltine population is described by the equations in Appendix B.

For small selection coefficients, the relative annual fitness of the a-allele is approximately equal to

$${\mathrm{w}}_{\mathrm{a}}\approx 1+{\mathrm{s}}_1({\mathrm{h}}_1(2\mathrm{q}-1)-\mathrm{q})+{\mathrm{s}}_2({\mathrm{h}}_2(2\mathrm{q}-1)-\mathrm{q}+1),$$ (12)

where q is the a-frequency (see Equations A2.7–8 in Appendix B). Solving for ${\mathrm{w}}_{\mathrm{a}}$ = 1, I obtain the approximate equilibrium a-frequency at the start of season 1 ($\widehat{{\mathrm{q}}_1}$, Supplementary Fig. 1a–c):

$$\widehat{{\mathrm{q}}_1}={\displaystyle \frac{{\mathrm{h}}_1{\mathrm{s}}_1-(1-{\mathrm{h}}_2){\mathrm{s}}_2}{2{\mathrm{h}}_1{\mathrm{s}}_1-{\mathrm{s}}_1+2{\mathrm{h}}_2{\mathrm{s}}_2-{\mathrm{s}}_2}.}$$ (13)

As a matter of fact, $\widehat{{\mathrm{q}}_1}$ is the classical equilibrium for heterozygote advantage (Fisher 1922) as applied to the annual fitnesses and is equivalent to the expectation under two-trait antagonistic pleiotropy, in which the traits interact additively with respect to viability (Rose 1982). This is sensible since the assumption of small selection coefficients flattens the distinction between arithmetic and geometric mean fitnesses. A unique feature of seasonally balanced selection that transcends constant-selection models of heterozygote advantage is the allele frequency cycle owing to the alternation of fitnesses. The approximate magnitude of this oscillation can be derived by first solving for the a-allele frequency at the start of season 2,

$$\widehat{{\mathrm{q}}_2}={\displaystyle \frac{({\mathrm{h}}_1{\mathrm{s}}_1+({\mathrm{h}}_2-1){\mathrm{s}}_2)({\mathrm{s}}_1({\mathrm{h}}_1({\mathrm{h}}_1{\mathrm{s}}_1-2)+1)+{\mathrm{s}}_2({\mathrm{h}}_2(({\mathrm{h}}_1+1){\mathrm{s}}_1-2)-{\mathrm{s}}_1+1))}{(2{\mathrm{h}}_1-1){\mathrm{s}}_1^2({\mathrm{h}}_1({\mathrm{h}}_1{\mathrm{s}}_1-2)+1)+2(2{\mathrm{h}}_2-1){\mathrm{s}}_2{\mathrm{s}}_1({\mathrm{h}}_1({\mathrm{h}}_1{\mathrm{s}}_1-2)+1)+{\mathrm{s}}_2^2(({\mathrm{h}}_2-1)((2{\mathrm{h}}_1+1){\mathrm{h}}_2-1){\mathrm{s}}_1-{(1-2{\mathrm{h}}_2)}^2)}.}$$ (14)

Then the change in allele frequencies between seasons provides the oscillation:

$$\mathrm{\Delta }\hat{\mathrm{q}}=\widehat{{\mathrm{q}}_2}-\widehat{{\mathrm{q}}_1}.$$ (15)

Without loss of generality, consider the case ${\mathrm{s}}_1\le {\mathrm{s}}_2$. If the homozygous fitnesses are nearly equal, $|\mathrm{\Delta }\hat{\mathrm{q}}|$ is maximized in the nonreversing quadrant associated with dominance of the less-fit allele (Fig. 5a). Grossly asymmetric selection coefficients result in $|\mathrm{\Delta }\hat{\mathrm{q}}|$ being maximized by beneficial reversal (Fig. 5c). This pattern also holds under strong selection (Fig. 5d–f). Allele frequencies at equilibrium imply considerable levels of heterozygosity unless selection coefficients are especially disparate or the dominance scheme lies too close to the maximal boundary of the P₁ region (Supplementary Fig. 1).

Fig. 5.

Amplitude of allelic oscillation in a bivoltine equilibrium. a–f) Each point in the contour plots indicates the absolute value of the change in allele frequency between seasons 1 and 2 during the equilibrial cycle. a–c) Weak selection plots, using Equation 15. a) s₁ = 0.00475, s₂ = 0.00525, b) s₁ = 0.00425, s₂ = 0.00575, c) s₁ = 0.003, s₂ = 0.007. d–f) Strong selection plots, using numerical iteration. Allelic oscillation magnitudes were obtained for 2,101 parameter sets sampled from the region P₁. Recursions for the single-locus biallelic model were iterated for 10⁴ generations starting from an initial q = 0.5. d) s₁ = 0.32, s₂ = 0.35, e) s₁ = 0.26, s₂ = 0.35, f) s₁ = 0.15, s₂ = 0.35.

Evolution of dominance

Upon establishment of the bivoltine equilibrium cycle ($\widehat{{\mathrm{q}}_1},\widehat{{\mathrm{q}}_2})$, alleles outside of the set {A, a} may subsequently evolve by virtue of their effects on fitness. Consider a rare mutant (a_mut) that is descended from the a-allele but which has a modified season-specific dominance $({\mathrm{h}}_1^{*}$, ${\mathrm{h}}_2^{*})$ as compared to the resident population $({\mathrm{h}}_1$, ${\mathrm{h}}_2)$; selection coefficients remain unaltered (Table 4). The equations for the 3-allele system {A, a, a_mut} are provided in Appendix C, where p, q, and q_mut are the respective frequencies.

Table 4.

Fitnesses of diploid genotypes including dominance mutants of the a-allele.

	AA	Aa	Aamut	aa, aamut, amutamut
Season 1	1	$1-{\mathrm{h}}_1{\mathrm{s}}_1$	$1-{\mathrm{h}}_1^{*}{\mathrm{s}}_1$	$1-{\mathrm{s}}_1$
Season 2	$1-{\mathrm{s}}_2$	$1-{\mathrm{h}}_2{\mathrm{s}}_2$	$1-{\mathrm{h}}_2^{*}{\mathrm{s}}_2$	1

For small selection coefficients, the annual fitness of a_mut upon introduction into the population is (Appendix C):

$$\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})\approx 1+(\mathrm{q}-1)({\mathrm{h}}_1^{*}-2{\mathrm{h}}_1\mathrm{q}+\mathrm{q}){\mathrm{s}}_1+(\mathrm{q}-1)({\mathrm{h}}_2^{*}-2{\mathrm{h}}_2\mathrm{q}+\mathrm{q}-1){\mathrm{s}}_2.$$ (16)

Assuming ${\mathrm{s}}_1\le {\mathrm{s}}_2$, I consider three types of mutant effects and state the required conditions for allelic invasion into the resident equilibrium (using the Reduce function in Mathematica):

At $\mathrm{q}=\widehat{{\mathrm{q}}_1}$ and ${\mathrm{q}}_{\mathrm{mut}}=0,\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})>1$ if

Season-1-limited modification, $({\mathrm{h}}_1^{*},{\mathrm{h}}_2^{*})$ = $({\mathrm{h}}_1^{*},{\mathrm{h}}_2)$:

$${\mathrm{h}}_1^{*}<{\mathrm{h}}_1$$ (17)

Season-2-limited modification, $({\mathrm{h}}_1^{*},{\mathrm{h}}_2^{*})$ = $({\mathrm{h}}_1,{\mathrm{h}}_2^{*})$:

$${\mathrm{h}}_2^{*}<{\mathrm{h}}_2$$ (18)

Season-specific modification $({\mathrm{h}}_1^{*},{\mathrm{h}}_2^{*})$:

$${\mathrm{h}}_1^{*}<{\mathrm{h}}_1+{\displaystyle \frac{{\mathrm{s}}_2}{{\mathrm{s}}_1}{\mathrm{h}}_2\mathrm{and}{\mathrm{h}}_2^{*}<{\mathrm{h}}_2-{\displaystyle \frac{{\mathrm{s}}_1}{{\mathrm{s}}_2}({\mathrm{h}}_1^{*}-{\mathrm{h}}_1)}}$$ (19a)

Or equivalently:

$${\mathrm{h}}_2^{*}<{\mathrm{h}}_2+{\displaystyle \frac{{\mathrm{s}}_1}{{\mathrm{s}}_2}{\mathrm{h}}_1\mathrm{and}{\mathrm{h}}_1^{*}<{\mathrm{h}}_1-{\displaystyle \frac{{\mathrm{s}}_2}{{\mathrm{s}}_1}({\mathrm{h}}_2^{*}-{\mathrm{h}}_2)}}$$ (19b)

An increase in one season's dominance coefficient requires a decrease in that of the other season. In addition to oppositely directed changes, season-specific modification promotes reductions of dominance in both seasons.

It can be shown that the invasion of ${\mathrm{a}}_{\mathrm{mut}}$ implies the establishment of a new biallelic equilibrium composed of {A, ${\mathrm{a}}_{\mathrm{mut}}$} (Appendix C). The value of $\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})$ in an arbitrary resident equilibrium ($\mathrm{q}=\widehat{{\mathrm{q}}_1},{\mathrm{q}}_{\mathrm{mut}}=0$) is maximized by (${\mathrm{h}}_1^{*}=0,{\mathrm{h}}_2^{*}=0$), and so complete beneficial dominance reversal is a global evolutionary stable strategy, i.e. stable to the invasion of any possible mutants.

The same invasion and ESS conditions as above are obtained when considering the 3-allele system {A, a, A_mut} for the dynamics of an A-descended mutant (File S1). A notable difference between the two classes of mutants is that the invasion fitness of A_mut exceeds that of a_mut for identical dominance modifications, which ultimately owes to the greater strength of selection on A_mut in season 2 (due to assuming ${\mathrm{s}}_1\le {\mathrm{s}}_2$) (Fig. 6; note the contour legends). Overall, invading modifiers typically have effective selection coefficients [$\mathrm{s}({\mathrm{a}}_{\mathrm{mut}})=\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})-1]$ that are on the order of 10⁻² to 10⁰ times the magnitudes of s₁, s₂ (Fig. 6).

Fig. 6.

Examples of the selection strength on dominance modifiers. a–f) Contours plot the value W(a_mut) (top row) or W(A_mut) (bottom row) for rare mutants as a function of $({\mathrm{h}}_1^{*},{\mathrm{h}}_2^{*})$, assuming s₁ = 0.004, s₂ = 0.005. Resident values of (h₁, h₂) are depicted by pink points. The region P₁ is outlined in dashed boundaries. Note the distinct color scales. a, d) (h₁, h₂) = (0.35, 0.45), b, e) (h₁, h₂) = (0.05, 0.75), c, f) (h₁, h₂) = (0.75, 0.05).

Discussion

Having analyzed the size of geometrically overdominant parameter space, as well as its distribution over the major categories of season-specific dominance, I conclude the following: (1) seasonal fitness cycles are broadly favorable for the maintenance of single-locus polymorphism and (2) a good deal of the stabilizing potential owes to nonreversing allelic dominance. These conclusions apply for any strength of selection and counter the notion that the conditions required for temporally varying selection to act as a balancing mechanism are stringent (Dempster 1955). Moreover, these results challenge the role of dominance reversal as the primary stabilizing force acting on newly arisen allelic oscillations. Random sampling of dominance parameters readily results in an annual heterozygote advantage (with 10–50% probability) and among these stabilizing schemes one-quarter to one-half involve constant directions of dominance. Add to these conclusions the presumptive likelihood that nonreversals originate via mutation at a higher rate than reversals, then such considerations combine to elevate the role that constant-directional dominance is expected to play in maintaining newly established allelic variation (but see Karageorgi et al. 2024).

The bivoltine model of allele frequency change yielded additional insights into the characteristics of seasonally fluctuating equilibria. The model suggests that the evolution of large fluctuations is permitted broadly across most dominance schemes (Fig. 5d–f), with the main limiting factor being the magnitude of seasonal fitness changes (s₁, s₂) and, secondarily, their symmetry: values near parity result in the largest possible swings. While oscillation is maximized by constant dominance of the less-fit allele, it remains the case that much of the polymorphic region is associated with sizeable oscillation. Together with the invasion analysis of intralocus modifiers, the dynamical results suggest the following scenario for the history of seasonally balanced polymorphisms: Given an initially monomorphic locus, mutations with seasonally antagonistic effects (relative to the wild-type allele) originate in the population, and with the previously mentioned presumption about mutation rates, invading alleles will tend to display constant-directional dominance with a nonconstant (partial) magnitude (i.e. one of the nonreversing quadrants of Fig. 1). Over time, allele-specific expression changes accumulate via divergence in the cis-regulatory regions of the two alleles, resulting in various possible outcomes including transient reversals in constant dominance as well as changes that bring the dominance pattern closer to its ultimate and optimal fate: complete beneficial reversal.

In contrast to recent work (Wittmann et al. 2017; Bertram and Masel 2019; Nabutanyi and Wittmann 2021, present article), the classical treatments of deterministic cyclical selection (Haldane and Jayakar 1963; Hoekstra 1975) avoided the decomposition of heterozygous fitness into dominance and selection parameters in their formulation of the model. Sufficient conditions giving rise to protected polymorphism (Levene 1953; Prout 1968) were found to be neatly expressible in terms of relative viabilities (Haldane and Jayakar 1963; Hoekstra 1975; Ewing 1977; 1979; Reinhold 1999; 2000), but ultimately such a formulation has limited informativeness with regard to questions about the genetic architecture of seasonal fitness and how this architecture is modified over time. After all, these matters depend partly on an understanding of how allelic dominance shapes fitness and so reformulating the geometric mean overdominance criterion in terms of these parameters is essential (Wittmann et al. 2017; Bertram and Masel 2019; Gillespie and Langley 1974; Nagylaki 1975).

The present results emphasize that the genetic basis of seasonal evolution is predicted to owe to a mixture of effect sizes and modes of gene action, and so no special relation or category of dominance, be it cumulative overdominance or beneficial reversal, is expected to constitute an exclusive role in stabilizing allelic variation of any oscillation magnitude (contra Bertram and Masel 2019). Rather, seasonally balanced polymorphism is predicted to involve heterozygous genotypes that are characterized by a distribution of season-specific dominance coefficients (e.g. Equations 9–11 under the assumption of uniform mutation rates). It should be noted that it is important to distinguish between models of dominance with respect to fitness and models of dominance with respect to phenotype. For example, additive interactions (a case of nonreversal) within and among loci with respect to a phenotypic score may ultimately translate to dominance reversal with respect to fitness, as depends on the shape of the function mapping phenotype to fitness (Flintham et al 2024; see also Wittmann et al. 2017, Supplementary Fig. 2 therein). The theory merits elaboration for epistatically interacting sets of loci with general two-season dominance coefficients (for simulation results, see Wittmann et al. 2017, Figure 10 therein).

Stochastic effects in finite populations are likely to introduce additional bias on the expected distribution of dominance coefficients, perhaps most strongly for those polymorphisms that cycle near a fixation boundary (Robertson 1962). As mentioned, the contributions of underdominance and overdominance (within a generation) to the stability of cyclic polymorphism are ignored here on the presumption that they are uncommon. Nevertheless, karyotypic changes are known to interact with seasonally fluctuating alleles (Nunez et al. 2024), and so chromosomal variants characterized by one of these fitness schemes (or by directional selection) are likely to alter the dynamics.

Situating the results in the broader topic of antagonistic pleiotropy (Hedrick 1999), there is an affinity with earlier work that looked beyond the special dominance relations (e.g. additivity, cumulative overdominance, symmetric dominance reversal). Curtsinger et al. (1994) sampled a coarse grid of values from the dominance unit square in their numerical investigation of two-trait antagonistic pleiotropy. Despite finding a substantial proportion of stabilizing parameter sets across the unit square, they dismissed as implausible that antagonistic pleiotropy promotes variation based on empirical results showing that the variance of dominance components in quantitative genetics experiments is typically small. Fry (2010) cautioned against concluding from narrow assumptions on autosomal dominance that sexual antagonism should be concentrated on the X chromosome; investigating numerical cases with partial dominance favoring the more fit allele in each sex, Fry concluded there exists a broader basis for the maintenance of genetic variation (see also Jordan and Charlesworth 2012). Van Dooren (2006) formulated an adaptive-dynamics model to investigate the likelihood of protected polymorphism in a multidimensional phenotypic context, concluding that antagonistic pleiotropy and trait-specific dominance were necessary for polymorphism, but did not necessarily require beneficial reversal. Furthermore, the maximum possible ratio of reversing to nonreversing dominance (among polymorphism-stabilizing parameter sets) “appears to be 3” (p. 2001). The present study explains why this ratio appears as an analytical result in the two-season model (Fig. 4a). Spencer and Priest (2016) analyzed a modifier model of sexual antagonism, concluding that the sexes are expected to evolve dominance in opposing directions. The large amount of sex-by-dominance variance discovered in the seed beetle Callosobruchus maculatus (Grieshop and Arnqvist 2018), a model of sexual antagonism, testifies to the potential for widespread violations of constant-magnitude assumptions when dealing with antagonistic selection (see also Connallon and Chenoweth 2019). Otto and Bourguet (1999) originally argued that balanced polymorphisms at intermediate frequencies were ripe for dominance modification. The strength of selection on modifiers derived for the seasonal model (Fig. 6) shores up this general conclusion regarding the unrestrictive prospects for the evolutionary adaptation of the genetic system to seasonality (see Rybnikov et al. 2024 on modifiers of recombination plasticity).

Appendix A

 

Deriving conditions on stabilizing dominance

The two-season formulation (Equation 3) of the geometric mean overdominance condition, G, is equal to

$$G=G_1\vee G_2,$$ (A1.1)

where $G_1=(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^r(1-{\mathrm{h}}_2{\mathrm{s}}_2)>(1-{\mathrm{s}}_1{)}^r\ge (1-{\mathrm{s}}_2)$, and $G_2=(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^r(1-{\mathrm{h}}_2{\mathrm{s}}_2)>(1-{\mathrm{s}}_2)>(1-{\mathrm{s}}_1{)}^r$.

The compound inequality $G_1$ can be written as

$$G_1=(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^r(1-{\mathrm{h}}_2{\mathrm{s}}_2)>(1-{\mathrm{s}}_1{)}^r\wedge (1-{\mathrm{s}}_1{)}^r\ge 1-{\mathrm{s}}_2$$ (A1.2a)

$$=(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^r>{\displaystyle \frac{{(1-{\mathrm{s}}_1)}^r}{1-{\mathrm{h}}_2{\mathrm{s}}_2}\wedge -1+(1-{\mathrm{s}}_1{)}^r\ge -{\mathrm{s}}_2}$$ (A1.2b)

$$=1-{\mathrm{h}}_1{\mathrm{s}}_1>{\displaystyle \frac{1-{\mathrm{s}}_1}{{(1-{\mathrm{h}}_2{\mathrm{s}}_2)}^{\frac{1}{r}}}\wedge 1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2}$$ (A1.2c)

$$G_1={\mathrm{h}}_1<{\displaystyle \frac{1-{\displaystyle \frac{1-{\mathrm{s}}_1}{{(1-{\mathrm{h}}_2{\mathrm{s}}_2)}^{\frac{1}{r}}}}}{{\mathrm{s}}_1}\wedge 1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2.}$$ (A1.2d)

An alternative formulation of $G_1$ is similarly obtained (i.e. starting from Equation A1.2a and rearranging terms to obtain the maximal h₂ value),

$$G_1={\mathrm{h}}_2<{\displaystyle \frac{1-{\left({\displaystyle \frac{1-{\mathrm{s}}_1}{1-{\mathrm{h}}_1{\mathrm{s}}_1}}\right)}^r}{{\mathrm{s}}_2}\wedge 1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2.}$$ (A1.3)

Likewise, $G_2$ can be written as

$$G_2=(1-{\mathrm{h}}_1{\mathrm{s}}_1{)}^r(1-{\mathrm{h}}_2{\mathrm{s}}_2)>(1-{\mathrm{s}}_2)\wedge (1-{\mathrm{s}}_2)>(1-{\mathrm{s}}_1{)}^r$$ (A1.4)

and is rearranged to obtain

$$G_2={\mathrm{h}}_1<{\displaystyle \frac{1-{\left({\displaystyle \frac{1-{\mathrm{s}}_2}{1-{\mathrm{h}}_2{\mathrm{s}}_2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\wedge 1-(1-{\mathrm{s}}_1{)}^r>{\mathrm{s}}_2}$$ (A1.5)

and

$$G_2={\mathrm{h}}_2<{\displaystyle \frac{1-{\displaystyle \frac{1-{\mathrm{s}}_2}{{(1-{\mathrm{h}}_1{\mathrm{s}}_1)}^r}}}{{\mathrm{s}}_2}\wedge 1-(1-{\mathrm{s}}_1{)}^r>{\mathrm{s}}_2.}$$ (A1.6)

Recall the bounds on parameter values assumed in Materials and methods,

$$B=0<{\mathrm{s}}_1<1\wedge 0<{\mathrm{s}}_2<1\wedge 0\le {\mathrm{h}}_1\le 1\wedge 0\le {\mathrm{h}}_2\le 1$$ (A1.7)

(To avoid degenerate cases in the subsequent analysis, all (${\mathrm{s}}_1,{\mathrm{s}}_2)$ associated with either fitness neutrality or homozygous lethality are excluded from B).

The conjunction of the geometric mean overdominance condition G with the parameter bounds B results in a sufficient condition for protected polymorphism that includes all cases of complete recessivity (of the seasonally deleterious allele) and intermediate dominance, but excludes all cases of complete dominance (of the seasonally deleterious allele), underdominance, and within-generation heterozygote advantage. And so we define the focal parameter region, $\mathrm{\Re }$, for r valued over the positive rational numbers,

$$\mathrm{\Re }:=\{(r,{\mathrm{s}}_1,{\mathrm{s}}_2,{\mathrm{h}}_1,{\mathrm{h}}_2)|G\wedge B\},\mathrm{\forall }r\in {\mathbb{Q}}^{+},$$ (A1.8)

where

$$\begin{array}{r}G\wedge B=\left(0<{\mathrm{s}}_1<1\wedge 0<{\mathrm{s}}_2<1-{(1-{\mathrm{s}}_1)}^r\wedge 0\le {\mathrm{h}}_2<1\wedge 0\le {\mathrm{h}}_1<\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-{\mathrm{h}}_2{\mathrm{s}}_2}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\right)\\ \vee \left(0<{\mathrm{s}}_1<1\wedge 1-{(1-{\mathrm{s}}_1)}^r\le {\mathrm{s}}_2<1\wedge 0\le {\mathrm{h}}_1<1\wedge 0\le {\mathrm{h}}_2<\frac{1-{\left(\frac{1-{\mathrm{s}}_1}{1-{\mathrm{h}}_1{\mathrm{s}}_1}\right)}^r}{{\mathrm{s}}_2}\right)\end{array}$$

or equivalently,

$$G\wedge B=\left(0<{\mathrm{s}}_1<1\wedge 0<{\mathrm{s}}_2<1-{(1-{\mathrm{s}}_1)}^r\wedge 0\le {\mathrm{h}}_1<{\displaystyle \frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\wedge 0\le {\mathrm{h}}_2<{\displaystyle \frac{1-{\displaystyle \frac{1-{\mathrm{s}}_2}{{(1-{\mathrm{h}}_1{\mathrm{s}}_1)}^r}}}{{\mathrm{s}}_2}}}\right)\vee \left(0<{\mathrm{s}}_1<1\wedge 1-{(1-{\mathrm{s}}_1)}^r\le {\mathrm{s}}_2<1\wedge 0\le {\mathrm{h}}_2<{\displaystyle \frac{1-{(1-{\mathrm{s}}_1)}^r}{{\mathrm{s}}_2}\wedge 0\le {\mathrm{h}}_1<{\displaystyle \frac{1-{\displaystyle \frac{1-{\mathrm{s}}_1}{{(1-{\mathrm{h}}_2{\mathrm{s}}_2)}^{\frac{1}{r}}}}}{{\mathrm{s}}_1}}}\right).$$

For any $(r,{\mathrm{s}}_1,{\mathrm{s}}_2)$, the region reduces to P_r (Equations 6a and b).

Remark. On the Cartesian plane, P_r is a geometric region with vertices $({\mathrm{h}}_1,{\mathrm{h}}_2)$ at ($0,0)$, $\left(\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1},0\right)$, and $(0,1)$  $\mathrm{if}0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$, or with vertices at ($0,0)$, $(1,0)$, and $\left(0,\frac{1-{(1-{\mathrm{s}}_1)}^r}{{\mathrm{s}}_2}\right)$ if $1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2<1$. The edges of P_r that include the origin as an endpoint lie entirely on either the h₁ or h₂ axis. The third edge of P_r encloses the region and its equation is described by ${\mathrm{h}}_1=\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-{\mathrm{h}}_2{\mathrm{s}}_2}\right)}^{\frac{1}{\mathrm{r}}}}{{\mathrm{s}}_1}$ and ${\mathrm{h}}_2=\frac{1-\frac{1-{\mathrm{s}}_2}{{(1-{\mathrm{h}}_1{\mathrm{s}}_1)}^{\mathrm{r}}}}{{\mathrm{s}}_2}$  $\mathrm{if}0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$, or by ${\mathrm{h}}_1=\frac{1-\frac{1-{\mathrm{s}}_1}{{(1-{\mathrm{h}}_2{\mathrm{s}}_2)}^{\frac{1}{r}}}}{{\mathrm{s}}_1}$ and ${\mathrm{h}}_2=\frac{1-{\left(\frac{1-{\mathrm{s}}_1}{1-{\mathrm{h}}_1{\mathrm{s}}_1}\right)}^r}{{\mathrm{s}}_2}$ if $1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2<1$.

Qualitative distribution of the stabilizing region across dominance categories

The unit square of dominance coefficients divides into the four quadrants:

Definition 1:

Beneficial reversal of dominance, b

$$b:=\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_1<{\displaystyle \frac{1}{2}\wedge 0\le {\mathrm{h}}_2<{\displaystyle \frac{1}{2}}}\right\},$$

Definition 2:

Deleterious reversal of dominance, d

$$d:=\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|{\displaystyle \frac{1}{2}<{\mathrm{h}}_1\le 1\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_2\le 1}}\right\},$$

Definition 3:

A-allele dominance, A

$$A:=\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_1<{\displaystyle \frac{1}{2}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_2\le 1}}\right\},$$

Definition 4:

a-allele dominance, a

$$a:=\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|{\displaystyle \frac{1}{2}<{\mathrm{h}}_1\le 1\wedge 0\le {\mathrm{h}}_2<{\displaystyle \frac{1}{2}}}\right\}.$$

These constitute the “major categories of season-specific dominance.” Cases of season-limited additivity (${\mathrm{h}}_1$ or ${\mathrm{h}}_2$ equal to one-half) and constant additivity $\left({\mathrm{h}}_1={\mathrm{h}}_2=\frac{1}{2}\right)$ constitute the remainder of the unit square.

The lemmas and definitions established directly below will be used to determine when $P_r$ is spread across 2, 3, or 4 quadrants of the unit square, each being a possibility depending on the parameters (r, s₁, s₂).

Lemma 1:

${\mathrm{h}}_{1,\max }$  is a concave, monotone-decreasing function of  ${\mathrm{h}}_2$.

Proof.

The first and second derivatives are

$${\displaystyle \frac{\mathrm{\partial }{\mathrm{h}}_{1,\max }}{\mathrm{\partial }{\mathrm{h}}_2}={\displaystyle \frac{-{\mathrm{s}}_2{\left({\displaystyle \frac{1-{\mathrm{s}}_2}{1-{\mathrm{h}}_2{\mathrm{s}}_2}}\right)}^{\frac{1}{r}+1}}{r{\mathrm{s}}_1(1-{\mathrm{h}}_2{\mathrm{s}}_2)}}}$$

$${\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{h}}_{1,\max }}{\mathrm{\partial }{\mathrm{h}}_2^2}={\displaystyle \frac{-(r+1){\mathrm{s}}_2^2{\left({\displaystyle \frac{1-{\mathrm{s}}_2}{1-{\mathrm{h}}_2{\mathrm{s}}_2}}\right)}^{{\displaystyle \frac{1}{r}}}}{r^2{\mathrm{s}}_1{({\mathrm{h}}_2{\mathrm{s}}_2-1)}^2}}}$$

It is obvious on inspection that the denominators are positive and that the numerators are negative for parameters in B and r > 0. ■

Lemma 2:

${\mathrm{h}}_{2,\max }$  is a concave, monotone-decreasing function of  ${\mathrm{h}}_1$.

Proof.

The first and second derivatives are

$${\displaystyle \frac{\mathrm{\partial }{\mathrm{h}}_{2,\max }}{\mathrm{\partial }{\mathrm{h}}_1}={\displaystyle \frac{-r{\mathrm{s}}_1{\left({\displaystyle \frac{1-{\mathrm{s}}_1}{1-{\mathrm{h}}_1{\mathrm{s}}_1}}\right)}^{r+1}}{(1-{\mathrm{s}}_1){\mathrm{s}}_2}}}$$

$${\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{h}}_{2,\max }}{\mathrm{\partial }{\mathrm{h}}_1^2}={\displaystyle \frac{-r(r+1){\mathrm{s}}_1^2{\left({\displaystyle \frac{1-{\mathrm{s}}_1}{1-{\mathrm{h}}_1{\mathrm{s}}_1}}\right)}^r}{{\mathrm{s}}_2{({\mathrm{h}}_1{\mathrm{s}}_1-1)}^2}}}$$

It is obvious on inspection that the denominators are positive and that the numerators are negative for parameters in B and r > 0. ■

Definition 5.

$P_r$  intersects dominance category C if  $P_r\cap C\ne \mathrm{\varnothing }$.

Lemma 3.

$P_r$ intersects at least two dominance categories; one of these is b.

Proof.

For any (r, s₁, s₂), the point $({\mathrm{h}}_1=0,{\mathrm{h}}_2=0)$ is an element of $P_r$, and therefore $P_r\cap b\ne \mathrm{\varnothing }$.

The following statements are implied by the conditions on $P_r$.

If $0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$, then the points $(0,{\mathrm{h}}_2)$ such that $\frac{1}{2}<{\mathrm{h}}_2<1$ are all elements of $P_r$, and so $P_r\cap A\ne \mathrm{\varnothing }$.

If $1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2<1$, then the points $({\mathrm{h}}_1,0)$ such that $\frac{1}{2}<{\mathrm{h}}_1<1$ are all elements of $P_r$, and so $P_r\cap a\ne \mathrm{\varnothing }$.

$$\therefore (P_r\cap b\ne \mathrm{\varnothing }\wedge P_r\cap A\ne \mathrm{\varnothing })\vee (P_r\cap b\ne \mathrm{\varnothing }\wedge P_r\cap a\ne \mathrm{\varnothing })\text{■}$$

Lemma 4.

If  $P_r\cap d\ne \mathrm{\varnothing }$, then  $P_r$  intersects all four quadrants.

Proof:

$$P_r\cap d\ne \mathrm{\varnothing }\wedge 0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$$

$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|\frac{1}{2}<{\mathrm{h}}_2<1\wedge \frac{1}{2}<{\mathrm{h}}_1<{\mathrm{h}}_{1,\max }\right\}\ne \mathrm{\varnothing }$ [by Definition 2 ]

$\Rightarrow {\mathrm{h}}_{1,\max }>\frac{1}{2}$ for some ${\mathrm{h}}_2$ in the interval $\left(\frac{1}{2},1\right)$

$\Rightarrow {\mathrm{h}}_{1,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_2$ in $[0,\frac{1}{2})$ [by Lemma 1]

$$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_2<{\displaystyle \frac{1}{2}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_1<{\mathrm{h}}_{1,\max }}}\right\}\ne \mathrm{\varnothing }$$

$\Rightarrow P_r\cap a\ne \mathrm{\varnothing }$ [by Definition 4]

$$P_r\cap d\ne \mathrm{\varnothing }\wedge 0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$$

$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|\frac{1}{2}<{\mathrm{h}}_1<\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\wedge \frac{1}{2}<{\mathrm{h}}_2<{\mathrm{\eta }}_{2,\max }\right\}\ne \mathrm{\varnothing }$ [by Definition 2]

$\Rightarrow {\mathrm{\eta }}_{2,\max }>\frac{1}{2}$ for some ${\mathrm{h}}_1$ in the interval $\left(\frac{1}{2},\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\right)$

$\Rightarrow {\mathrm{\eta }}_{2,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_1$ in the interval $\left(0,\frac{1}{2}\right)$ [applying Lemma 1 to ${\mathrm{\eta }}_{2,\max }$ ]

$$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0<{\mathrm{h}}_1<{\displaystyle \frac{1}{2}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_2<{\mathrm{\eta }}_{2,\max }}}\right\}\ne \mathrm{\varnothing }$$

$\Rightarrow P_r\cap A\ne \mathrm{\varnothing }$ [by Definition 3]

Similarly,

$$P_r\cap d\ne \mathrm{\varnothing }\wedge 1-(1-{\mathrm{s}}_1{)}^r<{\mathrm{s}}_2<1$$

$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|\frac{1}{2}<{\mathrm{h}}_1<1\wedge \frac{1}{2}<{\mathrm{h}}_2<{\mathrm{h}}_{2,\max }\right\}\ne \mathrm{\varnothing }$ [by Definition 2]

$\Rightarrow {\mathrm{h}}_{2,\max }>\frac{1}{2}$ for some ${\mathrm{h}}_1$ in $\left(\frac{1}{2},1\right)$

$\Rightarrow {\mathrm{h}}_{2,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_1$ in $[0,\frac{1}{2})$ [by Lemma 2]

$$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_1<{\displaystyle \frac{1}{2}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_2<{\mathrm{h}}_{2,\max }}}\right\}\ne \mathrm{\varnothing }$$

$\Rightarrow P_r\cap A\ne \mathrm{\varnothing }$ [by Definition 3]

$$P_r\cap d\ne \mathrm{\varnothing }\wedge 1-(1-{\mathrm{s}}_1{)}^r<{\mathrm{s}}_2<1$$

$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|\frac{1}{2}<{\mathrm{h}}_2<\frac{1-{(1-{\mathrm{s}}_1)}^r}{{\mathrm{s}}_2}\wedge \frac{1}{2}<{\mathrm{h}}_1<{\mathrm{\eta }}_{1,\max }\right\}\ne \mathrm{\varnothing }$ [by Definition 2]

$\Rightarrow {\mathrm{\eta }}_{1,\max }>\frac{1}{2}$ for some ${\mathrm{h}}_2$ in $\left(\frac{1}{2},\frac{1-{(1-{\mathrm{s}}_1)}^r}{{\mathrm{s}}_2}\right)$

$\Rightarrow {\mathrm{\eta }}_{1,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_2$ in $[0,\frac{1}{2})$ [applying Lemma 2 to ${\mathrm{\eta }}_{1,\max }$ ]

$$\Rightarrow \left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_2<{\displaystyle \frac{1}{2}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_1<{\mathrm{\eta }}_{1,\max }}}\right\}\ne \mathrm{\varnothing }$$

$\Rightarrow P_r\cap a\ne \mathrm{\varnothing }$ [by Definition 4]

It was just shown that $P_r\cap d\ne \mathrm{\varnothing }$ implies either that (i) ${\mathrm{h}}_{1,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_2$ in $[0,\frac{1}{2})$ and ${\mathrm{\eta }}_{2,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_1$ in $[0,\frac{1}{2})$, or (ii) ${\mathrm{h}}_{2,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_1$ in $[0,\frac{1}{2})$ and ${\mathrm{\eta }}_{1,\max }>\frac{1}{2}$ for all ${\mathrm{h}}_2$ in $[0,\frac{1}{2})$. In either case, b is subsumed by $P_r$ (from Definition 1). ■

The following Propositions 1–3 assume $0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$.

Proposition 1a.

If  $\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\le \frac{1}{2}$, then  $P_r$  intersects only b and A.

Proof.

Evaluating at ${\mathrm{h}}_2=0$, ${\mathrm{h}}_{1,\max }(0)=$  $\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}$.

$${\displaystyle \frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\le {\displaystyle \frac{1}{2}}}$$

$\Rightarrow {\mathrm{h}}_{1,\max }\le \frac{1}{2}$ at ${\mathrm{h}}_2=0$

$\Rightarrow {\mathrm{h}}_{1,\max }<\frac{1}{2}$ for all ${\mathrm{h}}_2\in (0,1)$ [by Lemma 1]

$$\Rightarrow P_r=\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_2<1\wedge 0\le {\mathrm{h}}_1<{\mathrm{h}}_{1,\max }\le {\displaystyle \frac{1}{2}}\right\}$$

$\Rightarrow P_r\cap (d\cup a)=\mathrm{\varnothing }$, $P_r\cap b\ne \mathrm{\varnothing }$, $P_r\cap A\ne \mathrm{\varnothing }$ [by Definitions 1–4] ■

Proposition 1b.

If  $P_r$  intersects only b and A, then  $\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\le \frac{1}{2}$.

Proof.

Suppose $\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}$. This implies ${\mathrm{h}}_{1,\max }(0)>\frac{1}{2}$. But then $P_r$ includes a counterexample (${\mathrm{h}}_1,0$) in the a quadrant, which implies the antecedent ($P_r$ intersects b and A only) is false. ■

$\therefore$ Proposition 1. $P_r$  intersects just two quadrants, b and A, if and only if  $\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\le \frac{1}{2}$ .

Proposition 2a.

If $P_r$  intersects all four quadrants, then  $\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}.$

Proof.

Using Lemma 4, it suffices to show that if $P_r$ intersects d, then $\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}.$

Recall that $P_r\cap d=\left\{({\mathrm{h}}_1,{\mathrm{h}}_2)|\frac{1}{2}\le {\mathrm{h}}_1<\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\wedge \frac{1}{2}\le {\mathrm{h}}_2<{\mathrm{\eta }}_{2,\max }\right\}$, and so

$$P_r\cap d\ne \mathrm{\varnothing }\Rightarrow {\displaystyle \frac{1}{2}<{\mathrm{h}}_1<{\displaystyle \frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_2<{\displaystyle \frac{1-{\displaystyle \frac{1-{\mathrm{s}}_2}{{(1-{\mathrm{h}}_1{\mathrm{s}}_1)}^r}}}{{\mathrm{s}}_2}}}}}$$

$$\Rightarrow {\displaystyle \frac{1-{\displaystyle \frac{1-{\mathrm{s}}_2}{{(1-{\mathrm{h}}_1{\mathrm{s}}_1)}^r}}}{{\mathrm{s}}_2}>{\displaystyle \frac{1}{2}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_1<{\displaystyle \frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}}}}}$$

$$={\mathrm{h}}_1<{\displaystyle \frac{1-{\left({\displaystyle \frac{1-{\mathrm{s}}_2}{1-{\displaystyle \frac{{\mathrm{s}}_2}{2}}}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\wedge {\displaystyle \frac{1}{2}<{\mathrm{h}}_1<{\displaystyle \frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}}}}$$

$$\Rightarrow {\displaystyle \frac{1-{\left({\displaystyle \frac{1-{\mathrm{s}}_2}{1-{\displaystyle \frac{{\mathrm{s}}_2}{2}}}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>{\displaystyle \frac{1}{2}\text{■}}}$$

Proposition 2b.

If  $\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}$, then  $P_r$  intersects all four quadrants.

Proof:

$\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}\Rightarrow {\mathrm{h}}_{1,\max }(1/2)>\frac{1}{2}$ [since ${\mathrm{h}}_{1,\max }\left(\frac{1}{2}\right)=\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}$ ]

$\Rightarrow \mathrm{\exists }{\mathrm{h}}_2\left({\mathrm{h}}_{1,\max }({\mathrm{h}}_2)=\frac{1}{2}\wedge {\mathrm{h}}_2>\frac{1}{2}\right)$ [by Lemma 1]

$$\Rightarrow \mathrm{\exists }({\mathrm{h}}_1,{\mathrm{h}}_2)\in P_r\left({\mathrm{h}}_1>{\displaystyle \frac{1}{2}\wedge {\mathrm{h}}_2>{\displaystyle \frac{1}{2}}}\right)$$

$\Rightarrow P_r\cap d\ne \mathrm{\varnothing }$ [by Definition 2, Lemma 4] ■

$\therefore$  Proposition 2.  $P_r$  intersects all four quadrants if and only if  $\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}$ .

Lemma 5.

$P_r$  cannot intersect only the two quadrants, b and a, if  $0<{\mathrm{s}}_2<1-(1-{\mathrm{s}}_1{)}^r$.

Proof.

By assumption, $P_r=\{({\mathrm{h}}_1,{\mathrm{h}}_2)|0\le {\mathrm{h}}_2<1\wedge 0\le {\mathrm{h}}_1<{\mathrm{h}}_{1,\max }\}$. Clearly, the points $(0,{\mathrm{h}}_2)$ are a subset of $P_r\cap A$ for all ${\mathrm{h}}_2\in \left(\frac{1}{2},1\right)$.■

Proposition 3.

If  $\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\le \frac{1}{2}$  and  $\frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}$, then  $P_r$  intersects only the 3 quadrants: b, A, and a.

Proof.  $\frac{1-{\left(\frac{1-{\mathrm{s}}_2}{1-\frac{{\mathrm{s}}_2}{2}}\right)}^{\frac{1}{r}}}{{\mathrm{s}}_1}\le \frac{1}{2}\wedge \frac{1-{(1-{\mathrm{s}}_2)}^{\frac{1}{r}}}{{\mathrm{s}}_1}>\frac{1}{2}\Rightarrow P_r\cap d=\mathrm{\varnothing }\wedge P_r\cap b\ne \mathrm{\varnothing }\wedge$  $P_r\cap A\ne \mathrm{\varnothing }\wedge P_r\cap a\ne \mathrm{\varnothing }$ as a consequence of Propositions 1–2 and Lemmas 3–5. ■

The following Propositions 4–6 assume $1-(1-{\mathrm{s}}_1{)}^r\le {\mathrm{s}}_2<1$. See File S1 for the corresponding proofs.

Proposition 4.

$P_r$  intersects just two quadrants, b and a, if and only if  $\frac{1-{(1-{\mathrm{s}}_1)}^r}{{\mathrm{s}}_2}\le \frac{1}{2}$.

Proposition 5.

$P_r$  intersects all four quadrants if and only if  $\frac{1-{\left(\frac{1-{\mathrm{s}}_1}{1-\frac{{\mathrm{s}}_1}{2}}\right)}^r}{{\mathrm{s}}_2}>\frac{1}{2}$.

Proposition 6.

If  $\frac{1-{\left(\frac{1-{\mathrm{s}}_1}{1-\frac{{\mathrm{s}}_1}{2}}\right)}^r}{{\mathrm{s}}_2}\le \frac{1}{2}$  and  $\frac{1-{(1-{\mathrm{s}}_1)}^r}{{\mathrm{s}}_2}>\frac{1}{2}$, then  $P_r$  intersects only the 3 quadrants: b, A, and a.

***

Appendix B

The classical bivoltine model (Haldane and Jayakar 1963) can be written in terms of the fitnesses in Table 1, resulting in the system of equations for season 1:

$${\mathrm{p}}^{\mathrm{\prime }}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_1}({\mathrm{p}}^2+\mathrm{pq}(1-{\mathrm{h}}_1{\mathrm{s}}_1))}$$ (A2.1)

$${\mathrm{q}}^{\mathrm{\prime }}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_1}({\mathrm{q}}^2(1-{\mathrm{s}}_1)+\mathrm{pq}(1-{\mathrm{h}}_1{\mathrm{s}}_1))}$$ (A2.2)

$${\overline{\mathrm{W}}}_1={\mathrm{p}}^2+2\mathrm{pq}(1-{\mathrm{h}}_1{\mathrm{s}}_1)+{\mathrm{q}}^2(1-{\mathrm{s}}_1)$$ (A2.3)

and season 2:

$${\mathrm{p}}^{\mathrm{″}}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_2}({({\mathrm{p}}^{\mathrm{\prime }})}^2(1-{\mathrm{s}}_2)+{\mathrm{p}}^{\mathrm{\prime }}{\mathrm{q}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2{\mathrm{s}}_2))}$$ (A2.4)

$${\mathrm{q}}^{\mathrm{″}}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_2}({({\mathrm{q}}^{\mathrm{\prime }})}^2+{\mathrm{p}}^{\mathrm{\prime }}{\mathrm{q}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2{\mathrm{s}}_2))}$$ (A2.5)

$${\overline{\mathrm{W}}}_2=({\mathrm{p}}^{\mathrm{\prime }}{)}^2(1-{\mathrm{s}}_2)+2{\mathrm{p}}^{\mathrm{\prime }}{\mathrm{q}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2{\mathrm{s}}_2)+({\mathrm{q}}^{\mathrm{\prime }}{)}^2$$ (A2.6)

where p, q is the frequency of A, a, respectively, at the start of season 1 and p + q = 1.

The ratio of allele frequencies is $\mathrm{q}/\mathrm{p}$ for the initial generation, and is equal to ${\mathrm{q}}^{\mathrm{″}}/{\mathrm{p}}^{\mathrm{″}}$ following an annual cycle. The ratio of these ratios is equal to the relative annual fitness of the a-allele (${\mathrm{w}}_{\mathrm{a}}$):

$${\mathrm{w}}_{\mathrm{a}}={\displaystyle \frac{\left({\displaystyle \frac{{\mathrm{q}}^{\mathrm{″}}}{{\mathrm{p}}^{\mathrm{″}}}}\right)}{\left({\displaystyle \frac{\mathrm{q}}{\mathrm{p}}}\right)}}$$ (A2.7)

The solution of ${\mathrm{w}}_{\mathrm{a}}=1$ in terms of q corresponds to the equilibrium frequency of the a-allele ($\widehat{{\mathrm{q}}_1}$). The exact solution for $\widehat{{\mathrm{q}}_1}$ is the real root of a cubic polynomial and its expression is unwieldy. The first-order Taylor series approximation for ${\mathrm{w}}_{\mathrm{a}}$ under weak selection coefficients is equal to

$${\mathrm{w}}_{\mathrm{a}}\approx {\mathrm{w}}_{\mathrm{a}}{|}_{\begin{array}{c}{\mathrm{s}}_1=0,\\ {\mathrm{s}}_2=0\end{array}}+\left({{\displaystyle \frac{\mathrm{\partial }{\mathrm{w}}_{\mathrm{a}}}{\mathrm{\partial }{\mathrm{s}}_1}}|}_{\begin{array}{c}{\mathrm{s}}_1=0,\\ {\mathrm{s}}_2=0\end{array}}\right){\mathrm{s}}_1+\left({{\displaystyle \frac{\mathrm{\partial }{\mathrm{w}}_{\mathrm{a}}}{\mathrm{\partial }{\mathrm{s}}_2}}|}_{\begin{array}{c}{\mathrm{s}}_1=0,\\ {\mathrm{s}}_2=0\end{array}}\right){\mathrm{s}}_2$$ (A2.8)

Equation A2.8 evaluates to Equation 12.

Appendix C

Evolutionary changes in the 3-allele system {A, a, a_mut} are governed by the following equations, which employ mutant dominance relations (${\mathrm{h}}_1^{*},{\mathrm{h}}_2^{*}$) for Aa_mut heterozygotes.

$${\mathrm{p}}^{\mathrm{\prime }}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_1}({\mathrm{p}}^2+\mathrm{pq}(1-{\mathrm{h}}_1{\mathrm{s}}_1)+\mathrm{p}{\mathrm{q}}_{\mathrm{mut}}(1-{\mathrm{h}}_1^{*}{\mathrm{s}}_1))}$$ (A3.1)

$${\mathrm{q}}^{\mathrm{\prime }}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_1}({\mathrm{q}}^2(1-{\mathrm{s}}_1)+\mathrm{q}{\mathrm{q}}_{\mathrm{mut}}(1-{\mathrm{s}}_1)+\mathrm{pq}(1-{\mathrm{h}}_1{\mathrm{s}}_1))}$$ (A3.2)

$${\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_1}({({\mathrm{q}}_{\mathrm{mut}})}^2(1-{\mathrm{s}}_1)+\mathrm{q}{\mathrm{q}}_{\mathrm{mut}}(1-{\mathrm{s}}_1)+\mathrm{p}{\mathrm{q}}_{\mathrm{mut}}(1-{\mathrm{h}}_1^{*}{\mathrm{s}}_1))}$$ (A3.3)

$$\begin{array}{rl}{\overline{\mathrm{W}}}_1=& {\mathrm{p}}^2+{\mathrm{q}}^2(1-{\mathrm{s}}_1)+({\mathrm{q}}_{\mathrm{mut}}{)}^2(1-{\mathrm{s}}_1)+2\mathrm{pq}(1-{\mathrm{h}}_1{\mathrm{s}}_1)\\ & +2\mathrm{p}{\mathrm{q}}_{\mathrm{mut}}(1-{\mathrm{h}}_1^{*}{\mathrm{s}}_1)+2\mathrm{q}{\mathrm{q}}_{\mathrm{mut}}(1-{\mathrm{s}}_1)\end{array}$$ (A3.4)

and for season 2:

$${\mathrm{p}}^{\mathrm{″}}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_2}({({\mathrm{p}}^{\mathrm{\prime }})}^2(1-{\mathrm{s}}_2)+{\mathrm{p}}^{\mathrm{\prime }}{\mathrm{q}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2{\mathrm{s}}_2)+{{\mathrm{p}}^{\mathrm{\prime }}\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2^{*}{\mathrm{s}}_2))}$$ (A3.5)

$${\mathrm{q}}^{\mathrm{″}}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_2}({({\mathrm{q}}^{\mathrm{\prime }})}^2+{{\mathrm{q}}^{\mathrm{\prime }}\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}+{\mathrm{p}}^{\mathrm{\prime }}{\mathrm{q}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2{\mathrm{s}}_2))}$$ (A3.6)

$${\mathrm{q}}_{\mathrm{mut}}^{\mathrm{″}}={\displaystyle \frac{1}{{\overline{\mathrm{W}}}_2}({({\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }})}^2+{{\mathrm{q}}^{\mathrm{\prime }}\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}+{{\mathrm{p}}^{\mathrm{\prime }}\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2^{*}{\mathrm{s}}_2))}$$ (A3.7)

$$\begin{array}{rl}{\overline{\mathrm{W}}}_2=& ({\mathrm{p}}^{\mathrm{\prime }}{)}^2(1-{\mathrm{s}}_2)+({\mathrm{q}}^{\mathrm{\prime }}{)}^2+({\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}{)}^2+2{\mathrm{p}}^{\mathrm{\prime }}{\mathrm{q}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2{\mathrm{s}}_2)\\ & +2{{\mathrm{p}}^{\mathrm{\prime }}\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }}(1-{\mathrm{h}}_2^{*}{\mathrm{s}}_2)+2{{\mathrm{q}}^{\mathrm{\prime }}\mathrm{q}}_{\mathrm{mut}}^{\mathrm{\prime }},\end{array}$$ (A3.8)

where p + q + q_mut = 1.

The annual fitness of a_mut is given by W(a_mut), and equals ${{\mathrm{q}}_{\mathrm{mut}}^{\mathrm{″}}}^{\text{″}}/{\mathrm{q}}_{\mathrm{mut}}$ by definition. The spread of ${\mathrm{a}}_{\mathrm{mut}}$ upon introduction is analyzed by the first-order Taylor series expansion of W(a_mut) at ${\mathrm{q}}_{\mathrm{mut}}=0$ for small selection coefficients:

$$\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})\approx \mathrm{W}({\mathrm{a}}_{\mathrm{mut}}){|}_{\begin{array}{c}{\mathrm{q}}_{\mathrm{mut}}=0\\ {\mathrm{s}}_1=0\\ {\mathrm{s}}_2=0\end{array}}+\left({{\displaystyle \frac{\mathrm{\partial }\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})}{\mathrm{\partial }{\mathrm{s}}_1}}|}_{\begin{array}{c}{\mathrm{q}}_{\mathrm{mut}}=0\\ {\mathrm{s}}_1=0\\ {\mathrm{s}}_2=0\end{array}}\right){\mathrm{s}}_1+\left({{\displaystyle \frac{\mathrm{\partial }\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})}{\mathrm{\partial }{\mathrm{s}}_2}}|}_{\begin{array}{c}{\mathrm{q}}_{\mathrm{mut}}=0\\ {\mathrm{s}}_1=0\\ {\mathrm{s}}_2=0\end{array}}\right){\mathrm{s}}_2+\left({{\displaystyle \frac{\mathrm{\partial }\mathrm{W}({\mathrm{a}}_{\mathrm{mut}})}{\mathrm{\partial }{\mathrm{q}}_{\mathrm{mut}}}}|}_{\begin{array}{c}{\mathrm{q}}_{\mathrm{mut}}=0\\ {\mathrm{s}}_1=0\\ {\mathrm{s}}_2=0\end{array}}\right){\mathrm{q}}_{\mathrm{mut}}$$ (A3.9)

The derivative in the fourth summand evaluates to 0. Equation A3.9 evaluates to Equation 16 (see Mathematica file). ■

The possibility for a triallelic polymorphism depends upon the set of quantities below all being positive (Theorem 5.3.1, Edwards 2000):

$${\mathrm{M}}_1={\mathrm{w}}_{22}{\mathrm{w}}_{33}-{\mathrm{w}}_{23}^2+{\mathrm{w}}_{13}{\mathrm{w}}_{23}-{\mathrm{w}}_{12}{\mathrm{w}}_{33}+{\mathrm{w}}_{12}{\mathrm{w}}_{23}-{\mathrm{w}}_{22}{\mathrm{w}}_{13}$$

$${\mathrm{M}}_2={\mathrm{w}}_{13}{\mathrm{w}}_{23}-{\mathrm{w}}_{12}{\mathrm{w}}_{33}+{\mathrm{w}}_{11}{\mathrm{w}}_{33}-{\mathrm{w}}_{13}^2+{\mathrm{w}}_{12}{\mathrm{w}}_{13}-{\mathrm{w}}_{11}{\mathrm{w}}_{23}$$

$${\mathrm{M}}_3={\mathrm{w}}_{12}{\mathrm{w}}_{23}-{\mathrm{w}}_{22}{\mathrm{w}}_{13}+{\mathrm{w}}_{12}{\mathrm{w}}_{13}-{\mathrm{w}}_{11}{\mathrm{w}}_{23}+{\mathrm{w}}_{11}{\mathrm{w}}_{22}-{\mathrm{w}}_{12}^2$$ (A3.11)

$${\sigma }_{23}=2{\mathrm{w}}_{23}-{\mathrm{w}}_{22}-{\mathrm{w}}_{33}$$

$${\sigma }_{13}=2{\mathrm{w}}_{13}-{\mathrm{w}}_{11}-{\mathrm{w}}_{33}$$

$${\sigma }_{12}=2{\mathrm{w}}_{12}-{\mathrm{w}}_{11}-{\mathrm{w}}_{22},$$

where

$${\mathrm{w}}_{11}=1-{\mathrm{s}}_2,{\mathrm{w}}_{12}=(1-{\mathrm{h}}_1{\mathrm{s}}_1)(1-{\mathrm{h}}_2{\mathrm{s}}_2),{\mathrm{w}}_{13}=(1-{\mathrm{h}}_1^{*}{\mathrm{s}}_1)(1-{\mathrm{h}}_2^{*}{\mathrm{s}}_2),$$

${\mathrm{w}}_{22}={\mathrm{w}}_{23}={\mathrm{w}}_{33}=1-{\mathrm{s}}_1$ for the system in A3.1–A3.8.

Since ${\mathrm{M}}_1={\sigma }_{23}=0$ (and assuming ${\mathrm{h}}_1\ne {\mathrm{h}}_1^{*}$ and/or ${\mathrm{h}}_2\ne {\mathrm{h}}_2^{*}$), a stable equilibrium with {A, a, a_mut} each at positive frequencies is impossible. Monomorphism of any allele or diallelic polymorphism of {a, a_mut} are unstable in the presence of geometric mean overdominance. Diallelic polymorphism of {A, a} is unstable in the presence of a_mut (Equations 17–19). The sole remaining possibility is a stable polymorphism of {A, a_mut}.

Analogous considerations establish the impossibility of a triallelic equilibrium for the system {A, a, A_mut}, and moreover, establish that the sole possibility for stability resides in diallelism of {a, A_mut} (Supplementary File 1).

Data availability

A Mathematica notebook (v.14.2; Wolfram Research, Inc. 2024) is available at https://doi.org/10.6084/m9.figshare.28329713.v1.

Supplemental material available at GENETICS online.

Funding

This work was supported by National Institutes of Health/National Institute of General Medical Sciences grant no. R35GM147107 and National Institutes of Health grant nos. R01GM100366 and R01GM137430.