Let the mutant allele be denoted as A, the resident allele as a, and their frequencies in the micro- and macroga-metes as pA, pa, respectively, qA, qa. The recurrences for the unstructured case are, with the next generation indicated by a prime, with n — f n mpA — mAApAqA + 1 m*A(pAq* + paqA)
m — pA qA mAA + (pAqa + paqA)™aA + paqmaa f — qApAfAA + (qApa + qapA)faA + qapfa
Substituting pa — 1 -^^ and ^ — 1 - ^^ and dropping quadratic terms gives for the frequencies of the two states in which allelic metaindividuals can be born:
2 maa 2 _/aa which on adding give
Hence, in unstructured populations p( j) — /lnf! f^1^ f ^
and in nonfluctuating environments
a result known as the Shaw-Mohler formula. Formulas  and  lie at the heart of the calculation of phenotypic ESSs and evolutionary trajectories (on the suppositions that the former are effectively genetically homozygous, respectively, of effective mutation limitation).
Polymorphic Residents, Allelic Evolution
Now assume that the resident population is dimorphic on the A-locus, and that the traits of the three corresponding genotypes are subject to modification either by the invasion of other alleles or by new alleles on other, previously monomorphic, loci, generically called B. To keep the formulas simple, the standard textbook assumption is made that m and f are proportional for all values of their arguments. Moreover, it will be assumed that the resident environment does not fluctuate. In that case, if mr and f denote the values of m and f at the resident equilibrium, m/mr — f /fr — f (since at equilibrium fr — 1). Their common value is indicated as w in accordance with population genetical tradition.
First consider allelic evolution, with the mutant allele denoted as a. For small pa p'a — W.apa with
W.a — w(Xaa|Xaa, XaA, XAA)pa w(XAa|Xaa, XaA, XAA)pA 
where the form ofthe formula brings out that the pheno-type engendered by an allele consists oftwo components, expressing the dependence on the two intraaggregate environments that it may encounter. In population genetics w.a is known as the marginal fitness of the a-allele.
Polymorphic Residents, Modifier-Driven Evolution
The final example considers a mutant B on a so-called modifier locus that previously was monomorphic for b. The rule that symbols playing no role in the argument are dropped lets aa stand for aabb, aaB for aabB, etc. When B is rigidly coupled to a the pair aB behaves like a new A-allele, with invasion fitness w.aB defined by  with a — aB. A similar consideration applies to a pair AB. In general the standard rules of transmission genetics give for the gamete frequencies paB and pAB,
paß — WaaBpapaB + WAB (1 - c)pApaB + WaABCpapAB pAB — WAABpApAB + WaAB (1 - c)papAB + WABCpApaB
with c the recombination probability and pa and pA calculated from the equilibrium equations corresponding to . In vector-matrix form
Evolutionarily Steady Polymorphisms
A polymorphism is evolutionarily steady if it is uninvad-able by both alternative alleles and modifiers. The first can be judged from , the second from .
Since q is linear in c, q is maximal for c = 0 or c = 1/2. Therefore, it suffices to consider q for those values of c only. The case c = 0 is already covered by the condition that no alternative allele is able to invade. So all that has to be done to deal with modifiers is to consider q for c = 1 /2 .
ESS arguments may often be further simplified by the observation, due to Eshel and Feldman, that (cf. )
= pa UaB WaaB + (pA UaB + pa UAB )WaAB + pA UAB WAAB 
Hence, if w(X|Xaa, XA, XAA) is maximal and equal to 1 at X = Xaa, X = XaA, and X = Xaa , the triple (Xaa, XaA, Xaa) is evolutionarily steady. Evolutionarily steady polymorphisms that equalize the reproductive output of the morphs are called ideal free. Hence ideal-free evolutionarily steady genetic polymorphisms can be calculated by treating each phenotype as if it reproduces clonally, with the additional constraint that at birth the phenotypes should occur in Hardy-Weinberg proportions.
Similar statements apply to the sexually differentiated case and to polymorphisms in more than one locus.
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