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Cell["\<\
Example of a genetic model: determining the optimum maternal sex ratio in \
diploids\
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Cell["\<\
aa = wild type mothers, produce sex ratio (female investment) F
Aa = rare mutant, mothers produce sex ratio (female investment) F+d, whereby \
we assume d to be small (i.e. allele has only a small effect) 
n = number of offspring produced\
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When A gene is rare we need to consider 2 types of matings:

wild type matings (FxM) : aa x aa                      
M type matings (FxM) : Aa x aa                     
P type matings (FxM) : aa x Aa

If the frequency of the A gene in females and males is pf and pm, then when \
rare its frequency in wild type, M type and P type colonies are approx. \
1-2pf-2pm, 2pf and 2pm :\
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Cell["\<\
Now let's calculate the frequency of the A gene in the gametes of next \
generation males and females. 
For males: number of Aa males produced x 0.5\
\>", "Text",
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An equilibrium occurs when the dominant eigenvalue = 1, and we can evaluate \
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The population dynamics of female grey squirrels of different ages were shown \
to be characterised by the following Leslie matrix:
(the 0.32, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57 are the age specific net \
fecundities of 1, 2, 3, 4, 5, 6 and 7 year old squirrels, ie how many \
offspring they produce that survive up to age 1 and
the 0.46, 0.77, 0.65, 0.67, 0.64 and 0.88 are the age specific survival \
probabilities, i.e. the probability that a squirrel from a given age class \
survives to the next year) \
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The dominant eigenvalue is the Eigenvalue with the largest absolute value, \
which is the first one, and that one determines overall population growth \
once the stable age distribution is reached :\
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i.e. when the stable age distribution is reached the population will increase \
by 3.85% per year.\
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The stable age distribution can be found from the eigenvector corresponding \
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stableagedistr=100*Eigenvectors[L][[1]]/Apply[Plus,Eigenvectors[L][[1]]]\
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i.e. the population will eventually contain 44%, 19%, 14%, 9%, 6%, 4% and 3% \
of 1, 2, 3, 4, 5, 6 and 7 year old female squirrels.\
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