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Is there any mathematical model to predict the behaviour and long-term consequence of counter-acting selection at different time scale?

For example, let's consider the bi-allelic gene A, with alleles A1 and A2. During a long period of n1 generations A1 is slightly beneficial (differential of selection: s1). After this period, follows a short period of n2 generations when A2 is highly beneficial (differential of selection: s2).

What mathematical model describes the frequency fluctuations of alleles and which allele will get fixed at the long term given the initial frequency ( f0 ), assuming infinite population size and random mating.

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Unsmooth changes like you describe are normally hard to use in an analytical model. You could build a model of smoothly changing selection using a sine wave or simulations (which would be easy to build). –  timcdlucas Aug 23 '13 at 7:25
That remainds me the famous case of Biston betularia –  Miguel Ángel Naranjo Ortiz Aug 24 '13 at 11:41
Biston betularia is a moth that lives in England and has a light color in order to mimetize with the bark of trees. However, about 1% of the population presents melanism, and its camouflage fails. During the industrial revolution, the trees became dark as a result of the pollution. When that happened, the proportion of melanistic moths inverted, to almost 99%. During the 20st century, when the industry independized of the coal, the trees got light again, and the proportions restored once again. –  Miguel Ángel Naranjo Ortiz Aug 25 '13 at 14:01
This might be a gross oversimplification but could you not use sequential univariate breeders equations, model the response during the s1 phase, this will erode variation in favour of the A1 allele, then you could then apply the univariate equation to the s2 phase. If all variation was lost in the s1 phase then the response in the s2 phase will be 0, and increase dependent on the strength of selection and size of remaining variation. –  GriffinEvo Apr 23 at 15:33
GriffinEvo's response sounds right. But does the fluctuation continue indefinitely? In that case you might use the idea of step functions from EE to model the process. –  daniel May 27 at 23:25

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