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.

  • $\begingroup$ 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). $\endgroup$
    – timcdlucas
    Aug 23, 2013 at 7:25
  • $\begingroup$ That remainds me the famous case of Biston betularia $\endgroup$ Aug 24, 2013 at 11:41
  • $\begingroup$ 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. $\endgroup$ Aug 25, 2013 at 14:01
  • $\begingroup$ 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. $\endgroup$
    – rg255
    Apr 23, 2014 at 15:33
  • $\begingroup$ 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. $\endgroup$
    – daniel
    May 27, 2014 at 23:25

1 Answer 1


The frequency fluctuations will be determined by a standard model of selection as found in any basic population genetics text. In this scenario they take a very basic form: during each long period $i$ the frequency of $A_1$ increases from $f_i$ to $f_i\cdot (1+s_1)^{n_1}$ and during each short period $j$ the frequency of $A_1$ decreases from $f_j$ to $f_j\cdot (1/(1+s_2))^{n_2}.$ Thus over each pair of periods the frequency of $A_1$ changes by $(1+s_1)^{n_1}/(1+s_2)^{n_2}$. If this quantity exceeds 1, $A_1$ goes to fixation; if it is less than one $A_2$ goes to fixation.

More generally, for an infinite population in a fluctuating environment, the allele with the higher geometric mean fitness will go to fixation. Early discussions of these results are due to Dempster (1955; Cold Spring Harbor Symp. Quant. Biol.), Haldane and Jayakar (1963; J. Genetics), and Lewontin and Cohen (1969; PNAS).

  • $\begingroup$ Nice answer to a very old question! Thank you. I edited your answer just to clarify that $f_i$ and $f_j$ multiply the thing in parenthesis and are not function of the thing in parenthesis. just in case someone could have a doubt... $\endgroup$
    – Remi.b
    Jan 18, 2015 at 16:16
  • $\begingroup$ If I am not mistaken, this model is a good approximation when the two phases are short enough. If the phases were to be too long then fixation may occur during the first phase and allele frequencies would never be affected by the second phase. This is even more important when $N$ and/or $s$ are high. It might be interesting to have such model for finite population. $\endgroup$
    – Remi.b
    Jan 18, 2015 at 16:39
  • 1
    $\begingroup$ Glad this was helpful. In the specific way that you formulated the question, we don't need to worry about fixation because the population is infinite and thus fixation can never occur. But as you note in the real world, or in a finite population model, fixation becomes a major issue. Off the top of my head I don't see how to derive a simple expression for the fixation probabilities of each allele in this case. $\endgroup$
    – Corvus
    Jan 18, 2015 at 17:25

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