evolution - Model for fluctuating selection

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.

Answer

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).