(See Arrival of the frequent for context)

If $NL\mu \gg 1$, the population naturally spreads over different genotypes, a regime called the polymorphic limit. See Polymorphic limit (Wright-Fisher model) tiddler for more.

To model neutral exploration, we let $1+s_p = \delta_{pq}$, where $\delta_{pq}$ is a Kronecker delta, so that only $q$ has some fitness, and all other phenotypes have $0$ fitness, and so, even if a mutation produces them, no offspring can inherit from them. At every generation, all offspring inherits from $\mathcal{N}_q$ only, and thus the population can only spread by mutations over a single generation jump, and it is most likely to stay mostly within $\mathcal{N}_q$, if $N$ is large enough.

We should note that equations, like Eq.3 would be the same, even though we assumed that all the individuals are in $\mathcal{N}_q$, because, as $1+s_p = \delta_{pq}$, all the selection weight is in $\mathcal{N}_q$, which produces the same results. More precisely, in the expression $\sum_{i=1}^N \tilde{\Phi_p}(g_i, s_i) \frac{N(1+s_i)}{\sum_{j=1}^N (1+s_j)}$ only $N'$ (the number of individuals in $\mathcal{N}_q$) elements are non-$0$ in the sum and so in the mean-field approx (where we assume $\tilde{\Phi_p}(g_i, s_i)$ is constant) the $N'$ from the sum cancels the $\sum_{j=1}^N (1+s_j) = \sum_{j=1}^N \delta_{pq} = N'$ from the denominator, leaving a $N$ on the top.

In the mean-field approximation the expected number of individuals with phenotype $p$ produced per generation is now independent of time, and given by Eq. 3. (we thus simply call $m_p(t) = m_p$), under the corresponding assumptions, because even if not all of the population are in $\mathcal{N}_q$, the assumption of fitness, we've made gives selective weight only to those in $\mathcal{N}_q$ (see Wright-Fisher model).

As we said above, the number of individuals with genotype $p$ (p-type) will follow a binomial distribution, with probability $m_p/N$ of success (getting p-type offspring), and number of trials $N$, and therefore the probability to get at least one such individual is:

$P(\text{at least on p-type offpsring}) = 1 - P(\text{no p-type offspring}) = 1 - (1 - m_p/N)^N \approx 1 - e^{-m_p}$

After $T$ generations, we have run the Bernoulli trial $TN$ times, and thus the number of p-type individuals we have gotten, summed over all the $T$ generation also follows a Binomial distribution, but with $NT$ samples, and same probability. Thus

$P(\text{at least on p-type offpsring over T generations}) \approx 1 - e^{-m_p T}$

Thus, the time when {{the probability of having discovered a p-type individual (produced a p-type offspring)} is $\alpha$} is found by:

$\alpha = 1 - e^{-m_p T}$

$e^{-m_p T} = 1- \alpha$

$-m_p T = \ln{(1- \alpha)}$

$T = \frac{-\ln{(1- \alpha)}}{N L\mu \Phi_{pq}}$

Eq. 4

Where we used Eq. 3 in Arrival of the frequent.