co.combinatorics - Archaeogenetics

I'm not sure what you mean by 'starts with a population formed by two empty sets'.

However, assuming you start with an initial population $P = leftlbrace A_1, ldots, A_n rightrbrace$ where each individual
$A_i = leftlbrace a_{i1}, ldots, a_{i{n_i}} rightrbrace $.

Let $|A_i|$ be the complexity (number of genes?) of individual $A_i$, i.e. $|A_1| = |leftlbrace a_11, a_12, a_13 rightrbrace | = 3$, and
$|P| = 1/n sum_{i=1}^n |A_i|$ be the average complexity of all of the individuals in the population. Let $P'$ be this population after applying the 'killing' operation and $P''$ be this population after applying the 'recombination' operation. Then, after some basic algebra / logic, we see that

$E(|P'|) = |P|$ and $E(|P''|) = |P|$ too.
Hence conversely, if $P_0$ was our initial population and $P_t$ is our population after $t$ applications of either the 'killing' operation or the 'recombination' operation, then

$E(|P_0|) = |P_t|$.

Not also that, as 'killing' removes one member of the population and 'recombination' adds one, we may model the total population as a random walk and so, for example, determine an expected initial population size based on current population and number of generations (off the top of my head it's also another invarient and so expected inital population = final population, although this needs checking.)

There are almost certainly other invarients.

On the more abstract side:

Call a population $P$ in which $forall i,j$, $A_i cap A_j = emptyset$ pure.

Note:

• Applying 'killing' to a pure population gives a pure population.


• Applying 'killing' to a non-pure population MAY give a pure population.


• Applying 'recombination' to a pure population gives a non-pure population.


• Applying 'recombination' to a non-pure population gives a non-pure population.

As the probability of them being the same is 0, wlog we may assume all 'genes' of all individuals are unique - i.e. the inital population is pure. This gives us a 2 state system that models the population, passing between the pure and non-pure states by these 2 operations. As generations pass I would expect the population to become 'more non-pure' as the probability of the application of a killing operation reverting it to a pure population reduces. This should give us a method by which to determine an expected value for the number of itterations that have occured by measuring the 'purity' of the population.