2
$\begingroup$

On page 24 of Gillespie's Population Genetics, 2nd ed, an equation for $H$, the probability that two randomly drawn alleles are different by state, is given.

$H$ is stated to be similar to the heterozygosity of the population, which I guess is expected to be $2pq$, where $p$ and $q$ are frequencies for different alleles. $ H′=(1−1/2N)×H$

On page 23, $H_t$, the probability that an individual chosen at random from the population after $t$ generations of random mating is heterozygous, is given : $$H_t = H_0 \times (1-\frac{1}{2N})^t$$

For a current time $t$, are $H$ and $H_t$ measurements of the same thing, or are they different ?

I interpreted them as being the same, because drawing two alleles at random is similar to drawing a diploid individual at random.

$\endgroup$
1
$\begingroup$

One is a recurrence equation and the other is a general solution. These are basic concepts of mathematical modelling.

Reccurence equation

A recurrent equation describes the state of a system (here, heterozygosity $H$) at the next time step given the state in the previous time step. The recurrence relation is $H' = (1-1/2N)\times H$. Given the state $H$, at a given generation, one can know the state $H'$ at the following generation (given the population size $N$).

General solution

A general solution describes the state of a system at any time step given an initial state. Not all recurrence relation have a general solution. Here there is a general solution and it is $H_t = H_0 \times (1-\frac{1}{2N})^t$. Given the initial state $H_0$, one can know the state $H_t$ after $t$ generations (given the population size $N$).

Two equations for the same model

The two equations both refer to the same model and therefore the same statistic of expected heterozygosity which, as you said, is $H = 2 p (1-p)$ (for a bi-allelic locus) by definition.

Understanding the basics of mathematical modeling in ecology and evolution

Have a look at A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by S. Otto for an introduction into mathematical modeling in the field of interest.

| improve this answer | |
$\endgroup$
  • $\begingroup$ @ghgh Please let me know if my answer helped you or if something is still unclear. $\endgroup$ – Remi.b Jul 20 '16 at 18:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.