Confusion over equations for population's heterozygosity vs. heterozygous individuals

Question

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.

Answer 1

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.