### Beverton–Holt: a discrete-time logistic model

The Beverton–Holt equation is a recurrence relation relating abundance of a population in generation $n$ to the abundance of the previous generation $n - 1$: \[ N_n = \frac{PN_{n-1}}{1 + PC^{-1}N_{n-1}}, \] where $N_n$ is the abundance at generation $n$, $P$ is the density-independent productivity of the population, and $C$ is the capacity of the population. This recurrence relation can be solved to explicitly relate the abundance at generation $n$ with the initial abundance $N_0$. Let $Q$ be the equilibrium abundance of the population given by the equation \[ Q = C \left (1-P^{-1} \right). \] Using this definition, the abunance at generation $n$ is given by the equation \[ N_n = \frac{QN_0}{N_0 + P^{-n}\left (Q - N_0 \right )}. \] The standard Beverton–Holt recurrence relation has an alternative form that is very similar to the solution of the recurrence relation. \begin{align} N_{n} &= \frac{P N_{n-1}}{1 + P C^{-1} N_{n-1}} \\ &= \frac{P N_{n-1}}{1 + P \left (1 - P^{-1 }