Beverton–Holt model

Example

Let $R_0 > 1$ and $M > 0$ be constants, and let $K = (R_0 - 1) M$. The Beverton–Holt model is the recurrence $$n_{t+1} = \frac{R_0\, n_t}{1 + n_t / M} \qquad (t \ge 0)$$ for a population size $n_t \ge 0$. It is a Riccati difference equation with $a = R_0$, $b = 0$, $c = 1/M$, $d = 1$, so $c \ne 0$ and $ad - bc = R_0 \ne 0$. For $n_0 > 0$, the solution is $$n_t = \frac{K\, n_0}{n_0 + (K - n_0)\, R_0^{-t}}.$$ Check: $z_t = 1/n_t$ satisfies the linear recurrence $z_{t+1} = z_t / R_0 + 1/(R_0 M)$, whose solution is $z_t = R_0^{-t} z_0 + (1 - R_0^{-t})/K$; inverting gives the formula. Since $R_0^{-t} \to 0$, every solution with $n_0 > 0$ tends to the equilibrium $K$, the carrying capacity.