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  "body": "\n## Example\n\nLet $R_0 \u003e 1$ and $M \u003e 0$ be constants, and let $K = (R_0 - 1) M$.\nThe *Beverton–Holt model* is the recurrence\n$$n_{t+1} = \\frac{R_0\\, n_t}{1 + n_t / M} \\qquad (t \\ge 0)$$\nfor a population size $n_t \\ge 0$.\nIt is a [[Riccati difference equation|D4Q8VM7826]] with $a = R_0$, $b = 0$, $c = 1/M$, $d = 1$, so $c \\ne 0$ and $ad - bc = R_0 \\ne 0$.\nFor $n_0 \u003e 0$, the solution is\n$$n_t = \\frac{K\\, n_0}{n_0 + (K - n_0)\\, R_0^{-t}}.$$\nCheck: $z_t = 1/n_t$ 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.\nSince $R_0^{-t} \\to 0$, every solution with $n_0 \u003e 0$ tends to the [[equilibrium|D62PEKHRV5]] $K$, the carrying capacity.\n\n## Explanations\n\n### L1 human\n\nThis is a discrete-time model of a population with limited resources, used in fisheries.\nWhen the population is small, it grows by the factor $R_0$ each generation.\nAs it grows, crowding reduces the growth, and the population levels off at the carrying capacity $K$.\nUnlike many nonlinear models, it has an exact formula for its solution, because taking reciprocals makes it linear.\n",
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