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  "body": "\n## Example\n\nLet $r \u003e 0$ and $K \u003e 0$ be constants.\nThe *Ricker model* is the [[nonlinear difference equation|7C7FBZGERM]]\n$$N_{t+1} = f(N_t), \\qquad f(N) = N\\, e^{r (1 - N/K)},$$\nfor a population size $N_t \\ge 0$.\nIts [[equilibria|D62PEKHRV5]] are the solutions of $f(N) = N$, namely $N = 0$ and $N = K$.\nThe derivative is $f'(N) = e^{r(1 - N/K)} (1 - r N / K)$, so $f'(0) = e^r \u003e 1$ and $f'(K) = 1 - r$.\nHence the equilibrium $0$ is unstable, and the equilibrium $K$ is asymptotically [[stable|CHEQ235HW9]] when $\\abs{1 - r} \u003c 1$, that is, when $0 \u003c r \u003c 2$.\nAt $r = 2$ one has $f'(K) = -1$, and as $r$ increases past $2$ the solutions go through a [[period-doubling cascade|NQDTTZ59X8]] towards chaotic behaviour.\n\n## Explanations\n\n### L1 human\n\nThe Ricker model describes a population, such as a fish stock, that reproduces once per season.\nWhen the population is far below $K$, it grows by about the factor $e^r$; above $K$, the exponential factor is less than $1$ and the population shrinks.\nFor a small growth rate $r$, the population settles at $K$.\nFor larger $r$, it overshoots and undershoots, first in a regular cycle of period $2$, then $4$, and finally erratically.\n",
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