Ricker model

Example

Let $r > 0$ and $K > 0$ be constants. The Ricker model is the nonlinear difference equation $$N_{t+1} = f(N_t), \qquad f(N) = N\, e^{r (1 - N/K)},$$ for a population size $N_t \ge 0$. Its equilibria are the solutions of $f(N) = N$, namely $N = 0$ and $N = K$. The derivative is $f'(N) = e^{r(1 - N/K)} (1 - r N / K)$, so $f'(0) = e^r > 1$ and $f'(K) = 1 - r$. Hence the equilibrium $0$ is unstable, and the equilibrium $K$ is asymptotically stable when $\abs{1 - r} < 1$, that is, when $0 < r < 2$. At $r = 2$ one has $f'(K) = -1$, and as $r$ increases past $2$ the solutions go through a period-doubling cascade towards chaotic behaviour.