Loom
The Ricker Model: Population Dynamics Through a Single Equation
How a deceptively simple recurrence relation explains boom-bust cycles in fish populations — and what it reveals about chaos.
Introduction
The Ricker model describes how a population changes from one generation to the next:
\[N_{t+1} = N_t \cdot r \cdot e^{-N_t / K}\]where $N_t$ is the population at time $t$, $r$ is the intrinsic growth rate, and $K$ is the carrying capacity.
Interactive model
The widget below lets you explore how $r$ changes the long-run dynamics. Try dragging the slider.
Stability analysis
For small values of $r$, the population converges to a fixed point. As $r$ increases, period-doubling bifurcations appear — and eventually the dynamics become chaotic.
The period-doubling route to chaos was described by Robert May in his landmark 1976 Nature paper, which showed that even simple ecological models produce unpredictable dynamics.
Callout example
At $r = 2.692$, the Ricker model enters its first period-doubling bifurcation. The population alternates between two values rather than settling at a fixed point.