Population Growth & Chaos

Population dynamics are rigidly governed by the logistic differential equation, factoring in carrying capacity constraints. As the reproductive rate parameter r increases in discrete-time models, the system transitions deterministically from stable equilibria to period-doubling bifurcations, ultimately arriving at mathematical chaos. This non-linear behavior demonstrates how simple deterministic rules inevitably generate complex, unpredictable long-term population states, strictly bounded by environmental resource limits.

In the continuous model a population grows fastest when it is only half-full. As numbers climb toward the carrying capacity K, crowding leaves less food and space per individual, so growth slows and the curve levels off into a smooth S-shape. The takeaway: switch to the Continuous tab and watch the fastest growth happen at the inflection point, exactly at N = K/2 — not at the start.

At r = 4 the discrete map is fully chaotic, yet it has an exact closed-form solution: writing x = sin²(π θ) turns each step into simply doubling θ and keeping the fractional part. In binary, doubling just shifts the digits left, so the "unpredictable" orbit is really reading off the successive bits of the starting value. It looks random only because pinning down θ would take infinite precision.