harvesting

Suppose a population is growing logistically, but people are also taking some of it away — fishing the pond, culling the herd, logging the forest. How much can you take and still have a population left next year? The harvesting model adds a removal term to the logistic equation, and it carries a sharp warning: take too much and the population collapses, with no gentle warning sign.

Start from logistic growth and subtract what is harvested. The simplest case is constant harvesting at rate H: dP/dt = rP(1 − P/K) − H. The growth curve rP(1 − P/K) is a downward parabola in P; subtracting H slides the whole curve down by H. As long as H is small, the curve still crosses zero at two points — a lower equilibrium (unstable) and an upper one (stable), and a population above the lower point settles to the upper one. But as you raise H, those two equilibria slide toward each other. At a critical harvest rate H = rK/4 they collide and vanish: above that rate there is no positive equilibrium at all, dP/dt is negative everywhere, and the population is driven to extinction no matter how big it started.

This is a vivid, real-world example of a saddle-node bifurcation: a parameter (the harvest rate) crosses a threshold and equilibria disappear. The lesson for fisheries and wildlife management is stark — the maximum sustainable yield sits right at that knife-edge, and pushing even slightly past it does not merely reduce the stock, it can wipe it out. Worse, the collapse can be sudden, because near the threshold the stable population is already low and weakly attracting.