Harvesting Models

Differential Equations · Axiom Academy

Harvesting Models: Sustainable Fishing Add a constant harvest to logistic growth and a fishery can settle into balance — or collapse outright. One number decides which. A fish population follows logistic growth toward a carrying capacity of K = 100 tons at rate r = 0.5/yr , but a fleet harvests a constant H tons/year regardless of how many fish remain. Push the harvest rate too high, and there's no population size left where growth can outrun it. Drag the harvest rate — where does the population settle? Start the fishery at 50 tons and pick a harvest rate. A light harvest lets the population settle at a new, lower balance. Push it far enough, and the population runs straight to zero. Where do the equilibria actually come from? Set the population change to zero and the model becomes a quadratic equation in P. Its discriminant is the whole story — positive means two equilibria exist, negative means none do. How much CAN you sustainably harvest? At any population P, the fishery naturally regrows at rate rP(1−P/K) — that's the most you could harvest and hold P steady. Drag along the curve and find its peak. Logistic growth minus a constant harvest, , has two equilibria exactly when — the maximum sustainable yield , reached at P=K/2 . Cross that line and there's no population size left where growth can outpace the harvest, so . The same balance governs hunting quotas, logging rates, and any renewable resource harvested faster than it can regrow.

This is the written version of the interactive lesson above. See the full Differential Equations course.