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Home/Biophysics, Fluids & Geoscience/Logistic Harvest & Maximum Sustainable Yield

Logistic Harvest & Maximum Sustainable Yield

Logistic growth with constant harvest: MSY = rK/4, stable stock, unstable collapse threshold, and population trajectory.

Population model

0.45 1/y
100
9/y
70

Measured values

Maximum sustainable yield11.25
Collapse threshold27.64
Stable stock72.36
Final N72.36

When H exceeds MSY no positive equilibrium exists. Below MSY there are two equilibria: the lower one is a collapse threshold and the upper one is stable.

Live graphs

About this model

A logistic population grows as rN(1-N/K). Constant harvest subtracts H, producing the classic maximum sustainable yield at N=K/2 with H_MSY=rK/4. Below MSY there are two equilibria: a lower unstable collapse threshold and an upper stable stock. Above MSY no positive equilibrium remains.

Who it's for: Ecology, fisheries science, environmental management, and differential equations courses.

Key terms

  • Logistic growth
  • Harvest
  • Maximum sustainable yield
  • Collapse threshold
  • Equilibrium
  • Carrying capacity

How it works

Logistic population growth with constant harvest, maximum sustainable yield, stable equilibrium, and collapse threshold.

Key equations

dN/dt = rN(1−N/K) − H
MSY occurs at N=K/2 with H_MSY=rK/4

Frequently asked questions

Why can a stock collapse even below MSY?
Below MSY the lower equilibrium is unstable. If the population starts below that threshold, harvest exceeds biological growth and the stock declines.
Is MSY a management recommendation?
It is a simple textbook benchmark. Real management adds uncertainty, age structure, economics, and precautionary buffers.