Harvesting & Fishing

Humans as the predator: how many fish can be caught every year without emptying the sea? Try a fixed catch quota or a fixed fishing effort, find the maximum sustainable yield, and see how a quota just a little too high makes the stock collapse.

EcologyFisheriesSustainability

Inputs

Input examples

The stock grows logistically with rate r up to its carrying capacity K (numbers or tonnes). A quota takes the same amount every year, however small the stock; a fixed effort catches the same share of whatever is there. Time is in years.

Growth rate r — per year

Carrying capacity K — stock without fishing

Stock at the start N₀

Catch per year H

Years to simulate

What is a Sustainable Yield?

A logistic population grows fastest at half its carrying capacity: near K there is no room, and a very small stock has too few parents. The growth is the surplus you can take away every year while the stock stays the same: the sustainable yield. Its highest value, rK/4 at a stock of K/2, is the maximum sustainable yield (MSY).

A fixed quota is risky: if the stock drops below a threshold, the same catch takes a bigger and bigger share and the stock collapses. A fixed effort is self-correcting: fewer fish means a smaller catch. Real fisheries also deal with uncertain estimates, so they usually aim well below MSY.

MSY= rK / 4

at a stock ofK / 2

best effortE = r / 2

quota > MSYcollapse

Stock Over Time

years

stock with fishing

carrying capacity K and K/2

Growth Against Catch

yield

natural growth (sustainable yield) at each stock

catch

stable stock

tipping point

Result

Enter the values above

Worked Techniques

Maximum Sustainable Yield

The top of the growth parabola rN(1 − N/K).

Example: r = 0.4, K = 10 000: 1 000 per year at a stock of 5 000.

Stock under a Quota

Two equilibria: the upper one is stable, the lower one is a tipping point below which the stock collapses.

Example: quota 800: stable at 7 236, tipping point 2 764.

Stock under a Fixed Effort

Doubling the effort does not double the catch: it shrinks the stock it is taken from.

Example: E = 10% and E = 30% both give 750 per year.

Sources & Verification

References

N. J. Gotelli, A Primer of Ecology, 4th ed., Sinauer, 2008 — chapter “Logistic Population Growth”.