Harvesting
Harvesting models describe removal of individuals from a population through fishing, hunting, culling or resource collection.
Constant harvesting
A logistic population with constant harvest \(H\) can be written
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-H.\]Equilibria satisfy
\[rN\left(1-\frac{N}{K}\right)=H.\]Sustainable harvest
The logistic growth term reaches its maximum at \(N=K/2\), with value
\[\frac{rK}{4}.\]Therefore, in this idealised model, a constant harvest exceeding \(rK/4\) cannot be balanced by population growth.
Proportional harvesting
If removal is proportional to population size, another model is
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-hN,\]where \(h\) is a harvesting rate.
Key idea. Harvesting changes both equilibrium population size and extinction risk. A biologically sustainable policy depends on the form of harvesting and the population-growth mechanism.