Harvesting
Harvesting models add deliberate removal to population growth. Depending on the setting, this can represent fishing, hunting, culling, forestry or another process that removes individuals from a population.
Begin with the unharvested population
For logistic growth,
\[ \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right). \]The natural production rate is
\[ G(N)=rN\left(1-\frac{N}{K}\right). \]It is zero at \(N=0\) and \(N=K\), and reaches its maximum at \(N=K/2\):
\[ \boxed{G_{\max}=\frac{rK}{4}}. \]Constant harvesting
If the same number \(H\) is removed per unit time regardless of population size,
\[ \boxed{\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-H}. \]| Symbol | Meaning |
|---|---|
| \(N(t)\) | population size |
| \(r\) | intrinsic growth-rate parameter |
| \(K\) | carrying capacity |
| \(H\) | constant number removed per unit time |
Natural production and harvest
Equilibria under constant harvesting
At equilibrium,
\[ rN\left(1-\frac{N}{K}\right)=H. \]The two possible equilibrium populations are
\[ \boxed{N_{\pm}=\frac{K}{2}\left(1\pm\sqrt{1-\frac{4H}{rK}}\right)}. \]Real positive equilibria exist only when
\[ \boxed{H\leq\frac{rK}{4}}. \]| Harvest level | Model behaviour |
|---|---|
| \(0\lt H\lt rK/4\) | two positive equilibria |
| \(H=rK/4\) | one repeated equilibrium at \(K/2\) |
| \(H\gt rK/4\) | no positive equilibrium |
The lower equilibrium is a threshold
When \(0\lt H\lt rK/4\), the upper equilibrium \(N_+\) is stable and the lower equilibrium \(N_-\) is unstable. Below \(N_-\), natural production is too small to balance the fixed harvest, so the population continues to decline in this deterministic model.
Maximum sustainable yield in the simple model
Logistic production reaches its maximum when
\[ N=\frac{K}{2}, \]and the corresponding production is
\[ \boxed{\mathrm{MSY}=\frac{rK}{4}}. \]Why harvesting at the theoretical maximum is fragile
At \(H=rK/4\), the harvest line touches the production curve at its maximum. If abundance falls below \(K/2\), natural production becomes smaller than the fixed harvest. There is therefore no safety margin.
A numerical example
Suppose
\[ r=0.4\text{ per year},\qquad K=1000. \]Then
\[ \frac{rK}{4}=100\text{ individuals per year}. \]If \(H=60\), the equilibria are approximately
\[ N_-\approx184,\qquad N_+\approx816. \]Population trajectories
A modelling issue at zero
The mathematical term \(-H\) continues removing individuals even when the population is extremely small. An unconstrained ODE can therefore pass below zero, which is biologically impossible. In an applied model the harvesting rule or boundary treatment must prevent this.
Proportional harvesting
If a fixed proportion \(h\) of the population is removed per unit time,
\[ \boxed{\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-hN}. \]Now the amount removed is \(hN\), so harvesting automatically decreases when the population becomes smaller.
Equilibrium under proportional harvesting
For a positive equilibrium,
\[ r\left(1-\frac{N}{K}\right)-h=0, \]which gives
\[ \boxed{N^*=K\left(1-\frac{h}{r}\right)}. \]A positive equilibrium requires \(h\lt r\).
Constant and proportional harvesting
| Feature | Constant harvest \(H\) | Proportional harvest \(hN\) |
|---|---|---|
| amount removed | fixed number | depends on population size |
| when abundance falls | removal stays fixed | removal decreases |
| positive equilibria | two, one or none | one when \(h\lt r\) |
| low-population threshold | can occur | not generated in the same way |
Effort and catchability
In fisheries, harvest is often modelled as
\[ H(N)=qEN, \]where \(E\) is effort and \(q\) is catchability. This corresponds to proportional harvesting with \(h=qE\).
Deterministic sustainability and real risk
A positive equilibrium in a deterministic model does not guarantee persistence in a real population. Environmental variation, uncertain parameters and random births and deaths can push abundance away from equilibrium. Management questions may therefore require extinction or threshold-crossing probabilities as well as deterministic equilibria.