$$\frac{dX}{dt} = rX \left(\frac{X}{K_1} – 1\right) \left(1 – \frac{X}{K_2}\right) – qXE$$ $$\frac{dE}{dt} = \alpha[(p – s)qX – c]E$$

The Variables

The equation for the change in population over time (X˙) and the change in effort over time (E˙) uses the following parameters:

• X: The renewable resource population level (e.g., the number of passenger pigeons).
• E: The level of effort devoted to harvest.
• r: The intrinsic growth rate (r > 0).
• K1​: The minimum viable population level (also known as the extinction threshold).
• K2​: The environmental carrying capacity (K2 > K1 > 0).
• q: The catchability coefficient (q > 0), representing how easily the resource is harvested.
• α: An adjustment coefficient (α > 0) for the effort.
• (p−s): The market price net of shipping cost (where p is the price and s is the shipping cost).
• c: The unit cost of effort at the harvest site (c > 0).

Key Ecological Concepts

• Critical Depensation: The model assumes that the resource’s net growth exhibits critical depensation, meaning if the population falls below K₁, it will naturally decline to extinction.
• Open Access Equilibrium (X∞): This is the population level where the system stabilizes, calculated as $$X_{\infty} = \frac{c}{(p – s)q}$$.
• Hopf Bifurcation: As economic conditions change (like the introduction of the telegraph or rail), X∞ can move from a stable level to an unstable level, causing wild oscillations (limit cycles) that often lead to extinction.