Natural Resource Open Access Model – Good Chicken Gardens, LLC

Work in progress!

Natural Resource Open Access Model

System State: Stable

Parameters

Initial Resource Population (X₀): 872

Habitat Area (Acres): 7328

Species Density (Indiv/Acre): 0.15

Net Profit (p – s): 1

Catchability (q): 0.066

Unit Cost of Effort (c): 163

Min Viable Population (K₁): 750

Current Indicators: Target Equilibrium (X\infty): — Stability Midpoint: —

Scientific Context

This model simulates a renewable resource system with critical depensation. The system enters a “Hopf Bifurcation” (wild oscillations) when the economic target falls below the biological midpoint.

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

The Variables

X: The renewable resource population.
E: The level of effort devoted to harvest.
r: The intrinsic growth rate.
K₁: The minimum viable population level.
K₂: The environmental carrying capacity.
q: The catchability coefficient.
α: An adjustment coefficient for effort.
(p−s): The market price net of shipping cost.
c: The unit cost of effort.

Key Ecological Concepts

Critical Depensation: If population falls below K₁, it will decline to extinction.
Open Access Equilibrium (X∞): The population level where the system stabilizes: $$X_{\infty} = \frac{c}{(p – s)q}$$.
Hopf Bifurcation: As economic conditions change, X∞ can move from stable to unstable, causing wild oscillations that lead to extinction.