History & Framework

Project Introduction

This project models the complex bioeconomic dynamics of a single-species fishery, specifically focusing on the relationship between biological harvesting and the resulting commercial profit. While traditional biological models evaluate population sustainability in a vacuum, real-world systems are fundamentally driven by economic incentives.

By merging these two disciplines, we can analyze how a population of fish behaves under continuous harvesting pressure while simultaneously tracking the profit yielded per unit catch.

The objective is to progressively advance from a deterministic baseline model to highly complex, stochastic economic environments—evaluating sinusoidal cost fluctuations, abrupt market shocks, and sudden demand surges over a standard season.

The original systemic architecture was programmed in Python, utilizing the SciPy library for differential equation integration. This web application translates that quantitative mathematical foundation directly into an interactive, browser-based numerical solver.

Mathematical Framework

Biological Population Dynamics

The biological dynamics of the fish population are governed by a logistic growth model combined with a harvesting term. The resulting system is described by the following differential equation:

\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) - qPE(t)

\( r \): The intrinsic biological growth rate of the species.
\( K \): The carrying capacity of the ecosystem.
\( q \): The catchability coefficient.
\( E(t) \): The fishing effort deployed at time \( t \).

Economic Volatility & Option Valuation

To accurately simulate real-world fisheries, the economic parameters cannot remain static. The baseline profit function is evaluated as the market price per unit minus the operational cost required to harvest that unit.

However, as the models advance, we introduce Geometric Brownian Motion to simulate random, realistic fluctuations in market demand and price. Ultimately, the framework bridges resource economics and quantitative finance by applying the Black-Scholes analytical formula to the fishery. In this paradigm, unharvested biomass is treated mathematically as a European call option, where the expected discounted profit \( V \) is evaluated based on market volatility and time to maturity.

V = P \cdot \Phi(d_1) - X e^{-\delta \tau} \cdot \Phi(d_2)

The interactive implementation of these models can be explored within the Model Playground, where users can adjust parameters and observe the resulting system dynamics in real time.