Biomass Optimization Model

This project implements a numerical optimization model for biomass management, considering both Volterra integral equations (with memory effects) and differential equations approaches.

The Volterra integral equation case is formulated based on my doctorate thesis in 1991, fourth application (pp. 159-169): Thèse de doctorat Yezza Abdel - Ph.D Yezza Abdel, 1991.

Description

The model optimizes the harvesting effort to maximize profit while maintaining sustainable biomass levels. It supports two different modeling approaches:

• Volterra integral equation (with memory/dissipation effects)
• Standard differential equation (logistic growth with harvesting)

🎯 Key Features

• Biomass dynamics simulation using either Volterra or differential equations
• Profit optimization with harvesting constraints
• Detailed visualization of results (5 different plots)
• Export of results to CSV and Excel files
• Configurable parameters via JSON file

🛠️ Installation

1. Clone this repository
2. Install required dependencies:

pip install numpy pandas scipy matplotlib openpyxl

📦 Configuration

Create a params.json file with the following parameters that can be modified from which the main program will extract the parameters:

{
    "T": 20,                  // Time horizon generally in years
    "N": 240,                // Number of time steps for simulation
    "x0": 500.0,             // Initial biomass
    "r_growth": 5.0,         // Growth rate
    "K": 5000.0,             // Carrying capacity
    "dissipation_rate": 0.05,// Memory effect parameter (applies to Volterra model only)
    "interest_rate": 0.05,   // Economic discount rate
    "unit_cost": 2.0,          // Cost per unit effort
    "unit_price": 10.0,        // Price per unit biomass
    "E": 1.0,                  // Maximum harvesting effort
    "model_type": "volterra" // Model type: "volterra" or "differential"
}

🚀 Usage

Run the model:

python main.py

The program will:

1. Load parameters from params.json
2. Optimize the harvesting strategy depending of the model type (Volterra or Differential)
3. Generate visualizations and save them as png images in the output directory
4. Save results to CSV and Excel files in the output directory

📋 Output

Visualizations

These graphs are generated in the same figure:

• Biomass trajectory
• Control effort
• Extracted biomass
• Instantaneous profit
• Cumulative profit

Files Generated

All output files are prefixed by a unique GUID:

• biomass_optimization_results_GUID.png: Combined visualization of all results
• biomass_optimization_timeseries_GUID.csv: Time series data
• detailed_results_GUID.xlsx: Detailed results with multiple sheets including:
  • Detailed_Results: Time series of all variables
  • Parameters: Model configuration

Examples of output files

Graphs (Volterra Model and Differential Model)

CSV

Excel

Mathematical Model

Objective Function

Maximize the discounted profit:

$$J = ∫0^T e^{-δt} (c - p * x(t))u(t) dt$$

Constraints

• 0 ≤ u(t) ≤ E (Harvesting effort bounds)
• x(t) ≥ 0 (Non-negative biomass)

Dynamics

Volterra Model

$$x(t) = x₀ + ∫₀ᵗ[rx(s)(1-x(s)/K)e⁻ᵖ⁽ᵗ⁻ˢ⁾ - u(s)x(s)]ds$$

Differential Model

$$dx/dt = rx(1-x/K) - ux$$

📁 Project Structure

modele_dissipation_ressources/
├── main.py              # Main application file with BiomassModel class
├── params.json          # Configuration parameters
├── README.md           # Project documentation
├── requirements.txt    # Python dependencies
└── output/            # Generated output directory
    ├── biomass_optimization_results_GUID.png    # Visualization plots
    ├── biomass_optimization_timeseries_GUID.csv # Time series data
    └── detailed_results_GUID.xlsx              # Detailed analysis results

Key Files Description

• main.py: Contains the core BiomassModel class implementing:

  • Biomass dynamics simulation (Volterra/Differential)
  • Optimization algorithms
  • Visualization methods
  • Results generation

• params.json: Configuration file containing:

  • Model parameters
  • Simulation settings
  • Economic parameters

• output/: Directory containing generated results:

  • PNG files for visualizations
  • CSV files for time series data
  • Excel files with detailed analysis

📜 License

This project is open source and available under the MIT License.

🤝 Contributing

Contributions are welcome! Please feel free to submit a Pull Request.