A neural network model for managing renewable resources with population growth

Area of Science:

• Ecology
• Mathematical Biology
• Dynamical Systems

Background:

• Population-resource dynamics are crucial for ecological sustainability.
• Traditional models often lack the capacity to capture complex memory effects inherent in biological systems.
• Fractional calculus offers a powerful framework for modeling systems with hereditary properties.

Purpose of the Study:

• To develop a novel fractional-order modeling framework for population-resource dynamics.
• To incorporate memory effects using the ψ-Hilfer derivative and fractional calculus.
• To validate the model using neural networks and analyze its stability and harvesting implications.

Main Methods:

• Application of the ψ-Hilfer fractional derivative to logistic population growth coupled with renewable resource biomass.
• Qualitative analysis including existence, uniqueness, and Ulam-Hyers stability.
• Development and neural network validation of a linearised quadrature numerical scheme.

Main Results:

• Fractional orders ([Formula: see text]) yield smoother population and resource transients compared to integer-order models.
• The fractional order ζ controls memory strength, while the type parameter ω influences response patterns.
• A sustainable harvesting threshold ([Formula: see text]) is identified, with fractional memory offering partial damping against resource collapse.

Conclusions:

• Fractional-order modeling provides a more nuanced understanding of population-resource dynamics, particularly regarding memory effects.
• The ψ-Hilfer derivative framework offers insights into optimal harvesting limits and stabilization strategies for sustainable resource management.
• Neural network validation confirms the accuracy and applicability of the proposed numerical scheme for fractional dynamics.