A neural network model for managing renewable resources with population growth

This paper presents a fractional-order modelling framework for population-resource dynamics using the ψ-Hilfer derivative with neural network-based validation. The model captures logistic population growth coupled to renewable resource biomass while incorporating memory effects through fractional calculus. Novel contributions include the application of the ψ-Hilfer operator to population-resource systems, representing hereditary dynamics via fractional order ζ and type parameter ω. A detailed qualitative analysis establishes existence, uniqueness, and Ulam-Hyers (UH) stability with explicit parameter conditions. A linearised quadrature numerical scheme for the ψ-Hilfer problem is developed and validated through neural networks, achieving [Formula: see text]. Simulations reveal that fractional orders ([Formula: see text]) produce smoother transients than integer-order counterparts, with ζ governing memory strength and ω modulating response patterns. Critically, we identify [Formula: see text] as a sustainable harvesting threshold, beyond which resource collapse occurs for integer-order systems, while fractional memory ([Formula: see text]) provides partial damping. These findings offer actionable policy insights, including optimal harvest limits and stabilisation strategies, advancing both theoretical understanding and practical tools for sustainable resource management.