Dynamical analysis of a fractional order management and renewable resources model with population growth using a dual computational approach

This study develops and analyzes a fractal-fractional mathematical model to investigate the interaction between global population density, which typically grows logarithmically with the density of resource biomass. The authors develop a set of fractional-order differential equations to represent a time-fractional-order global population density model that incorporates the effects of resource biomass. The well-posedness of the model is examined, focusing on positivity and boundedness of its solutions. Equilibrium analysis identifies three states that characterize the system’s long-term dynamics. The authors calculate the reproductive number and examine the sensitivity indices of the relevant factors. Additionally, the authors confirm that the equilibrium points are locally asymptotically stable. The study analyzes the Ulam-Hyers stability of the proposed model and develops fractional-order control techniques to stabilize chaotic dynamics, guiding the system toward sustainable equilibrium states. Using error matrices, the model’s controllability and observability are investigated. A generalized fractal fractional Adams-Bashforth numerical scheme with an exponential decay kernel is employed for numerical simulations. Several fractional parameters are also explored in the deterministic mathematical model to assess the impact of resource biomass. Numerical simulations provide insights into the behavior of global population density in relation to resource biomass across various fractional parameters.