A Numerical Investigation of Fractional-Order Nonlinear Models in Population Dynamics

Abstract

This study presents a numerical approach for solving fractional-order formulations of classical population dynamics models (the Logistic, Richards, Gompertz, and Predator-Prey models), with the aim of verifying the impact of incorporating non-integer orders in these models. The methodology is based on a discretization scheme tailored to fractional differential equations, combined with Newton’s iterative method for solving the resulting nonlinear systems. Stability, convergence, and robustness of the proposed method were verified by means of mesh refinement and sensitivity tests. Comparisons with classical analytical solutions and high-precision numerical results also demonstrated the method’s accuracy. The results highlight the significant role of the fractional order α in modulating system behavior: smaller values of α lead to memory effects that slow population growth and attenuate oscillations in predator-prey interactions. These results indicate the potential of fractional calculus to improve the modeling of complex population dynamics.