Application of parameterized method to a fractional order prey-predator model

In this thesis, a comprehensive theoretical and numerical investigation is conducted on differential equations expressed through contemporary definitions of fractional derivatives, such as those of Caputo, Atangana–Baleanu, and Caputo–Fabrizio, thereby extending beyond the scope of classical derivative-based systems. In the first stage of the study, existence and uniqueness conditions for solutions of these memory-influenced fractional differential equations are presented, aiming to ensure the mathematical consistency of the associated derivative formulations. Subsequently, a parametric numerical approach is employed to solve these fractional systems. This proposed method is compared with the well-known predictor–corrector Euler scheme through detailed error analysis in terms of both accuracy and computational efficiency. The comparative results demonstrate that the proposed method yields more stable and accurate solutions, particularly for systems governed by memory effects. Finally, the developed numerical approach is applied to a predator–prey dynamic model incorporating various fractional derivative types. The temporal evolution of the system is explored through a series of simulations. The findings reveal that fractional derivatives not only enrich the theoretical framework but also enhance the realism and flexibility of dynamic system modeling. In this respect, the thesis offers significant contributions by highlighting the potential of fractional calculus in the fields of applied mathematics and mathematical modeling.