Investigating the dynamics, synchronization and control of chaos within a transformed fractional Samardzija–Greller framework

In this article, in response to the limitations of existing ecological models, we address the critical need for a more comprehensive understanding of predator–prey dynamics by presenting a modified fractional Samardzija–Greller model that incorporates intra- and inter-species competitions within two predator populations. Our model stands out for being more realistic because it considers the natural competition that occurs among and between two predator species when they share a common prey We derived the local stability conditions at equilibrium points using Routh–Hurwitz conditions for the modified model. With the help of a suitably chosen Lyapunov function, we also obtained the global stability condition for our fractional model. The existence of chaos has been confirmed through Lyapunov exponents and bifurcation in the new system for two distinct sets of initial conditions for different fractional orders. Employing the active control method, we establish conditions for synchronization between these two fractional systems and introduce control functions for chaos management in the modified model. Numerical simulations, utilizing the generalized Adams–Bashforth–Moulton method, support the theoretical findings across a spectrum of fractional orders ranging from 0 to 1. We demonstrated the adaptability of the active control method for different fractional orders. A fractional order of α equal to 1 for synchronization shows rapid convergence, but a drop to α equal to 0.80 causes a substantial slowdown that takes almost six times more number of iterations to complete. Thus, we shed light on how the fractional order of the system plays a pivotal role in determining the speed of synchronization, with lower orders leading to a noticeable delay and higher fractional orders favoring faster synchronization. Our thorough investigation contributes to the understanding of complex ecological systems and offers practical insights into fractional chaos control mechanisms within the context of predator–prey models.