Unveiling periodic structures, period-bubbling, and multistability in a fractional-order system with Allee effect.

Nonlinear ecological systems exhibit rich dynamical behaviors, such as organized periodic structures, sudden transitions between attractors, period-bubbling, and multistability. While these features are well-documented in integer-order and discrete-time ecological models, their persistence under memory-driven dynamics remains largely unexplored. In this study, we investigate these nonlinear phenomena in a tri-trophic fractional-order ecological system incorporating an Allee effect. Analytically, the well-posedness of the system is established, and the existence of equilibria, along with the conditions for the local and global stability, is explored. The existence criteria for Hopf bifurcations are also discussed with respect to the fractional-order and the Allee parameter. To characterize the global dynamics, we have constructed two-parameter largest Lyapunov exponent plots for a fractional-order ecological system, together with high-resolution isospike diagrams. These diagrams reveal that the fractional-order system preserves the classical organized periodic windows, such as shrimp-shaped structures, double fish-hook patterns, and periodic strips. The system also shows a transition from periodicity to chaos through Feigenbaum-type scaling of periods. Most importantly, our results show that the fractional-order ecological system exhibits period-bubbling and multistability, two hallmark features of nonlinear dynamics that have been rarely explored in memory-driven ecological systems. Thus, although memory smoothens the transitions between attractors, the introduction of memory into the ecological systems does not eliminate the core features of nonlinearity known from integer-order and discrete-time systems.