Stability and Period-Doubling Bifurcation of Fractional-Order Commensal Symbiosis Model with Allee Effect

Differential equations with fractional order play an important role in modeling some natural phenomena. This paper investigates the dynamics of the fractional-order commensal symbiosis model with the Allee effect. This model describes the relationship between prey and predator populations. The piecewise-constant approximation technique is applied to discretize this model. Equilibrium points are established, and local stability conditions are calculated using fractional-order linearization and eigenvalue-based arguments. Moreover, the bifurcation theory is successfully invoked to discuss the period-doubling bifurcation. In particular, sufficient conditions are effectively determined for the emergence of the period-doubling bifurcation. We utilize the hybrid control approach to control the behavior of the considered system. Then, some numerical examples are presented to demonstrate the accuracy and validity of the theoretical results. The findings indicate that fractional order and Allee effects improve system dynamics and substantially improve stability limits and bifurcation structures, providing new insights into how to handle the dynamics of ecological systems.