Bifurcation Analysis of a Fractional‐Order Differential Algebraic Predator‐Prey System

ABSTRACT In this paper, we study a fractional‐order differential algebraic predator‐prey bioeconomy model with time delay. Firstly, by analyzing the characteristic equation, the critical time delay of system instability and the existence of Hopf bifurcation are studied. Then, it reveals that how time delay becomes the switch of system stability: there is a critical time delay, once exceeded, the system will turn from a stable equilibrium to a periodic oscillation bifurcation state. Subsequently, it is found that the order of fractional order is negatively correlated with the stability region of the system the lower the order, the stronger the ability of the system to resist instability. The study further treats the fractional order itself as a bifurcation parameter, elucidating a novel regulatory mechanism governing system stability. These results consistently demonstrate that fractional‐order model outperforms the traditional integer‐order counterpart in terms of stability. Finally, the obtained results are verified by numerical simulations.