Time-Delay-Induced Hopf Bifurcation in a Fractional-Order Ecological System

Abstract

In this paper, we investigate a fractional-order delayed dynamical system. Taking the time delay τ as the bifurcation parameter, we establish that the equilibrium is asymptotically stable for all τ<τ₀, and a Hopf bifurcation occurs as τ passes through the critical value τ₀. We derive the characteristic equation and verify the transversality condition, which identifies the onset of oscillations and characterizes the stability of the emerging periodic solutions. Numerical experiments carried out in MATLAB with a predictor–corrector scheme support the analysis and illustrate the resulting oscillatory behavior.