Dynamic analysis of the fractional-order logistic equation with two different delays

In this paper, we analyze the stability and Hopf bifurcation of the fractional-order logistic equation with two different delays \(\tau _{1}, \tau _{2}>0\): \(D^{\alpha }y(t)=\rho y(t-\tau _{1})\left( 1-y(t-\tau _{2})\right) \), \(t>0\), \(\rho >0\). We describe stability regions by using critical curves. We explore how the fractional order \(\alpha \), \(\rho \), and time delays influence the stability and Hopf bifurcation of the model. Then, by choosing \(\rho \), fractional order \(\alpha \), and time delays as bifurcation parameters, the existence of Hopf bifurcation is studied. An Adams-type predictor–corrector method is extended to solve fractional-order differential equations involving two different delays. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results.