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 τ ₁, τ ₂>0 τ 1 , τ 2 > 0 : Dαy(t)=ρ y(t-τ ₁)( 1-y(t-τ ₂)) D α y ( t ) = ρ y ( t - τ 1 ) 1 - y ( t - τ 2 ) , $$t>0$$ t > 0 , ρ >0 ρ > 0 . We describe stability regions by using critical curves. We explore how the fractional order α α , ρ ρ , and time delays influence the stability and Hopf bifurcation of the model. Then, by choosing ρ ρ , fractional order α α , 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.