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

Abstract

AbstractIn this paper, we analyze the stability and Hopf bifurcation of the fractional-order logistic equation with two different delays $$\tau _{1}, \tau _{2}>0$$ τ 1 , τ 2 > 0 : $$D^{\alpha }y(t)=\rho y(t-\tau _{1})\left( 1-y(t-\tau _{2})\right) $$ D α y ( t ) = ρ y ( t - τ 1 ) 1 - y ( t - τ 2 ) , $$t>0$$ t > 0 , $$\rho >0$$ ρ > 0 . We describe stability regions by using critical curves. We explore how the fractional order $$\alpha $$ α , $$\rho $$ ρ , and time delays influence the stability and Hopf bifurcation of the model. Then, by choosing $$\rho $$ ρ , fractional order $$\alpha $$ 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.