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

• Published in: Computational & applied mathematics Vol. 43; no. 6
• Main Authors: El-Saka H A A, El A, El-Sherbeny D, El-Sayed A M A
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Differential equations; Hopf bifurcation; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Predictor-corrector methods; Stability; Fractional-order logistic equation; Time delays; Numerical solutions; 26A33; 34K37; 92-10; Stability analysis; Hopf bifurcation; 91B62; 65H05; 03C45; 34A08
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-024-02877-2

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