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

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2024年8月13日

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H. A. A. El-Saka,
D. El. A. El-Sherbeny,
A. M. A. El-Sayed

摘要

In this paper, we analyze the stability and Hopf bifurcation of the fractional-order logistic equation with two different delays $\tau _{1}, \tau _{2}>0$ $τ_{1}, τ_{2} > 0$ : $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.