Dynamic analysis of the fractional distributed delay models

www.nature.com/scientificreports OPEN Dynamic analysis of the fractional distributed delay models H. A. A. El-Saka1, D. El. A. El-Sherbeny1,2 & A. M. A. El-Sayed3 In this paper, we analyze the stability of the fractional distributed delay models. We use the linear chain trick to convert these models into an incommensurate fractional order systems. We get the stability regions by studying the characteristic equation around equilibrium points. We investigate how the fractional order α1 , ρ and a affect the stability of the models. We study the fractional order delay logistic equation and compare the influence of the distributed delay on the stability regions. Numerical simulations are exhibited to confirm the analytical results. Keywords Distributed delay models, Incommensurate fractional order systems, Stability analysis, Numerical solutions Fractional calculus (FC) is one of the most important fields of mathematics. It’s an effective tool for explaining a wide range of scientific and engineering phenomena. Fractional order differential equations proposed natural models to study real-world problems, such as signal processing, biological mathematics, control processing, viscoelastic system1–4. The differential equation is known as a delayed differential equation (DDE) if the state variable appears with a delayed argument. Differential equation models with delay have become common in many technical and scientific fields in recent decades, such as biology, physics, neutral networks, and epidemiology5–11. The fractional delay differential equation (FDDE) has been applied for several years in many fields3,4,12–15. Fractional order delay differential equations can be better applied in biological, economic and social systems where memory effects play a role than in integer order equations. Delay differential equations with distributed delay can be represented in general as ˆ ∞ ẋ = F (t, x(t), µ(τ )x(t − τ )). 0 The distributed fractional-order nonlinear dynamical systems were first studied by Caputo in 196916. Distributed delay differential equations are particularly attractive for modeling evaluation due to their long delays and mathematical tractability5. Recently, the application of distributed delay models has been used in a number of fields. For example, in biology, Compbell et al.17 studied how distributed delays arise in biological models and reviewed the literature on such models. Cushing18 discussed the stability and bifurcation analysis of some specific biological models. Some linear stability results for general distribution have been obtained by N. MacDonald19,20. In neural network, Liu et al.21 investigated the dynamical behaviors of a fractional-order neural network with leakage, discrete and distributed delays. Zhou et al.22 discussed stability and Hopf bifurcation analysis on a two neuron network with both discrete and distributed delays. Karaoğlu et al.23 investigated local stability and Hopf bifurcation in a two—neuron network model with multiple discrete and distributed delays. In ecological model, Yu et al.24 studied the dynamic complexities of an ecological model with impulsive control strategy and distributed time delay. AV. Paparao25 investigated the stability analysis of three species ecological model with distributed time delay. In control systems, Riccati equations with distributed delay are necessary for developing robust control systems and improving their performance. El-Sayed et al.26 investigated the dynamic properties of Riccati differential equation with distributed delay. In neurology, Karmeshu et al.27 analyzed a neuronal model with distributed delay. Liao et al.28 investigated a general two neuron model with distributed delays and a weak kernel. Chen et al.29 explored the extended strictly positive realness of fractional order systems with distributed delays (FOSDDs) and the application to robust control. Kiskinov et al.30 studied a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. Chen et al.31 explored an order-dependent stability condition for nominal FOSDDs which is obtained through utilizing the small gain theorem. Popivanov et al.32 introduced an appropriate definition of the notion of a Lagrange (formal) adjoint system for homogeneous systems with incommensurate-order derivatives in Caputo’s sense and distributed delays. 1Mathematics Department, Faculty of Science, Damietta University, New Damietta 34517, Egypt. 2Faculty of Artificial Intelligence and Information, Horus University, New Damietta, Egypt. 3Faculty of Science, Alexandria University, Alexandria 21526, Egypt. email: Scientific Reports | (2026) 16:16252 | https://doi.org/10.1038/s41598-026-52327-8 1 The objective of this paper is to investigate the dynamic analysis of the fractional distributed delay models (20) and (24). In Section "Fractional order delay logistic equation", we obtained the stability analysis of the fractional order delay logistic equation by using the critical curve method. In Section "Fractional distributed delay models", by using the linear chain trick, we transform these models into an incommensurate fractional order systems. Section "Numerical simulations" contains the numerical simulations. Fractional order delay logistic equation Definition 2.1 33 The Riemann-Liouville fractional integral operator of order α ∈ R+ of the function f(t), t ≥ a, is defined by ˆ t 1 Iaα f (t) = (t − τ )α−1 f (τ )dτ, Γ(α) a where Γ(.) is the Gamma function. When a = 0, we set I0α f (t) = I α f (t) for simplicity. Definition 2.2 33 The Caputo fractional derivative of order α > 0 of f(t), t ≥ a, is defined by Daα f (t) = Ian−α Dn f (t), d where D = dt , and n − 1 < α ≤ n, n ∈ N . For simplicity, we set D0α f (t) = Dα f (t) when a = 0. For the main properties of fractional calculus, see33–38. In this section, we first consider the work of El-Saka et al.12, who studied the stability analysis and Hopf bifurcation of a fractional-order logistic equation with two different delays τ1 , τ2 > 0 Dα y(t) = ρy(t − τ1 ) (1 − y(t − τ2 )) , y(t) = ϕ(t), −τ ≤ t ≤ 0, t > 0, ρ > 0, (1) where Dα is a Caputo fractional derivative of order 0 < α ≤ 1, the initial condition ϕ(t) is continuous on [−τ, 0] and τ = max{τ1 , τ2 }. By using the results in12, we proposed the fractional-order delay logistic equation Dα y(t) = ρy(t) − ρy 2 (t − r), y(t) = ϕ(t), r, ρ > 0, t ∈ (0, T ],(2) t ⩽ 0.(3) The equilibrium points of Eq. (2) satisfy the equation ρy ∗ (1 − y ∗ ) = 0,(4) which yields y1∗ = 0, y2∗ = 1. We will investigate the stability of Eq. (2) by using the critical curve method8 and compare these results with the finding we will present in the next sections. The characteristic equation associated with the fractional order logistic equation with time delay Eq. (2) can be written as ( ) λα + ρ 2y ∗ e−λr − 1 = 0.(5) An equilibrium point y ∗ is asymptotically stable if all the roots λi of the c (...truncated)