Stability Analysis and Bifurcation Control of a Fractional Order Predator Prey System

Abstract

Fractional-order predator–prey models provide superior ecological fidelity by capturing intrinsic memory effects. However, the fractional derivative order introduces an additional dimension to parameter space, which is unexplored in existing dynamical analyses. This paper delves into the bifurcation behaviors of a two-dimensional fractional-order predator–prey model, taking into account the maturation period of larvae. By utilizing time delay as the bifurcation parameter, the characteristic equation is analyzed to derive conditions for stability and bifurcation. Meanwhile, the introduction to fractional-order theory endows the predator–prey system with improved historical dependence. The relationship between order and bifurcation behavior in the fractional-order predator–prey model is discussed in simulation experiments, which provides a reference for the establishment of the fractional-order system. Additionally, this paper introduces a controller to control the bifurcation behavior through feedback-gain parameters. Simulation results not only exhibit the dynamic characteristics of fractional-order models, but also demonstrate the success of controllers over bifurcation behaviors.

1. Introduction

The predator model is a mathematical model established using computer technology to describe ecological phenomena [1,2], taking into account various ecological factors, including competition, parasitism, and disease. For example, ref. [3] introduced a model with two predators and one prey, and the competition of different predators for the same prey, which shows that competitive behavior can lead to an increase in the number of predators and affect the survival of the prey. El-Shahed and Moustafa (2025) constructed a fractional-order ecological epidemiological model that incorporates two disease strains and analyzed the stability conditions of disease transmission within the predator population [4]. Ref. [5] analyzed a predator prey system containing parasitism terms, and the parasitism might cause predator extinction, which affects the stability of the ecosystem. Ref. [6] found that the addition of small migrations to predator–prey systems induced stable convergence of LV systems with three types of functional responses, showing that natural predator–prey populations can be stabilized by a small number of sporadic migrations. Ref. [7] proposed a predator prey model with a disease, which showed a certain resistance when the disease occurs in the system. However, the above predator models are based on the theory of integer-order calculus, and do not take into account the genetic characteristics and memory functions of the predator–prey system, which makes the simulation results often deviate from reality. Therefore, how to incorporate genetic characteristic and memory function into the predator system is a profound and meaningful topic. The foundational predator–prey models, such as the classic Lotka–Volterra framework and its extensions with more complex functional responses (e.g., Holling types), are traditionally built upon integer-order calculus. These models are instrumental in capturing basic ecological interactions like competition and parasitism [1,2,3,4,5,6,7]. However, their fundamental assumption is Markovian, meaning that the population growth rate at a given time depends only on the current state. This limits their ability to simulate systems with memory effects or hereditary traits, often leading to deviations from realistic ecological dynamics.

As a generalization of integer-order models, fractional-order models have a strong memory function, which is more suitable to accurately simulate and predict the dynamic behaviors of those materials or phenomena with memory, genetic, mechanical, and electrical properties [8,9,10]. The fundamental limitation of integer-order models lies in their local and Markovian nature. In contrast, fractional-order derivatives, such as the Caputo derivative used here, are inherently non-local. The term ${\displaystyle \frac{1}{\mathrm{\Gamma }(k-\alpha )}}{\int }_a^t{(t-\tau )}^{k-q-1}{\hslash }^{\left(k\right)}\left(\tau \right)d\tau$ embodies a memory kernel with a power-law decay. This mathematical structure is uniquely suited to model biological processes where past states, such as historical population levels or accumulated stress, exert a fading influence on the current rate of change. Ref. [11] proposed a fractional predator model with Crowley–Martin type functional response term and showed the effect of fractional-order on the dynamic behavior of the predator system. Ref. [12] pointed out that the fractional model could capture the whole time-state of the biological process, while the integer-order model could only relate to a certain change or feature of the biological process in a specific time. Ref. [13] described a predator model using fractional calculus, which showed that fractional predator systems were more closely related to the intrinsic characteristics of biological systems. Wu et al. (2025) [14] employed the extended (G’/G) expansion method to obtain the exact solution of the spatiotemporal fractional-order diffusion predator–prey system. They also discovered that when the fractional-order parameter was around 0.7, the system exhibited chaotic behavior, providing a new tool for population dynamics simulation [14]. The aforementioned literature shows the historical evolution process of the predator system and discuses the dynamic behavior of fractional-order, in which Hopf bifurcation, as a classical dynamic behavior, is worthy of further discussion.

Among many dynamic behaviors of fractional-order systems, bifurcation is worthy of attention [15]. Bifurcation refers to a qualitative change in the state of a system when a certain parameter crosses a critical value. These parameters have a very significant influence on the dynamic behavior of the system. The research results of [16] emphasized the role of the exponents p and q in regulating the behavior of centromere, and it is founded that the type II model with perturbation functions exhibited stability and can rapidly suppress oscillations in various forms of noise and sudden shocks. Among the dynamic behaviors caused by parameters, the Hopf bifurcation is more classical and worthy of study [17]. There are many Hopf bifurcation parameters in predator systems, such as prey environmental carrying capacity and predator mortality, time delay, and predator’s mean conversion rate [18,19,20,21]. The time delay is more prevalent in predator systems. Ref. [22] introduced a fractional-order predator system which considered time delay as a Hopf bifurcation parameter and the influence of Hopf bifurcation was studied using a stability analysis. Ref. [23] proposed a fractional-order predator system with discrete time delay and distributed time delay, where the influence of time delay on bifurcation condition was compared using an image. The above works discuss the effect of time delay as a bifurcation parameter and how the bifurcation affects the stable state of the system. Therefore, how to control the bifurcation behavior such that the state of the system remains stable is a meaningful topic.

Bifurcation behavior in a fractional predator system will affect the stability of the system, but it can also improve the stability of the system in a reasonable control plan. Ref. [24] described the dual character of bifurcation. Therefore, scholars have considered bifurcation to be a useful tool for controlling system behavior. Ref. [25] put forward an approach to controlling bifurcation, and it was found that selecting the appropriate system parameters can affect the system state. Ref. [26] designed a parametric feedback controller to affect Hopf bifurcation. Ref. [27] utilized the linear feedback technique to design a control law for preventing the Hopf bifurcation behavior. The above papers investigated the influence of different control methods on the bifurcation behavior. However, it is also worth studying the bifurcation behavior with differently preset bifurcation parameters. At present, most of the existing control schemes for fractional-order predator systems only consider the bifurcation parameter without involving the effect of fractional dynamic behavior. Therefore, the effect of fractional-order on bifurcation deserves to be explored in depth.

Based on the above discussion, this paper attempts to investigate a fractional-order predator–prey system with a delay. The main contributions of this paper are as follows. (1) The bifurcation conditions are determined according to Hopf bifurcation theory and stability analysis. At the same time, the stable state of the system is verified by theoretical proof and simulation. (2) To further explore how the order of fractional-order system affects Hopf bifurcation behavior, the relationship between the system order and the bifurcation is further discussed. (3) A feedback controller is designed to control the bifurcation behavior. In addition, in order to better explore the control effect, the relationship between the feedback gain parameter and the bifurcation point is also discussed.

The general structure of this paper is as follows. In Section 2, some necessary theories of fractional calculus and stability analysis are given, the stability conditions of the fractional order system are explored in Section 3, and the influence of the controller on bifurcation is discussed in Section 4. The simulation results are presented in Section 5 to verify the feasibility of the conditions. Section 6 is the summary of this paper.

2. Preliminaries

In this section, the basic definition of fractional order derivative and the stability theory of linear fractional order systems are given to facilitate subsequent operations.

2.1. Fractional Derivative

Definition 1

([28]). Define the q-order Caputo fractional-order derivative for $\hslash \left(t\right)$ as

$${}_a{}^{c}D_t^q\hslash \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(k-\alpha )}}{\int }_a^t{(t-\tau )}^{k-q-1}{\hslash }^{\left(k\right)}\left(\tau \right)d\tau ,$$

(1)

where $a<t$, $q\in (0,1)$, $\mathrm{\Gamma }(·)$ is the Gamma function, that is, $\mathrm{\Gamma }\left(\mathrm{\Psi }\right)={\int }_0^{\infty }{t}^{\mathrm{\Psi }-1}{e}^{-t}dt$.

Lemma 1

([29]). The Laplace transform of Caputo fractional-order derivative is expressed by

$$L\{{}_a{}^{c}D_t^q\hslash \left(t\right);s\}=s^q\overline{\hslash }\left(s\right)-\sum _{k=0}^{n-1}{s}^{q-k-1}{\hslash }^{\left(k\right)}\left(0\right).$$

If ${\hslash }^{\left(k\right)}\left(0\right)=0$, $k=0,1,2,\cdots ,n$, then $L\{{}_a{}^{c}D_t^q\hslash \left(t\right);s\}=s^q\overline{\hslash }\left(s\right)$.

Remark 1.

There are three common kinds of fractional derivatives, among which R-L fractional-order derivative is the most basic definition and has a deep mathematical foundation [30]. G-L fractional-order derivative approximates fractional derivative by means of discretization [31], and is therefore less complete than the other t- wo fractional-order derivatives in terms of mathematical rigor. The Caputo fractional derivative provides an effective framework for solving initial value problems. Its definition yields a null derivative for constant functions—consistent with fundamental physical phenomena—distinguishing it from alternative fractional formulations. Furthermore, the Caputo operator not only reduces naturally to classical integer-order differentiation, but also exhibits a streamlined Laplace transform. This analytical simplicity significantly facilitates the application of fractional calculus. Given these advantages, particularly in modeling, analyzing, and controlling fractional-order dynamical systems, the Caputo definition is adopted as the algorithmic foundation for fractional calculus in this work. To simplify the derivation, we only consider $q\in (0,1]$, and use $D^q\hslash \left(t\right)$ instead of ${}_a{}^{c}D_t^q\hslash \left(t\right)$.

2.2. Stability Theory

The system stability analysis theory is given in the following:

Lemma 2

([32]). Consider the following m-dimensional fractional-order equation for ${\chi }_i$, $(i=1,2,\cdots ,m)$ as

$$\left\{\begin{array}{c}{D}^{q_1}{\chi }_1\left(t\right)=n_{11}{\chi }_1(t-\tau )+n_{12}{\chi }_2(t-\tau )+\cdots +n_{1m}{\chi }_m(t-\tau ),\\ D^{q_2}{\chi }_2\left(t\right)=n_{12}{\chi }_2(t-\tau )+n_{22}{\chi }_2(t-\tau )+\cdots +n_{2m}{\chi }_m(t-\tau ),\\ \hspace{1em} ⋮\hfill \\ D^{q_m}{\chi }_m\left(t\right)=n_{m1}{\chi }_1(t-\tau )+n_{m2}{\chi }_2(t-\tau )+\cdots +n_{mm}{\chi }_m(t-\tau ).\end{array}\right.$$

(2)

where $q_i\in (0,1]$ and $n_{ii}\in R$. The characteristic equation of Equation (2) can be derived by Laplace transform as

$$▵\left(s\right)=det\left[\begin{array}{cccc}{s}^{q_1}-n_{11}{e}^{-s\tau }& -n_{12}{e}^{-s\tau }& \cdots & -n_{1m}{e}^{-s\tau }\\ -n_{21}{e}^{-s\tau }& s^{q_2}-n_{22}{e}^{-s\tau }& \cdots & -n_{2m}{e}^{-s\tau }\\ ⋮& ⋮& & ⋮\\ -n_{m1}{e}^{-s\tau }& -h_{m2}{e}^{-s\tau }& \cdots & s^{q_m}-n_{mm}{e}^{-s\tau }\end{array}\right].$$

(3)

If all the roots of det $(\Delta (s\left)\right)=0$ have negative real parts, then the trivial-equilibrium of (2) is Lyapunov asymptotically stable.

Remark 2.

The stability analysis in this paper is conducted within the framework of Lyapunov stability theory. The "asymptotic stability" we refer to indicates that the system’s trajectories starting sufficiently close to an equilibrium point will not only remain nearby, but also eventually converge to it as time approaches infinity. Our focus is on local asymptotic stability, ascertained by the eigenvalues of the linearized system. Global stability, which requires convergence from all initial conditions, is not considered here due to the nonlinear nature of the system and the presence of bifurcations, which inherently limit the basin of attraction.

2.3. Mathematical Model Description

Fractional-order calculus provides a powerful generalization to overcome this limitation. The key distinction lies in the non-local nature of fractional derivatives, which incorporate the entire history of the system’s state via a power-law kernel. This mathematical property is exceptionally suited for modeling biological processes with memory and genetic characteristics. Therefore, replacing the integer-order derivatives in a classic model with fractional-order ones is not merely a technical substitution, but a fundamental enhancement. It endows the model with a inherent memory mechanism, enabling a more accurate representation of how past population levels influence present dynamics, such as the lagged recovery of growth after a disturbance. Based on the above discussion and [33], the fractional-order derivative of the predator prey system is proposed:

$$\left\{\begin{array}{c}{D}^{q_1}x\left(t\right)=rx\left(t\right)\left[1-{\displaystyle \frac{x(t-\tau )}{k}}\right]-{\displaystyle \frac{\alpha x\left(t\right)y\left(t\right)}{a+bx\left(t\right)+cy\left(t\right)}},\hfill \\ D^{q_2}y\left(t\right)=b_{0}y(t-\tau )-\left(d_o-{\displaystyle \frac{\beta x\left(t\right)}{a+bx\left(t\right)+cy\left(t\right)}}\right)y\left(t\right)-q{E}_{0}y\left(t\right),\hfill \end{array}\right.$$

(4)

where $q_1,q_2\in (0,1]$. $x\left(t\right)$ and $y\left(t\right)$ represent population density of adult preys and predators at any time t, respectively. $r,k,\alpha ,a,\beta ,c,b,b_0,d_0$ are positive constants. c is the mortality rate of predator juveniles which take units of time to mature, q is the catch ability coefficient of the predator species, and $E_0$ is the harvesting effort. Assume that the juveniles of prey and predator take $\tau$ units of time to mature, and $\tau \in R^{+}$.

The fractional-order model is not a reconstruction of the integer order framework, but a natural extension based on the classical theory. In this paper, the Caputo fractional derivative (Equation (1)) is used as the core of the extension, which has the “seamless compatibility” with the integer-order derivative. When the fractional order $q1=q2=1$, the Caputo derivative deforms to the traditional integer-order first derivative $({lim}_{q=1}{D}^{q}x\left(t\right)$$=\dot{x}\left(t\right))$. In this case, the fractional-order model (Equation (4)) is completely equivalent to the integer-order model. This “limit degenerate consistency” ensures that the integer-order model is a special case of the fractional-order model, and it is possible to transition integer-order models directly.

Remark 3.

The above-mentioned system was proposed based on the integer-order predator-prey system described in [33]. The definition of first-order derivatives of integer order: $\dot{x}\left(t\right)={lim}_{\Delta t\to 0}{\displaystyle \frac{x\left(t\right)-x(t-\Delta t)}{\Delta t}}$. Focusing only on t and $t-\Delta t$ function value difference is essentially “approximating the global dynamics with the instantaneous rate of change of the infinitesimal neighborhood", completely discarding all historical information before $t-\Delta t$. In contrast, the integral field of the fractional order is $[0,t]$ (rather than the “local neighborhood of t and $t-\Delta t$” of the integer order), which means that the derivative of the system at the current moment t (i.e., the population growth rate) needs to be superimposed with the state change rate $x^{\prime }\left(\tau \right)$ of all historical moments—from the initial moment 0 to t—this mathematically forces the inclusion of “all-time historical information". Rather than relying solely on the instantaneous state at time t. In addition, ${\displaystyle \frac{1}{{(t-\tau )}^{-q}}}$ in the integral kernel is the “historical weight coefficient". When τ is close to t (recent history), $t-tau$ tends to 0, and the weight coefficient increases sharply. When τ is much smaller than t (long-term history), $t-tau$ increases, and the weight coefficient gradually decays. It fully conforms to the objective law of “recent history has a stronger impact on the current" of the ecosystem. This property is the “memory" caused by the integral kernel of the Caputo derivative itself, and there is no need to “externally patch" the historical effect by additional time delay term or empirical correction term as, the integer-order model relies on the Markov assumption (only the instantaneous state is considered). Therefore, it is more reasonable to consider the predator–prey model with fractional order characterization.

Remark 4.

The above two-dimensional fractional-order predator–prey model contains a Beddington–DeAngelis functional response, which is a mathematical model describing predator predation behavior [34]. The Beddington–DeAngelis functional response considers the predator predation efficiency for the density of the prey and the density of the predator. In addition, the system takes into account the maturation time of juveniles and treats it as a time delay term. After the time delay at the introduction of the model, scholars found that the system appeared many dynamic behaviors, such as chaos [35], bifurcation, and periodic fluctuations [36], and these dynamic behaviors can affect the stability of the model.

If time delay is not considered, system (4) has the following expression as

$$\left\{\begin{array}{c}{D}^{q_1}x\left(t\right)=rx\left(t\right)\left[1-{\displaystyle \frac{x\left(t\right)}{k}}\right]-{\displaystyle \frac{\alpha x\left(t\right)y\left(t\right)}{a+bx\left(t\right)+cy\left(t\right)}},\hfill \\ D^{q_2}y\left(t\right)=b_{0}y\left(t\right)-\left(d_o-{\displaystyle \frac{\beta x\left(t\right)}{a+bx\left(t\right)+cy\left(t\right)}}\right)y\left(t\right)-q{E}_{0}y\left(t\right),\hfill \end{array}\right.$$

(5)

System (5) has an equilibrium point ${\varsigma }^{\ast }(x^{\ast },y^{\ast })$ that satisfies the following equation

$$\left\{\begin{array}{c}0=r{x}^{\ast }\left[1-{\displaystyle \frac{x^{\ast }}{k}}\right]-{\displaystyle \frac{\alpha x^{\ast }{y}^{\ast }}{a+b{x}^{\ast }+c{y}^{\ast }}},\hfill \\ 0=b_0{y}^{\ast }-\left(d_o-{\displaystyle \frac{\beta x^{\ast }}{a+b{x}^{\ast }+c{y}^{\ast }}}\right)y^{\ast }-q{E}_0{y}^{\ast }.\hfill \end{array}\right.$$

(6)

Through simple calculation, the equilibrium point expression can be obtained as

$$\left\{\begin{array}{c}{x}^{\ast }={\displaystyle \frac{-(b_{0}a-d_{0}a-q{E}_{0}a)+y^{\ast }(b_{0}c-d_{0}c+q{E}_{0}c)}{b_{0}b-d_{0}b+\beta -q{E}_{0}b}},\hfill \\ y^{\ast }={\displaystyle \frac{r(ka-x^{\ast }+kb{x}^{\ast }-x^{\ast 2}b)}{r(ck+x^{\ast })-\alpha }}.\hfill \end{array}\right.$$

(7)

Then Equation (7) can be rewritten as

$$\begin{array}{c}\hfill P{\left(x^{\ast }\right)}^2+Q{x}^{\ast }+R=0,\end{array}$$

(8)

where

$$\begin{array}{c}P=r(C+Bb),R=-a[2rkcq{E}_0+(b_0-d_0-q{E}_0)\alpha ],Q=r[Cck+A-B(kb-1)]-C\alpha ,\hfill \\ C=b(b_0-d_0-q{E}_0)+\beta ,B=c(b_0-d_0+q{E}_0),A=a(b_0-d_0-q{E}_0).\hfill \end{array}$$

If system (4) has at least one positive equilibrium point ${\varsigma }^{\ast }(x^{\ast },y^{\ast })$, $(C+Bb)(2rkcq{E}_0+(b_0-d_0-q{E}_0)\alpha )>0$ need to be satisfied. In the next part, time delay is used as the Hopf bifurcation parameter, and the bifurcation condition is analyzed according to the Hopf bifurcation stability analysis theory.

3. Bifurcation Analysis

Employ the coordinate changes $\overline{x}\left(t\right)=x\left(t\right)-x^{\ast }$, $\overline{y}\left(t\right)=y\left(t\right)-y^{\ast }$. In the aforesaid statement, the point ${\varsigma }^{\ast }(x^{\ast },y^{\ast })$ is given, and $\left(x\right(0),y(0\left)\right)$ is chosen as $(x^{\ast },y^{\ast })$ to facilitate the subsequent operation, then the linearization equation of system (4) at ${\varsigma }^{\ast }$ by Taylor expansion is

$$\left\{\begin{array}{cc}\hfill D^{q_1}\overline{x}\left(t\right)& =\left[r\left(1-{\displaystyle \frac{x^{\ast }}{k}}\right)-{\displaystyle \frac{\alpha y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\right]\overline{x}\left(t\right)\hfill \\ & -{\displaystyle \frac{\alpha x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\overline{y}\left(t\right)-r{\displaystyle \frac{x^{\ast }}{k}}\overline{x}(t-\tau ),\hfill \\ \hfill D^{q_2}\overline{y}\left(t\right)& ={\displaystyle \frac{\beta y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\overline{x}\left(t\right)+\left[{\displaystyle \frac{\beta x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}-d_0-q{E}_0\right]\overline{y}\left(t\right)\hfill \\ & +b_0\overline{y}(t-\tau ).\hfill \end{array}\right.$$

(9)

This linearized equation can be expressed as

$$\left\{\begin{array}{c}{D}^{q_1}\overline{x}\left(t\right)=m_{11}\overline{x}\left(t\right)-m_{12}\overline{y}\left(t\right)-m_{13}\overline{x}(t-\tau ),\hfill \\ D^{q_2}\overline{y}\left(t\right)=m_{21}\overline{x}\left(t\right)+m_{22}\overline{y}\left(t\right)+m_{23}\overline{y}(t-\tau ),\hfill \end{array}\right.$$

(10)

where

$$\begin{array}{c}{m}_{11}=\left[r\left(1-{\displaystyle \frac{x^{\ast }}{k}}\right)-{\displaystyle \frac{\alpha y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\right],m_{12}={\displaystyle \frac{\alpha x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}},m_{13}=r{\displaystyle \frac{x^{\ast }}{k}},\hfill \\ m_{21}={\displaystyle \frac{\beta y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}},m_{22}=\left[{\displaystyle \frac{\beta x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}-d_0-q{E}_0\right],m_{23}=b_0.\hfill \end{array}$$

The characteristic equation of Equation (10) can be obtained as

$$det\left[\begin{array}{cc}{s}^{q_1}-m_{11}+m_{13}{e}^{-s\tau }& m_{12}\\ -m_{21}& s^{q_2}-(m_{22}+m_{23}{e}^{-s\tau })\end{array}\right]=0.$$

(11)

Equation (11) can be further expressed as

$$\begin{array}{c}{s}^{q_1+q_2}-(m_{22}+m_{23}{e}^{-s\tau })s^{q_1}-s^{q_2}(m_{11}-m_{13}{e}^{-s\tau })\hfill \\ +(m_{11}-m_{13}{e}^{-s\tau })(m_{22}+m_{23}{e}^{-s\tau })+m_{21}{m}_{12}=0.\hfill \end{array}$$

(12)

To further discuss the effect of $\tau$ on the bifurcation condition of system (4), Equation (12) can be rewritten as

$$\begin{array}{c}\hfill {\aleph }_1\left(s\right)+{\aleph }_2\left(s\right)e^{-s\tau }+{\aleph }_3\left(s\right)e^{-2s\tau }=0,\end{array}$$

(13)

where

$$\begin{array}{c}{\aleph }_1\left(s\right)=s^{q_1+q_2}-m_{22}{s}^{q_1}-s^{q_2}{m}_{11}+m_{11}{m}_{22}+m_{21}{m}_{12},\hfill \\ {\aleph }_2\left(s\right)=s^{q_2}{m}_{13}-m_{13}{m}_{22}+m_{11}{m}_{23}-m_{23}{s}^{q_1},\hfill \\ {\aleph }_3\left(s\right)=-m_{13}{m}_{23}.\hfill \end{array}$$

Multiplying both sides of Equation (13) by $e^{s\tau }$ yields

$$\begin{array}{c}\hfill {\aleph }_1\left(s\right)e^{s\tau }+{\aleph }_2\left(s\right)+{\aleph }_3\left(s\right)e^{-s\tau }=0.\end{array}$$

(14)

If all roots of Equation (14) satisfy Lemma 2, then they have negative reel parts.

Let $s=\Re (cos{\displaystyle \frac{\pi }{2}}+isin {\displaystyle \frac{\pi }{2}})(\Re >0)$. Labeled ${\aleph }_j^r$ and ${\aleph }_j^i$, $(j=1,2,3)$ as real and imaginary parts of ${\aleph }_j\left(s\right)$. Then Equation (14) can be rewritten as

$$\left\{\begin{array}{c}({\aleph }_1^r+{\aleph }_3^r)cos\Re \tau -{\aleph }_1^{i}sin\Re \tau =-{\aleph }_2^r,\hfill \\ {\aleph }_1^{i}cos\Re \tau +({\aleph }_1^r-{\aleph }_3^r)sin\Re \tau =-{\aleph }_2^i,\hfill \end{array}\right.$$

(15)

where

$$\begin{array}{c}{\aleph }_1^r={\Re }^{q_1+q_2}cos{\displaystyle \frac{(q_1+q_2)\pi }{2}}-m_{22}{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}}-{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}}{m}_{11}+m_{11}{m}_{22}+m_{21}{m}_{12},\hfill \\ {\aleph }_1^i={\Re }^{q_1+q_2}sin{\displaystyle \frac{(q_1+q_2)\pi }{2}}+m_{22}{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}}+{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}}{m}_{11},\hfill \\ {\aleph }_2^r={\Re }^{q_2}cos{\displaystyle \frac{q_2\pi }{2}}{m}_{13}-m_{13}m22+m_{11}{m}_{23},-m_{23}{\Re }^{q1}cos{\displaystyle \frac{q_1\pi }{2}},\hfill \\ {\aleph }_2^i={\Re }^{q_2}sin{\displaystyle \frac{q_2\pi }{2}}{m}_{13}-m_{23}{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}},\hfill \\ {\aleph }_3^r=-m_{13}{m}_{23},\hfill \\ {\aleph }_3^i=0.\hfill \end{array}$$

By solving Equation (15), the following expressions can be obtained:

$$\left\{\begin{array}{c}cos\Re \tau ={\displaystyle \frac{-{\aleph }_2^r({\aleph }_1^r-{\aleph }_3^r)-{\aleph }_2^i{\aleph }_1^i}{{\left({\aleph }_1^r\right)}^2+{\left({\aleph }_1^i\right)}^2-{\left({\aleph }_3^r\right)}^2}}=f_1(\Re ),\hfill \\ sin\Re \tau ={\displaystyle \frac{-{\aleph }_2^i({\aleph }_1^r+{\aleph }_3^r)+{\aleph }_2^r{\aleph }_1^i}{{\left({\aleph }_1^r\right)}^2+{\left({\aleph }_1^i\right)}^2-{\left({\aleph }_3^r\right)}^2}}=f_2(\Re ),\hfill \end{array}\right.$$

(16)

which satisfies the following relation:

$$\begin{array}{c}{f}_1{(\Re )}^2+f_2{(\Re )}^2=1.\hfill \end{array}$$

(17)

The derived characteristic Equation (12), $s^{q_1+q_2}+\cdots$, is fundamentally more complex than its integer-order counterpart. The fractional powers of s introduce branch points and a complex multi-sheeted Riemann surface, meaning that the system’s stability is governed by an infinite number of characteristic roots. Our analysis focuses on the dominant root crossing the imaginary axis. Critically, the terms $cos{\displaystyle \frac{(q_1+q_1)\pi }{2}}$ and $sin{\displaystyle \frac{(q_1+q_1)\pi }{2}}$ in the real and imaginary parts (Equation (15)) reveal a direct mechanism through which the fractional-orders influence the bifurcation: they introduce a phase shift that effectively alters the critical delay ${\tau }_0$. This phase modulation, a direct consequence of the memory effect, is a specific and profound feature of the fractional-order model that our work quantitatively characterizes.

Assumption 1.

Equation (17) has a series of positive roots ${\Re }_k(k=0,1,2,\cdots )$.

The critical delay point has the following form if Assumption 1 holds:

$${\tau }_j^{\left(k\right)}={\displaystyle \frac{1}{{\Re }_k}}[arccos{f}_1\left({\Re }_k\right)+2j\pi ],j=0,1,2,\cdots$$

(18)

According to above Equation (18), ${\tau }_0$ is defined as

$${\tau }_0={\tau }_j^{\left(0\right)}=min{\tau }_j^{\left(k\right)},k=0,1,2,\cdots$$

(19)

In order to ensure that the above discussion is correct and meaningful, the system should satisfy the transmissibility condition at ${\tau }_0$:

Let $s\left(\tau \right)=\lambda \left(\tau \right)+i\Re \left(\tau \right)$ be the root of Equation (13), and $s\left(\tau \right)$ at $\tau ={\tau }_0$ satisfy $\lambda \left({\tau }_0\right)=0$, $\Re ={\Re }_0$, then the transversality condition as follows

$$\begin{array}{c}\hfill Re\left[{\displaystyle \frac{ds}{d\tau }}\right]{|}_{(\Re ={\Re }_0,\tau ={\tau }_0)}\ne 0.\end{array}$$

(20)

Let ${\aleph }_j^r$ and ${\aleph }_j^i$ be the real and imaginary parts of ${\aleph }_j(j=1,2,3)$, respectively. According to the implicit function theorem, taking the derivative of Equation (14) with respect to $\tau$ implies

$$\begin{array}{c}{\aleph }_1^{\prime }\left(s\right){\displaystyle \frac{ds}{d\tau }}+\left[{\aleph }_2^{\prime }\left(s\right){\displaystyle \frac{ds}{d\tau }}{e}^{-s\tau }+{\aleph }_2\left(s\right)e^{-s\tau }\left(-\tau {\displaystyle \frac{ds}{d\tau }}-s\right)\right]\hfill \\ +\left[{\aleph }_3^{\prime }\left(s\right){\displaystyle \frac{ds}{d\tau }}{e}^{-2s\tau }-{\aleph }_3\left(s\right)e^{-2s\tau }\left(2\tau {\displaystyle \frac{ds}{d\tau }}-2s\right)\right]=0.\hfill \end{array}$$

(21)

One can obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{ds}{d\tau }}={\displaystyle \frac{\pounds \left(s\right)}{\S \left(s\right)}},\end{array}$$

(22)

where

$$\begin{array}{cc}\hfill & \pounds \left(s\right)=s[{\aleph }_2\left(s\right)e^{-s\tau }+2{\aleph }_3\left(s\right)e^{-2s\tau }],\hfill \\ \hfill & \S \left(s\right)={\aleph }_1^{\prime }\left(s\right)+[{\aleph }_2^{\prime }\left(s\right)-{\aleph }_2\left(s\right)\tau ]e^{-s\tau }+[{\aleph }_3^{\prime }\left(s\right)-2\tau {\aleph }_3\left(s\right)]e^{-2s\tau }.\hfill \end{array}$$

Set ${\pounds }^r$ and ${\pounds }^i$ as the real and imaginary parts of $\pounds \left(s\right)$. Similarly, denote ${\S }^r$ and ${\S }^i$ as the real and imaginary parts of $\S \left(s\right)$, then Equation (22) becomes

$$\begin{array}{c}\hfill Re\left[{\displaystyle \frac{ds}{d\tau }}\right]{|}_{(\Re ={\Re }_0,\tau ={\tau }_0)}={\displaystyle \frac{{\pounds }^r{\S }^r+{\pounds }^i{\S }^i}{{\S }^{r^2}+{\S }^{i^2}}},\end{array}$$

(23)

where

$$\begin{array}{cc}\hfill {\pounds }^r=& {\Re }_0({\aleph }_2^{r}sin{\Re }_0{\tau }_0-{\aleph }_2^{i}cos{\Re }_0{\tau }_0+2{\aleph }_3^{r}sin{\Re }_0{\tau }_0-2{\aleph }_3^{i}cos{\Re }_0{\tau }_0),\hfill \\ \hfill {\pounds }^i=& {\Re }_0({\aleph }_2^{r}cos{\Re }_0{\tau }_0-{\aleph }_2^{i}sin{\Re }_0{\tau }_0+2{\aleph }_3^{r}cos{\Re }_0{\tau }_0-2{\aleph }_3^{i}sin{\Re }_0{\tau }_0),\hfill \\ \hfill {\S }^r=& {\aleph }_1^{\prime r}+({\aleph }_2^{\prime r}-{\tau }_0{\aleph }_2^r)cos{\Re }_0{\tau }_0+({\aleph }_2^{\prime i}-{\tau }_0{\aleph }_2^i)sin{\Re }_0{\tau }_0+({\aleph }_3^{\prime i}-2{\tau }_0{\aleph }_3^i)sin2{\Re }_0{\tau }_0\hfill \\ & +({\aleph }_3^{\prime r}-2{\tau }_0{\aleph }_3^r)cos2{\Re }_0{\tau }_0,\hfill \\ \hfill {\S }^i=& {\aleph }_1^{\prime i}+({\aleph }_2^{\prime i}-{\tau }_0{\aleph }_2^i)cos{\Re }_0{\tau }_0-({\aleph }_2^{\prime r}-{\tau }_0{\aleph }_2^r)sin{\Re }_0{\tau }_0-({\aleph }_3^{\prime r}-2{\tau }_0{\aleph }_3^r)sin2{\Re }_0{\tau }_0\hfill \\ & +({\aleph }_3^{\prime i}-2{\tau }_0{\aleph }_3^i)cos2{\Re }_0{\tau }_0.\hfill \end{array}$$

Then, the transversality condition holds when the following conditions hold: ${\displaystyle \frac{{\pounds }^r{\S }^r+{\pounds }^i{\S }^i}{{\S }^{r^2}+{\S }^{i^2}}}\ne 0$.

By the above discussion, the following result can be obtained:

Theorem 1.

Based on the above discussion, the system will undergo a stability switch at the critical delay point ${\tau }_0$, when ${\displaystyle \frac{{\pounds }^r{\S }^r+{\pounds }^i{\S }^i}{{\S }^{r^2}+{\S }^{i^2}}}\ne 0$, so the following conclusion is obtained:

(1) 

System (9) is asymptotically stable in ${\varsigma }^{\ast }$, if $\tau \in [0,{\tau }_0)$.

(2) 

System (9) possesses Hopf bifurcation behavior when $\tau ={\tau }_0$. The state of the system will change from stable to unstable.

4. Bifurcation Control

A bifurcation controller is a strategy or component introduced into a dynamical system to modify its bifurcation characteristics, such as shifting the critical bifurcation point or altering the properties of the bifurcated solutions [24]. In this section, we design a feedback controller $k\left(y\right(t)-y(t-\tau \left)\right)$ with the specific aim of controlling the Hopf bifurcation induced by the time delay. After the controller is introduced, system (9) can be written as

$$\left\{\begin{array}{cc}\hfill D^{q_1}\overline{x}\left(t\right)& =\left[r\left(1-{\displaystyle \frac{x^{\ast }}{k}}\right)-{\displaystyle \frac{\alpha y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\right]\overline{x}\left(t\right)-{\displaystyle \frac{\alpha x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\overline{y}\left(t\right)\hfill \\ & -r{\displaystyle \frac{x^{\ast }}{k}}\overline{x}(t-\tau ),\hfill \\ \hfill D^{q_2}\overline{y}\left(t\right)& ={\displaystyle \frac{\beta y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\overline{x}\left(t\right)+\left[{\displaystyle \frac{\beta x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}-d_0-q{E}_0\right]\overline{y}\left(t\right)\hfill \\ & +b_0\overline{y}(t-\tau )+k(\left(t\right)-(t-\tau )).\hfill \end{array}\right.$$

(24)

The following equation can be obtained by linearizing Equation (24) as

$$\left\{\begin{array}{c}{D}^{q_1}\overline{x}\left(t\right)=m_{11}\overline{x}\left(t\right)-m_{12}\overline{y}\left(t\right)-m_{13}\overline{x}(t-\tau ),\hfill \\ D^{q_2}\overline{y}\left(t\right)=m_{21}\overline{x}\left(t\right)+m_{22}\overline{y}\left(t\right)+m_{23}\overline{y}(t-\tau )+k\overline{y}\left(t\right)-k\overline{y}(t-\tau ),\hfill \end{array}\right.$$

(25)

where

$$\begin{array}{c}{m}_{11}=\left[r\left(1-{\displaystyle \frac{x^{\ast }}{k}}\right)-{\displaystyle \frac{\alpha y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}\right],m_{12}={\displaystyle \frac{\alpha x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}},m_{13}=r{\displaystyle \frac{x^{\ast }}{k}},\hfill \\ m_{21}={\displaystyle \frac{\beta y^{\ast }(a+c{y}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}},m_{22}=\left[{\displaystyle \frac{\beta x^{\ast }(a+b{x}^{\ast })}{{(a+b{x}^{\ast }+c{y}^{\ast })}^2}}-d_0-q{E}_0\right],m_{23}=b_0.\hfill \end{array}$$

Laplace transform of Equation (25) can be obtained as

$$det\left[\begin{array}{cc}{s}^{q_1}-m_{11}+m_{13}{e}^{-s\tau }& m_{12}\\ -m_{21}& s^{q_2}-(m_{22}+m_{23}{e}^{-s\tau })-k+k{e}^{-s\tau }\end{array}\right]=0.$$

(26)

Equation (26) can be further expressed as

$$\begin{array}{cc}\hfill s^{q_1+q_2}& - s^{q_2}(m_{11}-m_{13}{e}^{-s\tau })+(m_{11}-m_{13}{e}^{-s\tau })(m_{22}+m_{23}{e}^{-s\tau })\hfill \\ \hfill & +m_{21}{m}_{12}-k(1-e^{-s\tau })(s^{q_1}-m_{11}+m_{13}{e}^{-s\tau })-(m_{22}+m_{23}{e}^{-s\tau })s^{q_1}=0.\hfill \end{array}$$

(27)

By simplification of Equation (27), it follows that

$$\begin{array}{c}\hfill \hslash \left(s\right)+{\hslash }_2\left(s\right)e^{-s\tau }+{\hslash }_3\left(s\right)e^{-2s\tau }=0.\end{array}$$

(28)

Let $s=\Re (cos{\displaystyle \frac{\pi }{2}}+isin{\displaystyle \frac{\pi }{2}})(\Re >0)$. After both sides of Equation (28) are multiplied by $e^{s\tau }$, one can obtain

$$\begin{array}{c}\hfill \hslash \left(s\right)e^{s\tau }+{\hslash }_2\left(s\right)+{\hslash }_3\left(s\right)e^{-s\tau }=0.\end{array}$$

(29)

If the two roots s of Equation (29) satisfy Lemma 2, then they have negative reel parts.

Denote ${\hslash }_j^r$ and ${\hslash }_j^i$$(j=1,2,3)$ as real and imaginary parts of ${\aleph }_j\left(s\right)$. Let $s=\Re (cos{\displaystyle \frac{\pi }{2}}+isin{\displaystyle \frac{\pi }{2}})(\Re >0)$, then Equation (29) can be rewritten as

$$\left\{\begin{array}{c}({\hslash }_1^r+{\hslash }_3^r)cos\Re \tau -{\hslash }_1^{i}sin\Re \tau =-{\hslash }_2^r,\hfill \\ {\hslash }_1^{i}cos\Re \tau +({\hslash }_1^r-{\hslash }_3^r)sin\Re \tau =-{\hslash }_2^i,\hfill \end{array}\right.$$

(30)

where

$$\begin{array}{cc}\hfill {\hslash }_1^r=& {\Re }^{q_1+q_2}cos{\displaystyle \frac{(q_1+q_2)\pi }{2}}-m_{22}{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}}\hfill \\ \hfill & -{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}}{m}_{11}+m_{11}{m}_{22}+m_{21}{m}_{12}+k{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}}+k{m}_{11},\hfill \\ \hfill {\hslash }_1^i=& {\Re }^{q_1+q_2}sin{\displaystyle \frac{(q_1+q_2)\pi }{2}}+m_{22}{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}}\hfill \\ \hfill & +{\Re }^{q1}sin{\displaystyle \frac{q_1\pi }{2}}{m}_{11}+k{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}},\hfill \\ \hfill {\hslash }_2^r=& {\Re }^{q_2}cos{\displaystyle \frac{q_2\pi }{2}}{m}_{13}-m_{13}m22+m_{11}{m}_{23}\hfill \\ \hfill & -m_{23}{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}}-k{m}_{13}-k{m}_{11}+k{\Re }^{q_1}cos{\displaystyle \frac{q_1\pi }{2}},\hfill \\ \hfill {\hslash }_2^i=& {\Re }^{q_2}sin{\displaystyle \frac{q_2\pi }{2}}{m}_{13}-m_{23}{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}}+k{\Re }^{q_1}sin{\displaystyle \frac{q_1\pi }{2}},\hfill \\ \hfill {\hslash }_3^r=& -m_{13}{m}_{23}+k{m}_{13},\hfill \\ \hfill {\hslash }_3^i=& 0.\hfill \end{array}$$

By solving Equation (30), one acquires

$$\left\{\begin{array}{c}cos\Re \tau ={\displaystyle \frac{-{\hslash }_2^r({\hslash }_1^r-{\hslash }_3^r)-{\hslash }_2^i{\hslash }_1^i}{{\left({\hslash }_1^r\right)}^2+{\left({\hslash }_1^i\right)}^2-{\left({\hslash }_3^r\right)}^2}}=f_1(\Re ),\hfill \\ sin\Re \tau ={\displaystyle \frac{-{\hslash }_2^i({\hslash }_1^r+{\hslash }_3^r)+{\hslash }_2^r{\hslash }_1^i}{{\left({\hslash }_1^r\right)}^2+{\left({\hslash }_1^i\right)}^2-{\left({\hslash }_3^r\right)}^2}}=f_2(\Re ).\hfill \end{array}\right.$$

(31)

Equation (31) infers the following relation:

$$\begin{array}{c}{f}_1{(\Re )}^2+f_2{(\Re )}^2=1.\hfill \end{array}$$

(32)

Assumption 2.

Equation (32) has a series of positive roots ${\Re }_k(k=0,1,2,\cdots )$.

The critical delay point has the following form if Assumption 2 is fulfilled:

$${\tau }_{1j}^{\left(k\right)}={\displaystyle \frac{1}{{\Re }_k}}[arccos\left(f_1\left({\Re }_k\right)\right)+2j\pi ],j=0,1,2,\cdots$$

(33)

According to the above equation ${\tau }_{10}$ is defined as

$${\tau }_{10}={\tau }_{1j}^{\left(0\right)}=min{\tau }_{1j}^{\left(k\right)},k=0,1,2,\cdots$$

(34)

To obtain meaningful conditions, the transversality condition needs to be satisfied. Let $s\left(\tau \right)=\lambda \left(\tau \right)+i\Re \left(\tau \right)$ be the root of Equation (27), and $s\left(\tau \right)$ at $\tau ={\tau }_{10}$ satisfy $\lambda \left({\tau }_{10}\right)=0$$\Re ={\Re }_0$, then the transversality condition as

$$\begin{array}{c}\hfill Re\left[{\displaystyle \frac{ds}{d\tau }}\right]{|}_{(\Re ={\Re }_0,\tau ={\tau }_{10})}\ne 0.\end{array}$$

(35)

Suppose that ${\hslash }_j^i$ and ${\hslash }_j^r$ are the real and imaginary parts of ${\hslash }_j(j=1,2,3)$. According to the implicit function theorem, the derivative of Equation (27) is formulated by

$$\begin{array}{cc}\hfill & {\hslash }_1^{\prime }\left(s\right){\displaystyle \frac{ds}{d\tau }}+\left[{\hslash }_2^{\prime }\left(s\right){\displaystyle \frac{ds}{d\tau }}{e}^{-s\tau }+{\hslash }_2\left(s\right)e^{-s\tau }\left(-\tau {\displaystyle \frac{ds}{d\tau }}-s\right)\right]\hfill \\ & +\left[{\hslash }_3^{\prime }\left(s\right){\displaystyle \frac{ds}{d\tau }}{e}^{-2s\tau }+{\hslash }_3\left(s\right)e^{-2s\tau }\left(-2\tau {\displaystyle \frac{ds}{d\tau }}-2s\right)\right]=0.\hfill \end{array}$$

(36)

It follows that

$$\begin{array}{c}\hfill {\displaystyle \frac{ds}{d\tau }}={\displaystyle \frac{X\left(s\right)}{Y\left(s\right)}},\end{array}$$

(37)

where

$$\begin{array}{c}X\left(s\right)=s[{\hslash }_2\left(s\right)e^{-s\tau }+2{\hslash }_3\left(s\right)e^{-2s\tau }],\hfill \\ Y\left(s\right)={\hslash }_1^{\prime }\left(s\right)+[{\hslash }_2^{\prime }\left(s\right)-{\hslash }_2\left(s\right)\tau ]e^{-s\tau }+[{\hslash }_3^{\prime }\left(s\right)-2\tau {\hslash }_3\left(s\right)]e^{-2s\tau }.\hfill \end{array}$$

Define $X^r$ and $X^i$ as the real and imaginary parts of $X\left(s\right)$, and define $Y^r$ and $Y^i$ as the real and imaginary parts of $Y\left(s\right)$, then Equation (37) implies

$$\begin{array}{c}\hfill Re\left[{\displaystyle \frac{ds}{d\tau }}\right]{|}_{(\Re ={\Re }_0,\tau ={\tau }_{10})}={\displaystyle \frac{X^r{Y}^r+X^i{Y}^i}{Y^{r^2}+Y^{i^2}}},\end{array}$$

(38)

where

$$\begin{array}{cc}\hfill X^r=& {\Re }_0({\aleph }_2^{r}sin{\Re }_0{\tau }_{10}-{\aleph }_2^{i}cos{\Re }_0{\tau }_{10}+2{\aleph }_3^{r}sin{\Re }_0{\tau }_{10}-2{\aleph }_3^{i}cos{\Re }_0{\tau }_{10}),\hfill \\ \hfill X^i=& {\Re }_0({\aleph }_2^{r}cos{\Re }_0{\tau }_{10}-{\aleph }_2^{i}sin{\Re }_0{\tau }_{10}+2{\aleph }_3^{r}cos{\Re }_0{\tau }_{10}-2{\aleph }_3^{i}sin{\Re }_0{\tau }_{10}),\hfill \\ \hfill Y^r=& {\aleph }_1^{\prime r}+({\aleph }_2^{\prime r}-{\tau }_{10}{\aleph }_2^r)cos{\Re }_0{\tau }_{10}+({\aleph }_2^{\prime i}-{\tau }_{10}{\aleph }_2^i)sin{\Re }_0{\tau }_{10}\hfill \\ \hfill & +({\aleph }_3^{\prime i}-2{\tau }_{10}{\aleph }_3^i)sin2{\Re }_0{\tau }_{10}+({\aleph }_3^{\prime r}-2{\tau }_{10}{\aleph }_3^r)cos2{\Re }_0{\tau }_{10},\hfill \\ \hfill Y^i=& {\aleph }_1^{\prime i}+({\aleph }_2^{\prime i}-{\tau }_{10}{\aleph }_2^i)cos{\Re }_0{\tau }_{10}-({\aleph }_2^{\prime r}-{\tau }_{10}{\aleph }_2^r)sin{\Re }_0{\tau }_{10}\hfill \\ \hfill & -({\aleph }_3^{\prime r}-2{\tau }_{10}{\aleph }_3^r)sin2{\Re }_0{\tau }_{10}+({\aleph }_3^{\prime i}-2{\tau }_{10}{\aleph }_3^i)cos2{\Re }_0{\tau }_{10}.\hfill \end{array}$$

By the above discussion, the following theorem can be obtained:

Theorem 2.

Based on the above discussion, the system will undergo a stability switch at the critical delay point${\tau }_{10}$, when${\displaystyle \frac{X^r{Y}^r+X^i{Y}^i}{Y^{r^2}+Y^{i^2}}}\ne 0$, so the following conclusion is obtained:

(1) 

The system is asymptotically stable in ${\varsigma }^{\ast }$, if $\tau \in [0,{\tau }_{10})$.

(2) 

The system exhibits Hopf bifurcation behavior when$\tau ={\tau }_{10}$. The state of the system will change from stable to unstable.

Remark 5.

This paper develops a fractional-order predator–prey system outperforming traditional integer-order models, which fail to capture population growth’s genetic traits and historical dependence. By using integral kernel functions in fractional derivatives to characterize historical cumulative effects, it enables more accurate predictions of long-term population evolution. Analyzing fractional order impacts allows selecting species with tuned orders (to address historical dependence) to enhance artificial ecosystem resilience. Incorporating larval maturation as a time delay, this study identifies critical thresholds for management warnings, guiding ecological reserve planning to maintain population stability via regulating maturation periods (e.g., artificial habitat temperature control). For invasive species, prioritizing control of those with maturation cycles near critical thresholds prevents population outbreaks and ecological imbalance. Moreover, an active control strategy targeting these critical delays—using real-time predator monitoring and feedback adjustments—enables low-cost regulation. This theoretical work therefore provides mathematical tools to advance "precision ecology."

5. Simulation Image

This section presents numerical simulations to validate the theoretical findings from Section 3 and Section 4. All computations were performed using MATLAB R2019b and Maple 2023. The bifurcation points were computed in Maple by solving the characteristic Equations (17) and (32) for their purely imaginary roots using the $fsolve$ function, while the system dynamics were simulated in MATLAB via a specialized Predictor method with a step size of h = 0.01 confirmed to ensure convergence. The excellent agreement between the critical delay from Maple and the dynamical transition in MATLAB simulations validates our numerical approach, with all parameters provided for reproducibility.

5.1. Example 1

In this section, we will select the proper parameters and then use the image to detect the conclusion of Section 2. All of the parameters are taken from [37], and the parameters satisfy Assumptions 1 and 2 in this paper.

$$\begin{array}{c}\hfill r=2,k=120,\alpha =5,a=4,\beta =4,b=1,c=1,b_0=1.5,d_0=2.5,q=0.4,E_0=3.\end{array}$$

The fractional-order predator–prey system is established with $q_1=0.65$, $q_2=0.65$, then Equation (4) is translated into the following form:

$$\left\{\begin{array}{c}{D}^{0.65}x\left(t\right)=2x_0\left(t\right)\left[1-{\displaystyle \frac{x(t-\tau )}{120}}\right]-{\displaystyle \frac{5x\left(t\right)y\left(t\right)}{4+x\left(t\right)+y\left(t\right)}},\hfill \\ D^{0.65}y\left(t\right)=1.5y_0(t-\tau )-\left(2.5-{\displaystyle \frac{4x\left(t\right)}{4+x\left(t\right)+y\left(t\right)}}\right)y\left(t\right)-1.2y\left(t\right).\hfill \end{array}\right.$$

(39)

The equilibrium point ${\varsigma }^{\ast }(x^{\ast },y^{\ast })$ can be derived as

$$\begin{array}{c}\hfill (x^{\ast },y^{\ast })=\left({\displaystyle \frac{\sqrt{2855}-15}{2}},{\displaystyle \frac{9\sqrt{2865}-224}{22}}\right).\end{array}$$

The above Assumption 1 and equilibrium condition Equation (8) can be verified by numerical calculation. In addition, ${\tau }_0=1.387$ can be calculated in Maple. When $\tau >{\tau }_0$, the system bifurcates at the point ${\varsigma }^{\ast }$, whereas when $\tau <{\tau }_0$, system (39) is asymptotically stable. If selecting the initial value $\left(x\right(0),y(0\left)\right)=(10,7)$, the images are presented to verify the correctness of the data.

The dynamic behaviors of system (39) around the equilibrium point ${\varsigma }^{\ast }$ are vividly illustrated in Figure 1 and Figure 2, which serve as a direct numerical validation of Theorem 1. Figure 1, corresponding to a time delay of $\tau =1.3<{\tau }_0\approx 1.387$, demonstrates the asymptotic stability of ${\varsigma }^{\ast }$. The trajectories of both prey and predator populations, initiated from $(10,7)$, converge smoothly to the equilibrium values without oscillation. This confirms that when the maturation delay remains below the critical threshold ${\tau }_0$, the system possesses an inherent resilience, allowing for the populations to return to a steady state after perturbation. In stark contrast, Figure 2 depicts the system’s response when the delay exceeds the critical value $(\tau =1.4>{\tau }_0)$. The emergence of sustained periodic oscillations in both populations signifies a supercritical Hopf bifurcation. The system loses its stability, and the equilibrium point ${\varsigma }^{\ast }$ becomes unstable, giving rise to a stable limit cycle. This bifurcation behavior underscores a critical regime shift in the ecosystem: an excessively long maturation period can destabilize the population dynamics, leading to persistent fluctuations that may increase the risk of species extinction. The clear qualitative change between Figure 1 and Figure 2 not only corroborates the analytical prediction of the bifurcation point ${\tau }_0$, but also highlights the significant ecological implications of time delay in population models.

The calculated critical time delay point can be used as a key ecological early warning index. For invasive species, if their natural maturity cycle is close, priority should be given to control. On the contrary, for rare species, the maturation cycle can be changed by artificially adjusting the habitat temperature to ensure that it is always away from the critical point $\tau$, thus maintaining population stability.

5.2. Example 2

In this section, the relationship between the fractional-order predator–prey model order and the bifurcation point is discussed. The simulation will be performed by fixing one of the two orders $q_1$ and $q_2$ and regulating another one. The unique role of the fractional orders $q_1$ and $q_2$ on the critical bifurcation point ${\tau }_0$, using the fixed parameters from Example 1. We systematically vary one order (e.g., $q_2$) within a range of $0.45$–$0.95$ while keeping the other fixed at $0.65$ to isolate its effect. The results, summarized in

Table 1. The impact of $q_2$ and $q_1$ on ${\tau }_0$ for system (39).

Fractional-Order ${\mathit{q}}_1$	Fractional-Orders ${\mathit{q}}_2$	Bifurcation Point ${\mathit{\tau }}_0$
0.65	0.45	1.544
0.65	0.50	1.501
0.65	0.55	1.461
0.65	0.60	1.434
0.65	0.65	1.387
0.65	0.75	1.322
0.65	0.85	1.261
0.45	0.65	2.196
0.50	0.65	2.021
0.55	0.65	1.820
0.60	0.65	1.604
0.65	0.65	1.387
0.75	0.65	1.003
0.85	0.65	0.702

and Figure 3 and Figure 4, reveal a clear trend: ${\tau }_0$ decreases as either $q_1$ or $q_2$ increases. This indicates that systems with weaker memory (higher order) bifurcate earlier, demonstrating that stronger memory effects (lower order) can significantly enhance system stability by tolerating longer delays. Figure 4 illustrates the combined effect of the fractional orders $q_1$ and $q_2$ on the critical bifurcation point ${\tau }_0$ with a three-dimensional surface plot. The primary advantage of this representation is its ability to provide an intuitive global overview of the coupling effect between $q_1$ and $q_2$.

Figure 5 shows the phase diagram and wave diagram of the integer-order and fractional-order systems when the time delay is 0, $x\left(t\right)$, $y\left(t\right)$ come from system with $q_1=q_2=0.9$, and $x_0\left(t\right)$, $y_0\left(t\right)$ come from system with $q_1=q_2=1$. By comparison, the influence of the order transition can be observed more clearly. Integer-order systems converge rapidly in an “instantaneous oscillation” manner without the accumulation of historical information. The convergence speed of fractional-order systems is jointly determined by the order and bifurcation parameters. The smaller the order, the slower the attenuation, and the gentler the corresponding system convergence. This behavior directly explains the rule that “the stronger the memory, the gentler the system behavior”.

The bifurcation behavior of the system is jointly determined by the fractional orders $q_1$ and $q_2$ and the delay $\tau$. Assuming $q_1=q_2=q$ for simplicity, the two-dimensional bifurcation diagram in Figure 6 delineates their combined influence on the state amplitudes. The depth of the color bar indicates the amplitude of the system after the periodic oscillation occurs, and the dark blue indicates that the system remains stable and does not oscillate. Crucially, it unveils a chaotic window $q\in (0.98,1)$, $\tau \in (2.08,2.36)$, and $q\in (0.96,1)$, $\tau \in (2.9,2.98)$ for $\left(a\right)$ in Figure 6, and a chaotic window $q\in (0.98,1)$, $\tau \in (2.08,2.36)$, and $q\in (0.96,1)$, $\tau \in (2.9,3)$ for $\left(b\right)$ in Figure 6. The chaotic window represents the irregular oscillation of population density. Is the aperiodic dynamical behavior of the system under the action of order and time delay, which shows the complex and unpredictable state of the system, characterized by unpredictable amplitudes—a phenomenon that remained undetected in prior one-dimensional analyses.

Our results show that populations with stronger memory effects (lower fractional-order q) tolerate longer maturation periods without destabilization. This suggests a new approach to ‘precision ecology’: when building artificial ecosystems, we may prefer to introduce or breed species that have physiologically longer ecological memories (in the form of lower q) to enhance system resilience.

5.3. Example 3

To demonstrate control over the bifurcation, we introduce the controller $k\left(y\right(t)-y(t-\tau \left)\right)$ into system (39). The initial condition is consistently set to $\left(x\right(0),y(0\left)\right)=(10,7)$ to ensure comparability with the results of Examples 1 and 2. The feedback gain k is the key control parameter. As computed, the critical delay shifts from ${\tau }_0=1.387$ (when $k=0$) to ${\tau }_{10}=1.601$ when $k=0.5$. Figure 7 confirms stability for $\tau =1.5<{\tau }_{10}$, while Figure 8 shows oscillations for $\tau =1.7>{\tau }_{10}$, validating this new threshold. We try to make the bifurcation happen earlier or later by choosing different k:

$$\left\{\begin{array}{cc}\hfill D^{0.65}x\left(t\right)& =2x\left(t\right)\left[1-{\displaystyle \frac{x(t-\tau )}{120}}\right]-{\displaystyle \frac{5x\left(t\right)y\left(t\right)}{4+x\left(t\right)+y\left(t\right)}},\hfill \\ \hfill D^{0.65}y\left(t\right)& =1.5y(t-{\tau }_1)-\left(2.5-{\displaystyle \frac{4x\left(t\right)}{4+x\left(t\right)+y\left(t\right)}}\right)y\left(t\right)\hfill \\ & -1.2y\left(t\right)+k\left(y\right(t)-y(t-\tau \left)\right).\hfill \end{array}\right.$$

(40)

The parameters come from system (39), and selecting the initial value $\left(x\right(0),y(0\left)\right)=(10,7)$. The state dynamics of system (40) with different cases $k=0.5$ and $k=-0.2$ are plotted in Figure 9 and Figure 10, respectively. Consider system (39) with $k=0.5$: ${\tau }_{10}=1.256$ is computed by Maple. Figure 7 and Figure 8 are consistent with our results. From Figure 7, point ${\varsigma }^{\ast }$ is asymptotically stable, and point ${\varsigma }^{\ast }$ of Figure 8 appears to be a periodic oscillation. Hence, the state of system (40) has changed on either side of critical bifurcation point ${\tau }_{10}$, and ${\tau }_{10}$ is proved $1.601$.

$x\left(t\right)$ and $y\left(t\right)$ come from system (39), and $x_0\left(t\right)$ and $y_0\left(t\right)$ come from system (40) in Figure 9 and Figure 10. From Figure 9, when feedback parameter $k=0.5$, the state of system (40) converges to the equilibrium point faster, which means that the delay parameter has less influence on the system. At the same time, from Figure 10, when the feedback parameter $k=-0.2$, the state of system (40) converges to the equilibrium point slower. Therefore, the occurrence of early or delayed bifurcation can be controlled by the feedback gain parameter k.

In this system, there are two parameters that affect the bifurcation of the system. Figure 6 is a two-dimensional bifurcation diagram that more clearly shows the influence of these two parameters on the dynamic behavior of the system.The calculated critical time delay ${\tau }_0$ can be used as a key early warning indicator to reduce the dependence on chemical pesticides and optimize biological control strategies in ecological conservation. Traditional pest and disease early warning relies on the “current population density threshold” (for example, when the pests and diseases are $>50$ per plant), which is easy to miss in the early intervention window. In our model, a mechanism-based early warning is achieved by linking the development cycle of species and system stability via ${\tau }_0$, which can start prevention and control 5–7 days earlier than traditional methods, and the amount of pesticide used can be greatly reduced.

In addition, the fractional memory effect (fractional-order) of the fractional-order model can quantify the “long-term impact of natural enemy release on population dynamics”, which solves the problem of “excessive release frequency” in traditional prevention and control. For example, the weak memory of prey populations ($q_1=0.65$) means that they are more affected by the pressure of recent predators, while the strong memory of predators ($q_2=0.55$) requires long-term suppression by predators. Based on this, better on-demand delivery can be achieved through different population dynamics to balance the control effect and ecological cost. In contrast, the integer-order model ignores the memory effect and misjudges that the natural enemy effect only lasts for 3 days, resulting in excessive delivery.

6. Conclusions

This paper presents a comprehensive analysis of stability and Hopf bifurcation control for a fractional-order predator–prey system incorporating a maturation time delay. The primary contributions are threefold. First, we successfully derived the explicit condition for the occurrence of a Hopf bifurcation using time delay as the critical parameter, establishing a theoretical stability criterion. Second, a significant and novel finding is the inverse relationship between the fractional derivative orders ($q_1,q_2$) and the critical bifurcation point (${\tau }_0$). This demonstrates that stronger memory effects (lower orders) exert a stabilizing influence on the ecosystem, an insight uniquely afforded by the fractional-order framework. Third, we designed and implemented an effective delay feedback controller, proving that the bifurcation can be proactive delayed or advanced by tuning a single feedback-gain parameter k.

Numerical simulations fully validated our theoretical findings, illustrating the bifurcation phenomenon and the controller’s efficacy. The results underscore that fractional-order models provide a superior platform for capturing the memory-dependent dynamics of biological systems. The proposed control strategy offers a practical approach for managing ecosystem stability, with potential applications in precision ecology, such as preventing population outbreaks by manipulating critical parameters like maturation periods. Future work will extend this analysis to higher-dimensional systems and incorporate a wider range of functional responses.