Bifurcation Analysis of a Predator–Prey Model with Coefficient-Dependent Dual Time Delays

Abstract

In this paper, a class of two-delay predator–prey models with coefficient-dependent delay is studied. It examines the combined effect of fear-induced delay and post-predation biomass conversion delay on the stability of predator–prey systems. By analyzing the distribution of roots of the characteristic equation, the stability conditions for the internal equilibrium and the existence criteria for Hopf bifurcations are derived. Utilizing normal form theory and the central manifold theorem, the direction of Hopf bifurcations and the stability of periodic solutions are calculated. Finally, numerical simulations are conducted to verify the theoretical findings. This research reveals that varying delays can destabilize the predator–prey system, reflecting the dynamic complexity of real-world ecosystems more realistically.

1. Introduction

Predator–prey interactions represent a core focus of ecological research, as their dynamic relationships directly influence ecosystem stability, species diversity, and resource management strategies. The classic Lotka–Volterra model elucidates the periodic oscillations between predators and prey [1,2] but fails to explain complex nonlinear phenomena observed in real-world ecosystems, such as fear-induced effects, delayed feedback, and population collapses. With ecosystems facing increasing human disturbances (e.g., habitat fragmentation and climate change), developing more sophisticated mathematical models to quantify multi-factor couplings has become crucial for predicting population dynamics and formulating conservation policies. In recent years, scholars have advanced beyond traditional models by integrating fear effects, diverse delay mechanisms, and functional responses [3,4,5,6,7], offering new perspectives for analyzing ecosystem vulnerability and resilience.

Traditional predator–prey models focus on direct predation. However, the mere presence of predators can exert a more profound impact on prey populations than actual killing. Zanette et al. rigorously demonstrated through experimentation that simply broadcasting predator calls caused a 40% decline in the reproductive success rate of white-throated sparrows [8]. This discovery highlights the critical role of fear, which reshapes population dynamics by altering prey behavior and physiology. Wang et al. were the first to incorporate this fear effect into the classical Logistic growth model. Fear reduces the prey’s effective birth rate via the function f(k,y). After incorporating predation, they proposed the following model [9]

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}=rxf(k,y)-dx-a{x}^2-g(x)y,\hfill \\ {\displaystyle \frac{dy}{dt}}=y(-m+cg(x)),\hfill \end{array}\right.$$

(1)

where $x$ and $y$ denote the population densities of prey and predators, respectively. Additionally, $r$ represents the prey’s birth rate, $k$ serves as the fear level parameter, and $f(k,y)$ denotes the fear effect function, $d$ signifies the prey’s natural mortality rate, $a$ denotes the mortality rate due to inter-species competition within the prey population, $m$ represents the natural mortality rate of the predator population, and $c$ denotes the conversion efficiency of predators who convert the captured prey into energy for their own growth. The predator’s functional response, denoted by $g(x)$, can adopt a linear form $g(x)=px$ or a Holling-II type function $g(x)={\displaystyle \frac{px}{1+qx}}$. By integrating fear effects with functional responses, system (1) elucidates the complex impacts of non-lethal predation risk on population dynamics, providing a mathematical framework for understanding the “ecology of fear”.

Since then, numerous studies on predator–prey models with fear effects emerged [10,11,12]. Building upon model (1), Panday et al. introduced a fear delay and selected the Holling-II type functional response function to establish the following model [13]

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{rx}{1+ky(t-\tau )}}(1-{\displaystyle \frac{x}{k_0}})-{\displaystyle \frac{pxy}{1+qx}},\hfill \\ {\displaystyle \frac{dy}{dt}}={\displaystyle \frac{cpxy}{1+qx}}-my,\hfill \end{array}\right.$$

(2)

where $x$ and $y$ denote the population densities of prey and predators, respectively. r, k, m and c have the same meanings as in model (1); $k_0$ signifies the prey’s environmental carrying capacity, $m$ stands for the predator’s natural mortality rate, ${\displaystyle \frac{px}{1+qx}}$ indicates the predator’s functional response, $p$ denotes the predator’s average predation efficiency, and $q$ denotes the fixed processing time for each prey capturing. Model (2) highlights the intricate roles of fear effects and delays in predator–prey interactions, thereby enhancing our understanding of ecological stability mechanisms and offering theoretical tools for predicting and managing population fluctuations in real-world ecosystems. Shi and Hu further proposed a nonlinear functional response model by incorporating fear-induced prey mortality [14]

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{rx}{1+ky(t-\tau )}}-(1+by(t-\tau ))\alpha x-a{x}^2-{\displaystyle \frac{pxy}{1+qx}}\hfill \\ {\displaystyle \frac{dy}{dt}}={\displaystyle \frac{cpxy}{1+qx}}-my-n{y}^2,\hfill \end{array}\right.,$$

(3)

where $x$ and $y$ represent the population densities of prey and predators, respectively. r, k, m c, p, q, and ${\displaystyle \frac{px}{1+qx}}$ have the same meanings as in model (2), $b$ reflects the level of fear effect impacting prey mortality, $\alpha$ denotes the prey’s natural mortality rate, $(1+by(t-\tau ))\alpha x$ stands for the mortality rate of the prey caused by fear effects, $a$ signifies the mortality rate resulting from inter-species competition among prey, $n$ denotes the mortality rate due to inter-species competition among predators, and $\tau$ denotes the delay associated with fear effects.

Wang introduced a post-predation biomass conversion delay into the fear-effect model [15]

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}=rxf(k,y)-\alpha x-a{x}^2-g(x)y,\hfill \\ {\displaystyle \frac{dy}{dt}}=\beta e^{-d\tau }g(x(t-\tau ))y(t-\tau )-my-n{y}^2,\hfill \end{array}\right.$$

(4)

where $x$ and $y$ denote the population densities of the prey and predator, respectively; r, k, m n, α, and a have the same meanings as in model (3); $f(k,y)$ denotes the fear effect function; $g(x)$ denotes the predator’s functional response; $\beta$ denotes the biomass conversion efficiency of biomass from prey to post-predation; $e^{-d\tau }$ signifies the predator’s survival rate during the biomass conversion process; and $\tau$ denotes the delay in biomass conversion from prey to post-predation.

The fear effect in predator–prey systems has been demonstrated to profoundly influence population dynamics, reshaping ecological balance by suppressing prey reproduction rather than through direct predation. However, in natural ecological processes, both the fear effect and energy conversion experience time delays. Current research predominantly focuses on simplified models incorporating only a single type of delay, making it difficult to reveal the intricate dynamics arising from the coupled effects of dual delays. Inspired by the aforementioned model, this paper delves into a double-delay predator–prey model incorporating the fear effect

$$\left\{\begin{array}{c}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{rx}{1+ky(t-{\tau }_1)}}-(1+by(t-{\tau }_1))\alpha x-a{x}^2-cxy,\hfill \\ {\displaystyle \frac{dy}{dt}}=\beta e^{-d{\tau }_2}x(t-{\tau }_2)y(t-{\tau }_2)-my-n{y}^2.\hfill \end{array}\right.$$

(5)

The notations $x,y,r,k,\alpha ,a,b,e^{-d{\tau }_2},m,n$ retain their meanings, as outlined in systems (3) and (4), with $c$ representing the predator’s average predation efficiency. For simplicity, the product of the two parameters $\beta$ in system (4) and $c$ in system (5) is still denoted as $\beta$ in system (5). Here, ${\tau }_1$ denotes the fear delay, and ${\tau }_2$ denotes the delay in biomass conversion from prey to post-predation. All parameters $r,k,\alpha ,a,b,e^{-d{\tau }_2},m,n,\beta ,c$ are positive. Five scenarios are examined based on different values of the two delays for interior equilibrium stability points and Hopf bifurcation conditions in these scenarios. Using normal form theory and the central manifold theorem, the bifurcation direction and periodic solution stability are studied, with MATLAB R2023b simulations verifying the theoretical results. By establishing bifurcation conditions and performing numerical simulations, this study reveals the modulation mechanisms through which dual delays govern the system’s stability. This work advances the analytical framework for multi-delay dynamical systems and provides practical guidance for designing pest control strategies and conservation measures for endangered species.

2. Stability of Equilibrium and Existence of Hopf Bifurcation

Based on references [9,16] and considering the biological significance of $k$ and $f(k,y),$ we make the following assumptions

$$(H_0):f(0,y)=1,f(k,0)=1,\underset{k\to \infty }{{\displaystyle \lim }}f(k,y)=0,\underset{y\to \infty }{{\displaystyle \lim }}f(k,y)=0,{\displaystyle \frac{\partial f(k,y)}{\partial k}}<0,{\displaystyle \frac{\partial f(k,y)}{\partial y}}<0,r>\alpha .$$

where parameter r denotes the prey’s birth rate and α denotes the prey’s natural mortality rate. A higher mortality rate is detrimental to any species and leads to its extinction, which aligns with biological intuition. When $r<\alpha$, both the prey and predator populations tend toward extinction regardless of fear effects and predation mechanisms. Therefore, the condition $r>\alpha$ is biologically essential and will be imposed throughout this study.

Setting the right-hand side of system (5) equal to zero yields its internal equilibrium $E(x^{*},y^{*})$, where

$$x^{*}={\displaystyle \frac{m+n{y}^{*}}{\beta }}{e}^{{d\tau }_2},y^{*}={\displaystyle \frac{-a_2+\sqrt{a_2^2-4a_1{a}_3}}{2a_1}},$$

and the coefficients are defined as

$$a_1=\beta ck+\beta \alpha kb+a{e}^{d{\tau }_2}kn,$$

$$a_2=\beta \alpha b+\beta \alpha k+a{e}^{d{\tau }_2}km+a{e}^{d{\tau }_2}n+\beta c,$$

$$a_3=-r\beta +\beta \alpha +a{e}^{d{\tau }_2}m.$$

When condition $(H_1):a_3<0$ holds, a positive internal equilibrium exists for system (5).

The linear part of system (5) at the internal equilibrium $E(x^{*},y^{*})$ is

$$\left\{\begin{array}{c}{\displaystyle \frac{dx}{dt}}=a_{11}x+a_{12}y+a_{12}^{\prime }y(t-{\tau }_1),\hfill \\ {\displaystyle \frac{dy}{dt}}=a_{22}y+a_{21}x(t-{\tau }_2)+a_{22}^{\prime }y(t-{\tau }_2),\hfill \end{array}\right.$$

where

$$a_{11}={\displaystyle \frac{r}{1+k{y}^{*}}}-\alpha (1+b{y}^{*})-2a{x}^{*}-cy*, a_{12}=-c{x}^{*}, a_{12}^{\prime }=-{\displaystyle \frac{rk{x}^{*}}{{(1+k{y}^{*})}^2}}-b\alpha x^{*},$$

$$a_{21}=\beta e^{-d{\tau }_2}{y}^{*}, a_{22}=-m-2n{y}^{*}, a_{22}^{\prime }=\beta e^{-d{\tau }_2}{x}^{*}.$$

The characteristic equation at the internal equilibrium $E(x^{*},y^{*})$ is

$$\left|\begin{array}{cc}\lambda -a_{11}\hfill & -a_{12}-a_{12}^{\prime }{e}^{-\lambda {\tau }_1}\hfill \\ -a_{21}{e}^{-\lambda {\tau }_2}\hfill & \lambda -a_{22}-a_{22}^{\prime }{e}^{-\lambda {\tau }_2}\hfill \end{array}\right|=0,$$

which is

$${\lambda }^2+p_1\lambda +e^{-\lambda {\tau }_2}(p_2\lambda +p_3)+e^{-\lambda ({\tau }_1+{\tau }_2)}{p}_4+p_5=0,$$

(6)

where

$$p_1=-a_{11}-a_{22}, p_2=-a_{22}^{\prime }, p_3=-a_{12}{a}_{21},$$

$$p_4=-a_{12}^{\prime }{a}_{21}, p_5=a_{11}{a}_{22}.$$

The stability of the internal equilibrium $E(x^{*},y^{*})$ and the existence of Hopf bifurcations for system (5) will be analyzed under the following five distinct cases.

• Case 1. ${\tau }_1={\tau }_2=0.$

Equation (6) becomes

$${\lambda }^2+(p_1+p_2)\lambda +(p_3+p_4+p_5)=0.$$

For analytical convenience, we introduce the assumption $(H_2):p_1+p_2>0,p_3+p_4+p_5>0.$ According to the Routh–Hurwitz criterion, we establish the following stability theorem.

Theorem 1.

If $(H_1)$ and $(H_2)$ hold, then the internal equilibrium of system (5) is asymptotically stable.

• Case 2. ${\tau }_1>0,{\tau }_2=0.$

Equation (6) becomes

$${\lambda }^2+b_1\lambda +b_2{e}^{-\lambda {\tau }_1}+b_3=0,$$

(7)

where

$$b_1=p_1+p_2, b_2=p_4, b_3=p_3+p_5.$$

Substituting $\lambda =i\omega (\omega >0)$ into Equation (7) and separating the real and imaginary parts yields

$$\left\{\begin{array}{l}-{\omega }^2+b_2\cos (\omega {\tau }_1)+b_3=0,\hfill \\ b_1\omega -b_2\sin (\omega {\tau }_1)=0.\hfill \end{array}\right.$$

(8)

Consequently, we obtain

$$\left\{\begin{array}{l}\sin (\omega {\tau }_1)={\displaystyle \frac{b_1\omega }{b_2}},\hfill \\ \cos (\omega {\tau }_1)={\displaystyle \frac{{\omega }^2-b_3}{b_2}},\hfill \end{array}\right.$$

squaring both equations in (8) and summing them leads to

$${\omega }^4+(b_1^2-2b_3){\omega }^2+b_3^2-b_2^2=0.$$

(9)

Lemma 1.

If $(H_3):b_3^2-b_2^2<0$ holds, then Equation (9) admits a positive root.

Proof. 

By direct computation, we obtain a root of Equation (9) as

$${\omega }_0=\sqrt{{\displaystyle \frac{-(b_1^2-2b_3)+\sqrt{{(b_1^2-2b_3)}^2-4(b_3^2-b_2^2)}}{2}}}.$$

When $(H_3)$ holds, the inequality

$$\sqrt{{(b_1^2-2b_3)}^2-4(b_3^2-b_2^2)}>\left|b_1^2-2b_3\right|>0$$

is satisfied. Hence

$$-(b_1^2-2b_3)+\sqrt{{(b_1^2-2b_3)}^2-4(b_3^2-b_2^2)}>0.$$

This completes the proof. □

Lemma 2.

If $(H_4):b_1^2-2b_3>0$ holds, then ${{\displaystyle \left.\mathrm{Re}{\left(\frac{d\lambda }{d{\tau }_1}\right)}^{-1}\right|}}_{\lambda =i{\omega }_0}>0.$

Proof. 

For the positive root, ${\omega }_0,$ of Equation (9), there exists a sequence of critical delay values ${\tau }_1^{(n)}(n=1,2,3,\cdots )$ given by

$${\tau }_1^{(n)}={\displaystyle \frac{1}{{\omega }_0}}(\arccos {\displaystyle \frac{{\omega }^2-b_3}{b_2}}+2n\pi ),n=1,2,3,\cdots .$$

Let $\lambda ({\tau }_1)=\alpha ({\tau }_1)+i\omega ({\tau }_1)$ as the root of Equation (7) near ${\tau }_1={\tau }_1^{(n)}$, satisfying $\alpha ({\tau }_1^{(n)})=0,$ $\omega \left({\tau }_1^{(n)}\right)={\omega }_0.$

Differentiating both sides of Equation (7) implicitly with respect to ${\tau }_1$ and rearranging terms yields

$${\left({\displaystyle \frac{d\lambda }{d{\tau }_1}}\right)}^{-1}={\displaystyle \frac{2\lambda +b_1-b_2{e}^{-\lambda {\tau }_1}{\tau }_1}{\lambda b_2{e}^{-\lambda {\tau }_1}}}.$$

(10)

Substituting $\lambda =i{\omega }_0$ into (10) and extracting the real part gives

$${{\displaystyle \left.\mathrm{Re}{\left(\frac{d\lambda }{d{\tau }_1}\right)}^{-1}\right|}}_{\lambda =i{\omega }_0}={\displaystyle \frac{2{\omega }_0^2+(b_1^2-2b_3)}{b_1^2{\omega }_0^2+{({\omega }_0^2-b_3)}^2}}.$$

Therefore, when $(H_4):b_1^2-2b_3>0$ holds, we have ${{\displaystyle \left.\mathrm{Re}{\left(\frac{d\lambda }{d{\tau }_1}\right)}^{-1}\right|}}_{\lambda =i{\omega }_0}>0.$ This completes the proof. □

From Lemmas 1 and 2, the following theorem is derived.

Theorem 2.

If $(H_1),(H_2),(H_3),(H_4)$ hold, then when ${\tau }_1\in [0,{\tau }_1^{(0)})$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is asymptotically stable. When ${\tau }_1>{\tau }_1^{(0)}$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is unstable. When ${\tau }_1={\tau }_1^{(n)}(n=1,2,3,\cdots ),$ system (5) exhibits a Hopf bifurcation at the internal equilibrium $E(x^{*},y^{*})$.

• Case 3. ${\tau }_1=0,{\tau }_2>0.$

Equation (6) becomes

$${\lambda }^2+p_1\lambda +e^{-\lambda {\tau }_2}[p_2\lambda +(p_3+p_4)]+p_5=0.$$

(11)

Let

$$\begin{array}{l}P(\lambda ,{\tau }_2)={\lambda }^2+p_1\lambda +p_5,\\ Q(\lambda ,{\tau }_2)=p_2\lambda +(p_3+p_4),\end{array}$$

then Equation (11) is transformed into

$$P(\lambda ,{\tau }_2)+e^{-\lambda {\tau }_2}Q(\lambda ,{\tau }_2)=0.$$

(12)

From hypothesis $(H_1)$, we obtain the constraint ${\tau }_2<{\displaystyle \frac{1}{d}}\ln {\displaystyle \frac{1}{am}}(r\beta -\beta \alpha ).$ Accordingly, we define the maximum admissible delay ${\tau }_{\max }={\displaystyle \frac{1}{d}}\ln {\displaystyle \frac{1}{am}}(r\beta -\beta \alpha ).$

Following the geometric criteria in [17], we now verify that Equation (12) satisfies the following five conditions

$(i) P(0,{\tau }_2)+Q(0,{\tau }_2)\ne 0,\forall {\tau }_2\in$ ${ℝ}_{+}$;

$(ii) P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)\ne 0,\forall \omega \in$ ${ℝ}_{+}$, $\forall {\tau }_2\in$ ${ℝ}_{+}$;

$(iii) \underset{\left|\lambda \right|\to \infty }{\lim }\sup \left\{\left|{\displaystyle \frac{Q(\lambda ,{\tau }_2)}{P(\lambda ,{\tau }_2)}}\right|:\mathrm{Re}\lambda \ge 0\right\}<1,\forall {\tau }_2\in$ ${ℝ}_{+}$;

$(iv)$ For each ${\tau }_2,$ the function $F(\omega ,{\tau }_2):={\left|P(i\omega ,{\tau }_2)\right|}^2-{\left|Q(i\omega ,{\tau }_2)\right|}^2$ has at most finitely many real zeros;

$(v) F(\omega ,{\tau }_2)=0$ has a positive root $\omega ({\tau }_2)$, this root is continuously differentiable with respect to ${\tau }_2$.

Condition $(v)$ holds trivially.

When $\lambda =0$, if $(H_2)$ is true, we can obtain

$$P(0,{\tau }_2)+Q(0,{\tau }_2)=p_3+p_4+p_5>0\ne 0.$$

So, condition $(i)$ is satisfied.

When $\omega \ne 0$, $P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)=-{\omega }^2+p_5+p_3+p_4+i\omega (p_1+p_2),$ if $(H_2)$ is true, $p_1+p_2>0,$ so $P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)\ne 0,$ and condition $(ii)$ is satisfied.

Because $\underset{\left|\lambda \right|\to \infty }{\lim }\left|{\displaystyle \frac{Q(\lambda ,{\tau }_2)}{P(\lambda ,{\tau }_2)}}\right|=\underset{\left|\lambda \right|\to \infty }{\lim }{\displaystyle \frac{p_2\lambda +(p_3+p_4)}{{\lambda }^2+p_1\lambda +p_5}}=0<1,$ condition $(iii)$ is satisfied.

From

$${\left|P(i\omega ,{\tau }_2)\right|}^2={(-{\omega }^2+p_5)}^2+p_1^2{\omega }^2,$$

$${\left|Q(i\omega ,{\tau }_2)\right|}^2={(p_3+p_4)}^2+p_2^2{\omega }^2,$$

we can obtain

$$F(\omega ,{\tau }_2)={\omega }^4+c_1{\omega }^2+c_2,$$

(13)

where

$$c_1=p_1^2-2p_5-p_2^2,c_2=p_5^2-{(p_3+p_4)}^2.$$

As a quartic polynomial, $F(\omega ,{\tau }_2)$ possesses at least four roots. Consequently, condition $(iv)$ is satisfied.

To ensure $F(\omega ,{\tau }_2)$ admits positive roots, we define the parameter set based on Equation (13)

$$I_1=\left\{{\tau }_2\left|c_2<0,\forall {\tau }_2\in \left[0,{\tau }_{\max }\right]\right.\right\}.$$

For ${\tau }_2\in I_1$, $F(\omega ,{\tau }_2)$ has a positive real zero

$${\omega }_1=\sqrt{{\displaystyle \frac{-c_1+\sqrt{c_1^2-4c_2}}{2}}}.$$

Now, assuming $\pm i\omega$ are pure imaginary roots of Equation (11), we substitute $\lambda =i\omega (\omega >0)$ into (11) and separate real and imaginary parts

$$\left\{\begin{array}{l}(p_3+p_4)\cos (\omega {\tau }_2)+p_2\omega \sin (\omega {\tau }_2)={\omega }^2-p_5,\hfill \\ p_2\omega \cos (\omega {\tau }_2)-(p_3+p_4)\sin (\omega {\tau }_2)=-p_1\omega ,\hfill \end{array}\right.$$

(14)

we obtain

$$\left\{\begin{array}{l}\sin (\omega {\tau }_2)={\displaystyle \frac{({\omega }^2-p_5)p_2\omega +(p_3+p_4)p_1\omega }{{(p_3+p_4)}^2+p_2^2{\omega }^2}},\hfill \\ \cos (\omega {\tau }_2)={\displaystyle \frac{({\omega }^2-p_5)(p_3+p_4)+p_2{p}_1{\omega }^2}{{(p_3+p_4)}^2+p_2^2{\omega }^2}}.\hfill \end{array}\right.$$

Squaring both equations in (14) and summing them leads to

$${\omega }^4+c_1{\omega }^2+c_2=0,$$

(15)

where

$$c_1=p_1^2-2p_5-p_2^2,c_2=p_5^2-{(p_3+p_4)}^2.$$

Since Equation (15) is identical to $F(\omega ,{\tau }_2)=0$, the existence of positive roots for (15) implies that Equation (11) possesses a pair of pure imaginary roots $\pm i\omega$. For ${\tau }_2\in I_1$, we define the phase function $\omega ({\tau }_2){\tau }_2=\theta ({\tau }_2)+2n\pi ,$ and construct the auxiliary function $S_n({\tau }_2)={\tau }_2-{\displaystyle \frac{\theta ({\tau }_2)+2n\pi }{\omega ({\tau }_2)}}.$ Then, $\pm i\omega ({\tau }_2)$ are pure imaginary roots of (11) if and only if ${\tau }_2$ satisfies $S_n({\tau }_2)=0$. Denoting the roots of $S_n({\tau }_2)=0$ by ${\tau }_2^{(i)}$, we establish the following theorem according to Theorem 2.2 in [17].

Theorem 3.

For ${\tau }_2\in I_1$, if there exists ${\tau }_2={\tau }_2^{(i)}$ satisfying $S_n({\tau }_2^{(i)})=0(n\in N),$ then Equation (11) possesses a pair of pure imaginary roots $\pm i\omega ({\tau }_2^{(i)})$ at ${\tau }_2={\tau }_2^{(i)}$. Furthermore, if $\delta ({\tau }_2^{(i)})>0(<0),$ then as ${\tau }_2$ increases through ${\tau }_2^{(i)}$, the characteristic roots corresponding to $\pm i\omega ({\tau }_2^{(i)})$ cross the imaginary axis from the left (right) half plane to the right (left) half plane of the complex plane, where the transversality condition is given by

$$\delta ({\tau }_2^{(i)})=sign\left\{{\left.{\displaystyle \frac{d\mathrm{Re}\lambda ({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}=sign\left\{F_{\omega }^{\prime }(\omega ({\tau }_2^{(i)}),{\tau }_2^{(i)})\right\}sign\left\{{\left.{\displaystyle \frac{d{S}_n({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}.$$

(16)

Since

$$F_{\omega }^{\prime }=4{\omega }^3+2c_1\omega ,$$

we introduce the assumption $(H_5):c_1>0,$ for analytical convenience. Under this condition, Equation (16) simplifies to

$$\delta ({\tau }_2^{(i)})=sign\left\{{\left.{\displaystyle \frac{d\mathrm{Re}\lambda ({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}=sign\left\{{\left.{\displaystyle \frac{d{S}_n({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}.$$

(17)

It is easy to identify that when ${\tau }_2\in I_1$, the auxiliary function $S_n({\tau }_2)={\tau }_2-{\displaystyle \frac{\theta ({\tau }_2)+2n\pi }{\omega ({\tau }_2)}}$ is monotonically decreasing with respect to $n$, that is, $S_n({\tau }_2)<S_{n+1}({\tau }_2)$. If $S_0({\tau }_2)$ has no zero point, $S_n({\tau }_2)=0(n\in N)$ has no zero point either. When ${\tau }_2=0$, obviously $S_n(0)=-{\displaystyle \frac{\theta (0)+2n\pi }{\omega (0)}}<0.$ When ${\tau }_2\to {\tau }_{\max }$, $c_2\to 0,$ then $\omega ({\tau }_2)\to 0.$ Additionally, according to $\sin (\omega {\tau }_2)\to 0,\cos \left(\omega {\tau }_2\right)\to 1,$ can determine that $\theta ({\tau }_2)\to 2\pi ,$ so $S_0({\tau }_2)\to -\infty .$ Therefore, $S_n({\tau }_2)(n\in N)$ intersects with the horizontal axis, and the number of intersections is even. Let the intersection be

$${\tau }_2^{\min }\stackrel{\Delta }{=}{\tau }_2^{(1)}<{\tau }_2^{(2)}<{\tau }_2^{(3)}<\cdots <{\tau }_2^{(i)}\stackrel{\Delta }{=}{\tau }_2^{\max },$$

where $i$ is an even number.

Theorem 4.

When ${\tau }_2\in I_1$, if $(H_1),(H_2),(H_5)$ hold, then we can determine the following:

$(i)$ If $S_0({\tau }_2)$ has no zeros, then the internal equilibrium $E(x^{*},y^{*})$ is asymptotically stable.

$(ii)$ If $S_0({\tau }_2)$ has at least one positive zero, then there exists ${\tau }_2^{(1)},$ and when ${\tau }_2\in [0,{\tau }_2^{(i)})$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is asymptotically stable. When ${\tau }_2\in ({\tau }_2^{(1)},{\tau }_2^{\max })$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is unstable. When ${\tau }_2\in ({\tau }_2^{\max },{\tau }_{\max })$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is asymptotically stable. At each ${\tau }_2={\tau }_2^{(i)}(i=1,2,3,\cdots )$, system (5) undergoes a Hopf bifurcation at $E(x^{*},y^{*})$.

• Case 4. ${\tau }_1>0,{\tau }_2\in [0,{\tau }_2^{\left(1\right)}),$ where ${\tau }_2$ is constrained within its stability interval, while ${\tau }_1$ is treated as a bifurcation parameter.

Equation (6) becomes

$${\lambda }^2+p_1\lambda +e^{-\lambda {\tau }_2}(p_2\lambda +p_3)+e^{-\lambda ({\tau }_1+{\tau }_2)}{p}_4+p_5=0.$$

(18)

Substituting $\lambda =i\omega (\omega >0)$ into Equation (18) and separating the real and imaginary parts yields

$$\left\{\begin{array}{l}{p}_4\cos (\omega {\tau }_1)\cos (\omega {\tau }_2)-p_4\sin (\omega {\tau }_1)\sin (\omega {\tau }_2)={\omega }^2-p_3\cos (\omega {\tau }_2)-p_2\omega \sin (\omega {\tau }_2)-p_5,\hfill \\ p_4\cos (\omega {\tau }_1)\sin (\omega {\tau }_2)+p_4\sin (\omega {\tau }_1)\cos (\omega {\tau }_2)=p_1\omega +p_2\omega \cos (\omega {\tau }_2)-p_3\sin (\omega {\tau }_2),\hfill \end{array}\right.$$

(19)

we obtain

$$\left\{\begin{array}{l}\sin (\omega {\tau }_1)={\displaystyle \frac{-{\omega }^2\sin (\omega {\tau }_2)+p_1\omega \cos (\omega {\tau }_2)+p_2\omega +p_5\sin (\omega {\tau }_2)}{p_4}},\hfill \\ \cos (\omega {\tau }_1)={\displaystyle \frac{{\omega }^2\cos (\omega {\tau }_2)+p_1\omega \sin (\omega {\tau }_2)-p_3-p_5\cos (\omega {\tau }_2)}{p_4}}.\hfill \end{array}\right.$$

Squaring both equations in (19) and summing them leads to

$${\omega }^4-d_1{\omega }^3+d_2{\omega }^2+d_3\omega +d_4=0,$$

(20)

where

$$d_1=-2p_2\sin (\omega {\tau }_2), d_2=p_2^2+p_1^2+2p_1{p}_2\cos (\omega {\tau }_2)-2p_3\cos (\omega {\tau }_2)-2p_5,$$

$$\begin{gathered} d_3=2p_2{p}_5\sin (\omega {\tau }_2)-2p_1{p}_3\sin (\omega {\tau }_2), \\ {d}_4=p_5^2+p_3^2-p_4^2+2p_3{p}_5\cos (\omega {\tau }_2). \end{gathered}$$

Lemma 3.

If $(H_6):d_4<0,f^{\prime }(\omega )>0$ holds, then Equation (20) admits a positive root, where $f^{\prime }(\omega )$ is demonstrated in the proof of Lemma 3.

Proof. 

Let

$$f(\omega )={\omega }^4-d_1{\omega }^3+d_2{\omega }^2+d_3\omega +d_4,$$

then

$$\begin{array}{ll}\hfill f^{\prime }(\omega )=& (4-2p_2{\tau }_2\cos (\omega {\tau }_2)){\omega }^3+(2p_3{\tau }_2-2p_1{p}_2{\tau }_2-6p_2)\sin (\omega {\tau }_2){\omega }^2+\hfill \\ & \omega (2p_1^2+2p_2^2+4p_1{p}_2\cos (\omega {\tau }_2)-4p_3\cos (\omega {\tau }_2)-4p_5+2p_2{p}_5{\tau }_2\cos (\omega {\tau }_2)\hfill \\ & -2p_1{p}_3{\tau }_2\cos (\omega {\tau }_2))+(2p_2{p}_5-2p_1{p}_3-2p_3{p}_5{\tau }_2)\sin (\omega {\tau }_2).\hfill \end{array}$$

Since $\underset{\omega \to +\infty }{{\displaystyle \lim }}f(\omega )=+\infty ,$ and $f(0)=d_4<0,f^{\prime }(\omega )>0$, the function $f(\omega )$ is strictly increasing in $\omega$. Consequently, when hypothesis $(H_6)$ holds, Equation (20) admits at least one positive root. This completes the proof. □

Lemma 4.

If $(H_7):M_1{M}_3+M_2{M}_4>0$ holds, then ${{\displaystyle \left.\mathrm{Re}{\left(\frac{d\lambda }{d{\tau }_1}\right)}^{-1}\right|}}_{\lambda =i{\omega }_2}>0$.

Proof. 

Let ${\omega }_2$ denotes a root of Equation (20). The corresponding critical delays ${\tau }_1^{(n)}(n=1,2,3,\cdots )$ are given by

$${\tau }_1^{(n)}={\displaystyle \frac{1}{{\omega }_2}}(\arccos {\displaystyle \frac{{\omega }^2\cos (\omega {\tau }_2)+p_1\omega \sin (\omega {\tau }_2)-p_3-p_5\cos (\omega {\tau }_2)}{p_4}}+2n\pi ),n=1,2,3,\cdots .$$

Let $\lambda ({\tau }_1)=\alpha ({\tau }_1)+i\omega ({\tau }_1)$ as the root of Equation (18) near ${\tau }_1={\tau }_1^{(n)}$ and satisfying $\alpha ({\tau }_1^{(n)})=0,$ $\omega \left({\tau }_1^{(n)}\right)={\omega }_0.$

Differentiating both sides of Equation (18) implicitly with respect to ${\tau }_1$ and rearranging terms yields

$${\left({\displaystyle \frac{d\lambda }{d{\tau }_1}}\right)}^{-1}={\displaystyle \frac{2\lambda +p_1+p_2{e}^{-\lambda {\tau }_2}+p_2\lambda {\tau }_2{e}^{-\lambda {\tau }_2}+p_3{\tau }_2{e}^{-\lambda {\tau }_2}-({\tau }_1+{\tau }_2)p_4{e}^{-\lambda ({\tau }_1+{\tau }_2)}}{e^{-\lambda ({\tau }_1+{\tau }_2)}{p}_4\lambda }}.$$

(21)

Substituting $\lambda =i{\omega }_2$ into (21) and extracting the real part gives

$${{\displaystyle \left.\mathrm{Re}{\left(\frac{d\lambda }{d{\tau }_1}\right)}^{-1}\right|}}_{\lambda =i{\omega }_2}={\displaystyle \frac{M_1{M}_3+M_2{M}_4}{M_3^2+M_4^2}},$$

where

$$M_1=p_1+p_2\cos ({\omega }_2{\tau }_2)+p_2{\tau }_2{\omega }_2\sin ({\omega }_2{\tau }_2)+p_3{\tau }_2\cos ({\omega }_2{\tau }_2)-({\tau }_1+{\tau }_2)p_4\cos ({\omega }_2({\tau }_1+{\tau }_2)),$$

$$M_2=2{\omega }_2-p_2\sin ({\omega }_2{\tau }_2)+p_2{\omega }_2\cos ({\omega }_2{\tau }_2)-p_3{\tau }_2\sin ({\omega }_2{\tau }_2)+({\tau }_1+{\tau }_2)\sin ({\omega }_2({\tau }_1+{\tau }_2)), M_3={\omega }_2\sin ({\omega }_2({\tau }_1+{\tau }_2)),$$

$$M_4={\omega }_2\cos ({\omega }_2({\tau }_1+{\tau }_2)).$$

Therefore, when condition $(H_7):M_1{M}_3+M_2{M}_4>0$ holds, we have ${{\displaystyle \left.\mathrm{Re}{\left(\frac{d\lambda }{d{\tau }_1}\right)}^{-1}\right|}}_{\lambda =i{\omega }_2}>0.$ This completes the proof. □

From Lemmas 3 and 4, the following theorem is derived.

Theorem 5.

If $(H_1),(H_2),(H_6),(H_7)$ hold, then when ${\tau }_1\in [0,{\tau }_1^{(0)})$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is asymptotically stable. When ${\tau }_1>{\tau }_1^{(0)}$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is unstable. When ${\tau }_1={\tau }_1^{(n)}(n=1,2,3,\cdots ),$ system (5) exhibits a Hopf bifurcation at the internal equilibrium $E(x^{*},y^{*})$.

• Case 5. ${\tau }_1\in [0,{\tau }_1^{\left(0\right)}),{\tau }_2>0,$ where ${\tau }_1$ is constrained within its stability interval, while ${\tau }_2$ is treated as a bifurcation parameter.

Equation (6) becomes

$${\lambda }^2+p_1\lambda +e^{-\lambda {\tau }_2}(p_2\lambda +p_3)+e^{-\lambda ({\tau }_1+{\tau }_2)}{p}_4+p_5=0.$$

(22)

Let

$$\begin{array}{l}P(\lambda ,{\tau }_2)={\lambda }^2+p_1\lambda +p_5,\\ Q(\lambda ,{\tau }_2)=p_2\lambda +p_4{e}^{-\lambda {\tau }_1}+p_3,\end{array}$$

then Equation (22) is transformed into

$$P(\lambda ,{\tau }_2)+e^{-\lambda {\tau }_2}Q(\lambda ,{\tau }_2)=0.$$

(23)

Following the geometric criteria in [17], we now verify that Equation (23) satisfies the following five conditions

$(i) P(0,{\tau }_2)+Q(0,{\tau }_2)\ne 0,\forall {\tau }_2\in$ ${ℝ}_{+}$;

$(ii) P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)\ne 0,\forall \omega \in$ ${ℝ}_{+}$, $\forall {\tau }_2\in$ ${ℝ}_{+}$;

$(iii) \underset{\left|\lambda \right|\to \infty }{\lim }\sup \left\{\left|{\displaystyle \frac{Q(\lambda ,{\tau }_2)}{P(\lambda ,{\tau }_2)}}\right|:\mathrm{Re}\lambda \ge 0\right\}<1,\forall {\tau }_2\in$ ${ℝ}_{+}$;

$(iv)$ For each ${\tau }_2,$ the function $F(\omega ,{\tau }_2):={\left|P(i\omega ,{\tau }_2)\right|}^2-{\left|Q(i\omega ,{\tau }_2)\right|}^2$ has at most finitely many real zeros;

$(v) F(\omega ,{\tau }_2)=0$ has a positive root $\omega ({\tau }_2)$, this root is continuously differentiable with respect to ${\tau }_2$.

Condition $(v)$ holds trivially. When $\lambda =0$, if $(H_2)$ is true, one can obtain $P(0,{\tau }_2)+Q(0,{\tau }_2)=p_3+p_4+p_5>0\ne 0,$ so condition $(i)$ is satisfied.

Because $\underset{\left|\lambda \right|\to \infty }{\lim }\left|{\displaystyle \frac{Q(\lambda ,{\tau }_2)}{P(\lambda ,{\tau }_2)}}\right|=\underset{\left|\lambda \right|\to \infty }{\lim }{\displaystyle \frac{p_2\lambda +(p_3+p_4)}{{\lambda }^2+p_1\lambda +p_5}}=0<1,$ condition $(iii)$ is satisfied.

Lemma 5.

If $(H_8):p_1+p_2-p_4{\tau }_1>0$ holds, then condition $(ii)$ is satisfied.

Proof. 

When $\omega \ne 0$, then

$$P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)=(-{\omega }^2+p_5+p_3+p_4\cos (\omega {\tau }_1))+i(p_1\omega +p_2\omega -p_4\sin (\omega {\tau }_1)).$$

When $\omega {\tau }_1>0$, obviously $\sin (\omega {\tau }_1)<\omega {\tau }_1,$ so

$$\mathrm{Im}[P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)]=p_1\omega +p_2\omega -p_4\sin (\omega {\tau }_1)>p_1\omega +p_2\omega -p_4\omega {\tau }_1.$$

Therefore, when $(H_8):p_1+p_2-p_4{\tau }_1>0$ holds, we have $P(i\omega ,{\tau }_2)+Q(i\omega ,{\tau }_2)>0\ne 0,$ and condition $(ii)$ is satisfied. This completes the proof. □

Lemma 6.

If $(H_9):p_5^2-p_3^2-p_4^2-2p_4{p}_3<0,2p_1^2-2p_2^2-4p_5+4p_2{p}_4{\tau }_1+2p_3{p}_4{\tau }_1^2>0$ holds, then $(iv)$ is satisfied.

Proof. 

From

$${\left|P(i\omega ,{\tau }_2)\right|}^2={(-{\omega }^2+p_5)}^2+p_1^2{\omega }^2,$$

$${\left|Q(i\omega ,{\tau }_2)\right|}^2={(p_3+p_4\cos (\omega {\tau }_1))}^2+{(p_2\omega -p_4\sin (\omega {\tau }_1))}^2,$$

can obtain

$$F(\omega ,{\tau }_2)={\omega }^4+(p_1^2-p_2^2-2p_5){\omega }^2+2p_2{p}_4\sin (\omega {\tau }_1)\omega +(p_5^2-p_3^2-p_4^2-2p_4{p}_3\cos (\omega {\tau }_1\left)\right).$$

(24)

Then,

$$F(0,{\tau }_2)=p_5^2-p_3^2-p_4^2-2p_4{p}_3,$$

$$F_{\omega }^{\prime }(\omega ,{\tau }_2)=4{\omega }^3+(2p_1^2-2p_2^2-4p_5+2p_2{p}_4{\tau }_1\cos (\omega {\tau }_1))\omega +(2p_2{p}_4+2p_3{p}_4{\tau }_1)\sin (\omega {\tau }_1).$$

When $\omega {\tau }_1>0$, obviously $\sin (\omega {\tau }_1)<\omega {\tau }_1,\left|\cos (\omega {\tau }_1)\right|\le 1,$ so

$$F_{\omega }^{\prime }(\omega ,{\tau }_2)>4{\omega }^3+(2p_1^2-2p_2^2-4p_5+4p_2{p}_4{\tau }_1++2p_3{p}_4{\tau }_1^2)\omega .$$

Therefore, when $(H_9)$ holds, we have $F(0,{\tau }_2)<0,F_{\omega }^{\prime }(\omega ,{\tau }_2)>0,$ so $F(\omega ,{\tau }_2)$ possesses at least four roots. Condition $(iv)$ is satisfied. This completes the proof. □

To ensure $F(\omega ,{\tau }_2)$ admits positive roots, we define the parameter set based on Equation (24)

$$I_2=\left\{{\tau }_2\left|\left(H_9\right)hold,\forall {\tau }_2\in \left[0,{\tau }_{\max }\right]\right.\right\}.$$

For ${\tau }_2\in I_2$, $F(\omega ,{\tau }_2)$ has a positive real zero ${\omega }_3({\tau }_2),$ where ${\omega }_3\left({\tau }_2\right)$ is determined by the equation

$${\omega }^4+(p_1^2-p_2^2-2p_5){\omega }^2+2p_2{p}_4\sin (\omega {\tau }_1)\omega +(p_5^2-p_3^2-p_4^2-2p_4{p}_3\cos (\omega {\tau }_1\left)\right)=0$$

When ${\tau }_2>0$, assuming $\pm i\omega$ are pure imaginary roots of Equation (23), we substitute $\lambda =i\omega (\omega >0)$ into (22) and separate real and imaginary parts

$$\left\{\begin{array}{l}(p_3+p_4\cos (\omega {\tau }_1))\cos (\omega {\tau }_2)+(p_2\omega -p_4\sin (\omega {\tau }_2))\sin (\omega {\tau }_2)={\omega }^2-p_5,\hfill \\ (p_3+p_4\cos (\omega {\tau }_1))\sin (\omega {\tau }_2)+(p_4\sin (\omega {\tau }_2)-p_2\omega )\cos (\omega {\tau }_2)=p_1\omega ,\hfill \end{array}\right.$$

(25)

we obtain

$$\left\{\begin{array}{l}\sin (\omega {\tau }_2)={\displaystyle \frac{N_1{N}_3+N_2{N}_4}{N_1^2+N_2^2}},\hfill \\ \cos (\omega {\tau }_2)={\displaystyle \frac{N_2{N}_3-N_1{N}_4}{N_1^2+N_2^2}},\hfill \end{array}\right.$$

where

$$N_1=p_4\cos (\omega {\tau }_1)+p_3, N_2=p_2\omega -p_4\sin (\omega {\tau }_1),$$

$$N_3={\omega }^2-p_5, N_4=-p_1\omega .$$

Squaring both equations in (25) and summing them leads to

$${\omega }^4+(p_1^2-p_2^2-2p_5){\omega }^2+2p_2{p}_4\sin (\omega {\tau }_1)\omega +(p_5^2-p_3^2-p_4^2-2p_4{p}_3\cos (\omega {\tau }_1\left)\right)=0.$$

(26)

Since Equation (26) is identical to $F(\omega ,{\tau }_2)=0$, the existence of positive roots for (26) implies that Equation (22) possesses a pair of pure imaginary roots $\pm i\omega$. For ${\tau }_2\in I_2$, we define the phase function $\omega ({\tau }_2){\tau }_2=\theta ({\tau }_2)+2n\pi ,$ and construct the auxiliary function $S_n({\tau }_2)={\tau }_2-{\displaystyle \frac{\theta ({\tau }_2)+2n\pi }{\omega ({\tau }_2)}}.$ Then, $\pm i\omega ({\tau }_2)$ are pure imaginary roots of (22) if and only if ${\tau }_2$ satisfies $S_n({\tau }_2)=0$. Denoting the roots of $S_n({\tau }_2)=0$ by ${\tau }_2^{(i)}$, we establish the following theorem according to Theorem 2.2 in [17].

Theorem 6.

For ${\tau }_2\in I_2$, if there exists ${\tau }_2={\tau }_2^{(i)}$ satisfying $S_n({\tau }_2^{(i)})=0(n\in N),$ then Equation (22) possesses a pair of pure imaginary roots $\pm i\omega ({\tau }_2^{(i)})$ at ${\tau }_2={\tau }_2^{(i)}$. Furthermore, if $\delta ({\tau }_2^{(i)})>0(<0),$ then as ${\tau }_2$ increases through ${\tau }_2^{(i)}$, the characteristic roots corresponding to $\pm i\omega ({\tau }_2^{(i)})$ cross the imaginary axis from the left (right) half plane to the right (left) half plane of the complex plane, where the transversality condition is given by

$$\delta ({\tau }_2^{(i)})=sign\left\{{\left.{\displaystyle \frac{d\mathrm{Re}\lambda ({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}=sign\left\{F_{\omega }^{\prime }(\omega ({\tau }_2^{(i)}),{\tau }_2^{(i)})\right\}sign\left\{{\left.{\displaystyle \frac{d{S}_n({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}.$$

(27)

Since

$$F_{\omega }^{\prime }(\omega ,{\tau }_2)=4{\omega }^3+(2p_1^2-2p_2^2-4p_5+2p_2{p}_4{\tau }_1\cos (\omega {\tau }_1))\omega +(2p_2{p}_4+2p_3{p}_4{\tau }_1)\sin (\omega {\tau }_1),$$

when $(H_9)$ holds, Equation (27) simplifies to

$$\delta ({\tau }_2^{(i)})=sign\left\{{\left.{\displaystyle \frac{d\mathrm{Re}\lambda ({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}=sign\left\{{\left.{\displaystyle \frac{d{S}_n({\tau }_2)}{d{\tau }_2}}\right|}_{{\tau }_2={\tau }_2^{(i)}}\right\}.$$

(28)

It is easy to identify that when ${\tau }_2\in I_2$, the auxiliary function $S_n({\tau }_2)={\tau }_2-{\displaystyle \frac{\theta ({\tau }_2)+2n\pi }{\omega ({\tau }_2)}}$ is monotonically decreasing with respect to $n$, that is, $S_n({\tau }_2)<S_{n+1}({\tau }_2)$. If $S_0({\tau }_2)$ has no zero point, $S_n({\tau }_2)=0(n\in N)$ has no zero point either. When ${\tau }_2=0$, obviously $S_n(0)=-{\displaystyle \frac{\theta (0)+2n\pi }{\omega (0)}}<0.$ When ${\tau }_2\to {\tau }_{\max }$, $p_5^2-p_3^2-p_4^2-2p_4{p}_3\cos (\omega {\tau }_1)\to 0,$ then $\omega ({\tau }_2)\to 0.$ Additionally, according to $\sin (\omega {\tau }_2)\to 0,\cos (\omega {\tau }_2)\to 1,$ we can determine that $\theta ({\tau }_2)\to 2\pi ,$ so $S_0({\tau }_2)\to -\infty .$ Therefore, $S_n({\tau }_2)(n\in N)$ intersects with the horizontal axis, and the number of intersections is even. Let the intersection be

$${\tau }_2^{\min }\stackrel{\Delta }{=}{\tau }_2^{(1)}<{\tau }_2^{(2)}<{\tau }_2^{(3)}<\cdots <{\tau }_2^{(i)}\stackrel{\Delta }{=}{\tau }_2^{\max },$$

where $i$ is an even number.

Theorem 7.

When ${\tau }_2\in I_1$, if $(H_1),(H_2),(H_8),(H_9)$ hold, then we can determine the following:

(i) If $S_0({\tau }_2)$ has no zeros, then the internal equilibrium $E(x^{*},y^{*})$ is asymptotically stable.

(ii) If $S_0({\tau }_2)$ has at least one positive zero, then there exists ${\tau }_2^{(1)},$ and when ${\tau }_2\in [0,{\tau }_2^{(i)})$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is asymptotically stable. When ${\tau }_2\in ({\tau }_2^{(1)},{\tau }_2^{\max })$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is unstable. When ${\tau }_2\in ({\tau }_2^{\max },{\tau }_{\max })$, the internal equilibrium $E(x^{*},y^{*})$ of system (5) is asymptotically stable. At each ${\tau }_2={\tau }_2^{(i)}(i=1,2,3,\cdots )$, system (5) undergoes a Hopf bifurcation at $E(x^{*},y^{*})$.

3. Computation of Hopf Bifurcation Properties

In this section, we take case 5 as an example to study the direction, stability, and period of Hopf bifurcation periodic solutions. Employing normal form theory and center manifold theorem, we fix ${\tau }_1={\tau }_1^{*}\in [0,{\tau }_1^{(0)})$ and treat ${\tau }_2$ as the bifurcation parameter. Other cases can be analyzed similarly and are omitted here. Without loss of generality, assume ${\tau }_2^{(1)}>{\tau }_1^{*},$ and by shifting the interior equilibrium $E(x^{*},y^{*})$ to the origin via the transformation $\overline{x}=x-x^{*}, \overline{y}=y-y^{*},$ system (5) becomes

$$\left\{\begin{array}{l}{\displaystyle \frac{d\overline{x}}{dt}}={\displaystyle \frac{r(\overline{x}+x^{*})}{1+k[\overline{y}(t-{\tau }_1)+y^{*}]}}-[1+b(\overline{y}(t-{\tau }_1)+y^{*})]\alpha (\overline{x}+x^{*})-a{(\overline{x}+x^{*})}^2-c(\overline{x}+x^{*})(\overline{y}+y^{*}),\hfill \\ {\displaystyle \frac{d\overline{y}}{dt}}=\beta e^{-d{\tau }_2}[\overline{x}(t-{\tau }_2)+x^{*}][\overline{y}(t-{\tau }_2)+y^{*}]-m(\overline{y}+y^{*})-n{(\overline{y}+y^{*})}^2.\hfill \end{array}\right.$$

For simplicity, we still denote $x=\overline{x},$ $y=\overline{y}$, and the system can then be expressed as

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}=({\displaystyle \frac{r}{1+k{y}^{*}}}-\alpha -\alpha b{y}^{*}-2a{x}^{*}-c{y}^{*})x-c{x}^{*}y-[{\displaystyle \frac{rk{x}^{*}}{{(1+k{y}^{*})}^2}}+b\alpha x^{*}]y(t-{\tau }_1),\hfill \\ {\displaystyle \frac{dy}{dt}}=(-m-2n{y}^{*})y+\beta e^{-d{\tau }_2}{y}^{*}x(t-{\tau }_2)+\beta e^{-d{\tau }_2}{x}^{*}y(t-{\tau }_2).\hfill \end{array}\right.$$

(29)

Let

$$h_1=X({\tau }_2,t),h_2=Y({\tau }_2,t),$$

$${{\displaystyle L}}_{{\tau }_2}:C([-{\tau }_2^{(1)},0],{{\displaystyle ℝ}}^2)\to {{\displaystyle ℝ}}^2,F:ℝ\times ([-{\tau }_2^{(1)},0],{{\displaystyle ℝ}}^2)\to {{\displaystyle ℝ}}^2,$$

then, system (29) can then be reformulated as

$$\dot{h}\left(t\right)=L_{{\tau }_2}\left(h_t\right)+F\left({\tau }_2,h_t\right),$$

(30)

where

$$h_t={\left(h_1,h_2\right)}^T\in C\left(\left[-{\tau }_2^{(1)},0\right],{ℝ}^2\right),$$

$$L_{{\tau }_2}\left(\phi \right)=A^{\prime }\phi \left(0\right)+B^{\prime }\phi (-{\tau }_1^{*})+C^{\prime }\phi (-{\tau }_2^{(1)})$$

$$F\left({\tau }_2,\phi \right)={\left(F_1,F_2\right)}^T,$$

$$A^{\prime }=\left(\begin{array}{ll}{a}_{11}\hfill & a_{12}\hfill \\ 0\hfill & a_{22}\hfill \end{array}\right),B^{\prime }=\left(\begin{array}{ll}0\hfill & a_{12}^{\prime }\hfill \\ 0\hfill & 0\hfill \end{array}\right),C^{\prime }=\left(\begin{array}{ll}0\hfill & 0\hfill \\ a_{21}\hfill & a_{22}^{\prime } \hfill \end{array}\right)$$

$$\left\{\begin{array}{l}{F}_1=g_1{x}^2+g_2{y}^2(t-{\tau }_1)+g_{3}xy(t-{\tau }_1),\hfill \\ F_2=g_{4}x(t-{\tau }_2)y(t-{\tau }_2)+g_5{y}^2.\hfill \end{array}\right.$$

According to the Riesz representation theorem, there exists a component consisting of a bounded variation function matrix $\eta (\theta , u)$ and a variable function $\theta \in [-{\tau }_2^{(1)},0],$ such that

$${{\displaystyle L}}_{{\tau }_2}(\phi )={\displaystyle {\int }_{-{\tau }_2^{(1)}}^{0}d\eta (\theta ,u)\phi (\theta )}, \theta \in ℝ.$$

In fact, we can choose

$$\eta (\theta ,{\tau }_2)=\left\{\begin{array}{ll}{A}^{\prime }+B^{\prime }+C^{\prime },\hfill & \theta =0,\hfill \\ A^{\prime }+B^{\prime }\hfill & \theta \in [-{\tau }_1^{*},0),\hfill \\ C^{\prime }\hfill & \theta \in [-{\tau }_2^{(1)},-{\tau }_1^{*}),\hfill \\ 0\hfill & \theta =-{\tau }_2^{(1)}.\hfill \end{array}\right.$$

For $\phi \in C\left(\left[-{\tau }_2^{(1)},0\right]\right.\left.,{ℝ}^2\right)$, we give the definitions

$$A({\tau }_2)\phi =\left\{\begin{array}{ll}{\displaystyle \frac{d\phi (\theta )}{d\theta }},\hfill & \theta \in \left[-{\tau }_2^{(1)},0\right),\hfill \\ {\displaystyle {\int }_{-{\tau }_2^{(1)}}^{0}d\eta (\theta ,{\tau }_2)\phi (\theta ),}\hfill & \theta =0, \hfill \end{array}\right.$$

$$R({\tau }_2)\phi =\left\{\begin{array}{ll}0,\hfill & \theta \in \left[-{\tau }_2^{(1)},0\right),\hfill \\ F({\tau }_2,\phi ),\hfill & \theta =0.\hfill \end{array}\right.$$

Equation (29) can be reformulated as

$$\dot{h}(t)=A({\tau }_2)h_t+R({\tau }_2)h_t.$$

(31)

Defining $A^{*}$ as

$$A^{*}({\tau }_2)s=\left\{\begin{array}{ll}-{\displaystyle \frac{d\psi (s)}{ds}},\hfill & s\in \left[0,{\tau }_2^{(1)}\right),\hfill \\ {\displaystyle {\int }_{-{\tau }_2^{(1)}}^{0}d{\eta }^T(\theta ,{\tau }_2)\psi (-s),}\hfill & s=0.\hfill \end{array}\right.$$

The bilinear form between $A$ and $A^{*}$ is given by

$$〈\psi ,\phi 〉=\overline{\psi }(0)\phi (0)-{\displaystyle {\int }_{-{\tau }_2^{(1)}}^0{\displaystyle {\int }_{\xi =0}^{\theta }{\overline{\psi }}^T(\xi -\theta )}}d\eta (\theta ,u)\phi (\xi )d\xi .$$

Clearly, $\pm i{\omega }_3$ are eigenvalues of $A({\tau }_2^{(1)})$ and also of ${{\displaystyle A}}^{*}({\tau }_2^{\left(1\right)})$. Through straightforward calculation, we obtain the corresponding eigenvectors

$$q(\theta )={(1,q_2(0))}^T{e}^{i{\omega }_3\theta },q^{*}(\theta )=D{(1,q_2^{*}(0))}^T{e}^{i{\omega }_3\theta },$$

where

$$q_2(0)={\displaystyle \frac{i{\omega }_3-a_{11}}{a_{12}-a_{12}^{\prime }{e}^{-i{\omega }_3{\tau }_1^{*}}}}, q_2^{*}(0)={\displaystyle \frac{-i{\omega }_3-a_{11}}{a_{21}{e}^{-i{\omega }_3{\tau }_2^{(1)}}}},$$

$$\overline{D}={[1+{\overline{q}}_2^{*}(0)q_2(0)+a_{12}^{\prime }{q}_2(0){{\displaystyle e}}^{-i{\omega }_3{\tau }_1^{*}}+(a_{21}{\overline{q}}_2(0)+a_{22}^{\prime }{\overline{q}}_2(0)q_2(0)){{\displaystyle e}}^{-i{\omega }_3{\tau }_2^{(1)}}]}^{-1}.$$

Let ${{\displaystyle h}}_t$ be the solution of Equation (31). We define

$$z(t)=〈q^{*},{{\displaystyle h}}_t〉, W(t,\theta )={{\displaystyle h}}_t(\theta )-2\mathrm{Re}\left\{z(t)q(\theta )\right\}={{\displaystyle h}}_t(\theta )-z(t)q(\theta )-\overline{z}(\theta )\overline{q}(\theta ).$$

On the center manifold theorem, we have

$$W(t,\theta )=W(z(t),\overline{z}(t),\theta )={{\displaystyle W}}_{20}(\theta ){\displaystyle \frac{z^2}{2}}+{{\displaystyle W}}_{11}(\theta )z\overline{z}+{{\displaystyle W}}_{02}(\theta ){\displaystyle \frac{{\overline{z}}^2}{2}}+\cdots ,$$

$$\dot{z}(t)=i{\omega }_{3}z(t)+{\overline{q}}^{*}(0)\left(\begin{array}{l}{{\displaystyle F}}_1(u,\phi )\hfill \\ {{\displaystyle F}}_2(u,\phi )\hfill \\ {{\displaystyle F}}_3(u,\phi )\hfill \end{array}\right)=i{\omega }_{3}z(t)+g(z,\overline{z}),$$

(32)

$$g(z,\overline{z})={{\displaystyle g}}_{20}(\theta ){\displaystyle \frac{z^2}{2}}+{{\displaystyle g}}_{11}(\theta )z\overline{z}+{{\displaystyle g}}_{02}(\theta ){\displaystyle \frac{{\overline{z}}^2}{2}}+{{\displaystyle g}}_{21}{\displaystyle \frac{z^2\overline{z}}{2}}+\cdots$$

(33)

From Equations (32) and (33), we obtain

$$\dot{W}=AW+H(z,\overline{z},\theta ),$$

(34)

where

$$H(z,\overline{z},\theta )={{\displaystyle H}}_{20}(\theta ){\displaystyle \frac{{{\displaystyle z}}^2}{2}}+{{\displaystyle H}}_{11}(\theta )z\overline{z}+{{\displaystyle H}}_{02}(\theta ){\displaystyle \frac{{{\displaystyle \overline{z}}}^2}{2}}+\cdots$$

By comparing the coefficients on both sides of Equation (34), we obtain

$$(2i{\omega }_3-A)W_{20}(\theta )=H_{20}(\theta ), -A{W}_{11}(\theta )=H_{11}(\theta ),$$

we have

$$\phi (0)=z\left(\begin{array}{l}1\hfill \\ q_2\hfill \end{array}\right)+\overline{z}\left(\begin{array}{l}1\hfill \\ {\overline{q}}_2\hfill \end{array}\right)+{{\displaystyle W}}_{20}(0){\displaystyle \frac{z^2}{z}}+{{\displaystyle W}}_{11}(0)z\overline{z}+{{\displaystyle W}}_{02}(0){\displaystyle \frac{{\overline{z}}^2}{2}}+\cdots ,$$

$$\begin{array}{ll}\hfill \phi (-{\tau }_1^{*})=& z\left(\begin{array}{l}1\hfill \\ q_2\hfill \end{array}\right){{\displaystyle e}}^{-i{\omega }_3{\tau }_1^{*}}+\overline{z}\left(\begin{array}{l}1\hfill \\ {\overline{q}}_2\hfill \end{array}\right){{\displaystyle e}}^{-i{\omega }_3{\tau }_1^{*}}+{{\displaystyle W}}_{20}(-{\tau }_1^{*}){\displaystyle \frac{z^2}{2}}+\hfill \\ & {{\displaystyle W}}_{11}(-{\tau }_1^{*})z\overline{z}+{{\displaystyle W}}_{02}(-{\tau }_1^{*}){\displaystyle \frac{{\overline{z}}^2}{2}}+\cdots , \hfill \end{array}$$

$$\begin{array}{ll}\hfill \phi (-{\tau }_2^{(1)})=& z\left(\begin{array}{l}1\hfill \\ q_2\hfill \end{array}\right){{\displaystyle e}}^{-i{\omega }_3{\tau }_2^{(1)}}+\overline{z}\left(\begin{array}{l}1\hfill \\ {\overline{q}}_2\hfill \end{array}\right){{\displaystyle e}}^{-i{\omega }_3{\tau }_2^{(1)}}+{{\displaystyle W}}_{20}(-{\tau }_2^{(1)}){\displaystyle \frac{z^2}{2}}+\hfill \\ & {{\displaystyle W}}_{11}(-{\tau }_2^{(1)})z\overline{z}+{{\displaystyle W}}_{02}(-{\tau }_2^{(1)}){\displaystyle \frac{{\overline{z}}^2}{2}}+\cdots .\hfill \end{array}$$

By comparing these with the coefficients in Equation (33), we obtain

$$g_{20}=2\overline{D}\{g_1{q}_1^2(0)+g_2{q}_2^2(-{\tau }_1^{*})+g_3{q}_1(0)q_2(-{\tau }_1^{*})+{\overline{q}}_2^{*}[g_4{q}_1(-{\tau }_2^{(1)})q_2(-{\tau }_2^{(1)})+g_5{q}_2^2(0)]\},$$

$$\begin{array}{ll}\hfill g_{11}=& \overline{D}\{2g_1{q}_1(0){\overline{q}}_1(0)+2g_2{q}_2(-{\tau }_1^{*}){\overline{q}}_2(-{\tau }_1^{*})+g_3[q_1(0){\overline{q}}_2(-{\tau }_1^{*})+{\overline{q}}_1(0)q_2(-{\tau }_1^{*})]+\hfill \\ & {\overline{q}}_2^{*}[g_4{q}_1(-{\tau }_2^{(1)}){\overline{q}}_2(-{\tau }_2^{(1)})+g_4{\overline{q}}_1(-{\tau }_2^{(1)})q_2(-{\tau }_2^{(1)})+2g_5{q}_2(0){\overline{q}}_2(0)]\}, \hfill \end{array}$$

$$g_{02}=2\overline{D}\{g_1{\overline{q}}_1^2(0)+g_2{\overline{q}}_2^2(-{\tau }_1^{*})+g_3{\overline{q}}_1(0){\overline{q}}_2(-{\tau }_1^{*})+{\overline{q}}_2^{*}[g_4{\overline{q}}_1^2(-{\tau }_2^{(1)})+g_5{\overline{q}}_2^2(0)]\},$$

$$\begin{array}{ll}\hfill g_{21}& =2\overline{D}\{g_1(W_{20}^{(1)}(0){\overline{q}}_1(0)+2W_{11}^{(1)}(0)q_1(0))+g_2(2W_{11}^{(2)}(-{\tau }_1^{*})q_2(-{\tau }_1^{*})+W_{20}^{(2)}(-{\tau }_1^{*}){\overline{q}}_2(-{\tau }_1^{*}))\hfill \\ & g_3(q_1(0)W_{11}^{(2)}(-{\tau }_1^{*})+W_{11}^{(1)}(0)q_2(-{\tau }_1^{*})+{\displaystyle \frac{1}{2}}{W}_{20}^{(1)}(0){\overline{q}}_2(-{\tau }_1^{*})+{\displaystyle \frac{1}{2}}{W}_{20}^{(2)}(-{\tau }_1^{*}){\overline{q}}_1(0))\hfill \\ & +{\overline{q}}_2^{*}[g_4(q_1(-{\tau }_2^{(1)})W_{11}^{(2)}(-{\tau }_2^{(1)})+{\displaystyle \frac{1}{2}}{\overline{q}}_1(-{\tau }_2^{(1)})W_{20}^{(2)}(-{\tau }_2^{(1)})+q_2(-{\tau }_2^{(1)})W_{11}^{(1)}(-{\tau }_2^{(1)})+\hfill \\ & {\displaystyle \frac{1}{2}}{\overline{q}}_2(-{\tau }_2^{(1)})W_{20}^{(1)}(-{\tau }_2^{(1)}))+g_5(2W_{11}^{(2)}(0)q_2(0)+W_{20}^{(2)}(0){\overline{q}}_2(0))]\},\hfill \end{array}$$

where

$$\begin{array}{l}{W}_{20}(\theta )={\displaystyle \frac{i{g}_{20}q(0)}{{\omega }_3}}{{\displaystyle e}}^{i{\omega }_3\theta }+{\displaystyle \frac{i{\overline{g}}_{02}\overline{q}(0)}{3{\omega }_3}}{{\displaystyle e}}^{-i{\omega }_3\theta }+E_1{{\displaystyle e}}^{2i{\omega }_3\theta },\\ W_{11}(\theta )={\displaystyle \frac{-i{g}_{11}q(0)}{{\omega }_3}}{{\displaystyle e}}^{i{\omega }_3\theta }+{\displaystyle \frac{i{\overline{g}}_{11}\overline{q}(0)}{{\omega }_3}}{{\displaystyle e}}^{-i{\omega }_3\theta }+E_2, \end{array}$$

$$\begin{array}{ll}\hfill E_1& ={(2i{\omega }_{3}I-{\displaystyle {\int }_{-{\tau }_2^{(1)}}^0{e}^{2i{\omega }_3\theta }d\eta (\theta )})}^{-1}{F}_{z^2}\hfill \\ & ={\left(\begin{array}{ll}2i{\omega }_3-a_{11}\hfill & -a_{12}-a_{12}^{\prime }{e}^{-2i{\omega }_3{\tau }_1^{*}}\hfill \\ -a_{21}{e}^{-2i{\omega }_3{\tau }_2^{(1)}}\hfill & 2i{\omega }_3-a_{22}-a_{22}^{\prime }{e}^{-2i{\omega }_3{\tau }_2^{(1)}}\hfill \end{array}\right)}^{-1}\cdot 2\left(\begin{array}{c}\begin{array}{l}{g}_1{q}_1^2(0)+g_2{q}_2^2(-{\tau }_1^{*})+\\ g_3{q}_1(0)q_2(-{\tau }_1^{*})\end{array}\\ g_4{q}_1(-{\tau }_2^{(1)})q_2(-{\tau }_2^{(1)})+g_5{q}_2^2(0)\end{array}\right), \hfill \end{array}$$

$$\begin{array}{ll}\hfill E_2& =-{\left({\displaystyle {\int }_{-{\tau }_2^{(1)}}^{0}d\eta (\theta )}\right)}^{-1}{F}_{z\overline{z}}\hfill \\ & =-{\left(\begin{array}{ll}{a}_{11}\hfill & a_{12}+a_{12}^{\prime }{e}^{-2i{\omega }_3{\tau }_1^{*}}\hfill \\ a_{21}{e}^{-2i{\omega }_3{\tau }_2^{(1)}}\hfill & a_{22}+a_{22}^{\prime }{e}^{-2i{\omega }_3{\tau }_2^{(1)}}\hfill \end{array}\right)}^{-1}\cdot \left(\begin{array}{c}\begin{array}{l}2g_1{q}_1(0){\overline{q}}_1(0)+2g_2{q}_2(-{\tau }_1^{*}){\overline{q}}_2(-{\tau }_1^{*})+\\ g_3(q_1(0){\overline{q}}_2(-{\tau }_1^{*})+{\overline{q}}_1(0)q_2(-{\tau }_1^{*}))\end{array}\\ \\ \begin{array}{l}{g}_4(q_1(-{\tau }_2^{(1)}){\overline{q}}_2(-{\tau }_2^{(1)})+\\ {\overline{q}}_1(-{\tau }_2^{(1)})q_2(-{\tau }_2^{(1)}))+2g_5{q}_2(0){\overline{q}}_2(0)\end{array}\end{array}\right).\hfill \end{array}$$

Thus, through calculation, we obtain

$$\begin{array}{l}{{\displaystyle c}}_1(0)={\displaystyle \frac{i}{2{{\displaystyle \omega }}_2{{\displaystyle \tau }}_2^{*}}}[{{\displaystyle g}}_{11}{{\displaystyle g}}_{20}-2{{\displaystyle \left|{{\displaystyle g}}_{11}\right|}}^2-{\displaystyle \frac{{{\displaystyle \left|{{\displaystyle g}}_{02}\right|}}^2}{3}}]+{\displaystyle \frac{{{\displaystyle g}}_{21}}{2}},\\ {{\displaystyle \mu }}_2={\displaystyle \frac{\mathrm{Re}\{{{\displaystyle c}}_1(0)\}}{\mathrm{Re}\{{\lambda }^{\prime }({{\displaystyle \tau }}_2^{*})\}}},\\ {{\displaystyle \beta }}_2=2\mathrm{Re}\{{{\displaystyle c}}_1(0)\},\\ {{\displaystyle T}}_2=-{\displaystyle \frac{\mathrm{Im}\{{{\displaystyle c}}_1(0)\}+{{\displaystyle \mu }}_2\mathrm{Im}\{{\lambda }^{\prime }({{\displaystyle \tau }}_2^{*})\}}{{{\displaystyle \omega }}_2}},\end{array}$$

where ${\mu }_2$ determines the direction of the Hopf bifurcation; when ${\mu }_2>0,{\tau }_2>{\tau }_2^{(1)}({\mu }_2<0,{\tau }_2<{\tau }_2^{(1)})$, the system undergoes a supercritical (subcritical) Hopf bifurcation near the equilibrium point. ${\beta }_2$ determines the stability of the bifurcated periodic solution, and ${\beta }_2>0 ({\beta }_2<0)$ indicates that the bifurcated periodic solution restricted to the center manifold is unstable (asymptotically stable). $T_2$ determines the increase or decrease in the period, and in $T_2>0(T_2<0)$, the part outside of the brackets indicates that the period increases, while that contained within the brackets indicates that the period decreases [18].

4. Numerical Simulations

In this section, numerical simulations are conducted under five scenarios with selected parameters.

• Case 1. ${\tau }_1={\tau }_2=0.$

Selecting parameters $r=0.5487,$ $a=0.0155,$ $m=0.0947,$ $b=0.9525,$ $n=0.5148,$ $c=0.8431,$ $k=0.9350,$ $\beta =0.9734,$ $\alpha =0.3725,$ and $d=0.3209,$ system (5) becomes

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{0.7803x}{1+0.9421y(t-{\tau }_1)}}-(1+0.4089y(t-{\tau }_1))0.5752x-0.3897x^2-0.1320xy,\hfill \\ {\displaystyle \frac{dy}{dt}}=0.9561e^{-0.0598{\tau }_2}x(t-{\tau }_2)y(t-{\tau }_2)-0.2417y-0.0965y^2.\hfill \end{array}\right.$$

(35)

Calculations confirm that system (35) possesses an internal equilibrium at $E(0.2099,0.5786)$, with conditions $(H_0),(H_1)$, and $(H_2)$ satisfied. We conclude from Theorem 1 that the interior equilibrium $E(0.2099,0.5786)$ is asymptotically stable, as demonstrated in Figure 1.

• Case 2. ${\tau }_1>0,{\tau }_2=0.$

Selecting parameters $r=0.9986,$ $a=0.3149,$ $m=0.2027,$ $b=0.2620,$ $n=0.8163,$ $c=0.8772,$ $k=0.6395,$ $\beta =0.9111,$ $\alpha =0.5648,$ and $d=0.5715,$ system (5) becomes

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{0.9986x}{1+0.6395y(t-{\tau }_1)}}-(1+0.2620y(t-{\tau }_1))0.3149x-0.3149x^2-0.8772xy,\hfill \\ {\displaystyle \frac{dy}{dt}}=0.9111e^{-0.5717{\tau }_2}x(t-{\tau }_2)y(t-{\tau }_2)-0.2027y-0.8163y^2.\hfill \end{array}\right.$$

(36)

Calculations confirm that system (36) possesses an internal equilibrium at $E(0.2156,0.5855)$, with conditions $(H_0),(H_1),(H_2),(H_3)$, and $(H_4)$ satisfied. The critical delay is ${\tau }_1^{(0)}=6.5$. According to Theorem 2, when ${\tau }_1=3<{\tau }_1^{(0)},$ the internal equilibrium $E(0.2156,0.5855)$ is asymptotically stable, as shown in Figure 2. When ${\tau }_1=9>{\tau }_1^{(0)},$ the internal equilibrium $E(0.2156,0.5855)$ becomes unstable, and a Hopf bifurcation occurs, as illustrated in Figure 3.

• Case 3. ${\tau }_1=0,{\tau }_2>0.$

Selecting parameters $r=0.8714,$ $a=0.1113,$ $m=0.1939,$ $b=0.7968,$ $n=0.0043,$ $c=0.0418,$ $k=0.2079,$ $\beta =0.9323,$ $\alpha =0.2393,$ and $d=0.4149,$ system (5) becomes

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{0.8714x}{1+0.2079y(t-{\tau }_1)}}-(1+0.7968y(t-{\tau }_1))0.2393x-0.1113x^2-0.0418xy,\hfill \\ {\displaystyle \frac{dy}{dt}}=0.9323e^{-0.4149{\tau }_2}x(t-{\tau }_2)y(t-{\tau }_2)-0.1939y-0.0043y^2.\hfill \end{array}\right.$$

(37)

Calculations confirm that system (37) possesses an internal equilibrium at $E(0.2156,1.6565)$, with conditions $(H_0),(H_1),(H_2)$, and $(H_5)$ satisfied. Figure 4 depicts the variation curves of $S_0$ and $S_1$ versus ${\tau }_2$. We observe that $S_0$ intersects the x-axis at two points, calculated as ${\tau }_2^{(1)}=0.76$ and ${\tau }_2^{(2)}=3.62$. According to Theorem 4, when ${\tau }_2=0.1<{\tau }_2^{(1)}$, the internal equilibrium $E(0.2156,1.6565)$ is asymptotically stable, as shown in Figure 5. When ${\tau }_2=2>{\tau }_2^{(1)}$, the internal equilibrium $E(0.2156,1.6565)$ becomes unstable, and a Hopf bifurcation occurs, as illustrated in Figure 6.

• Case 4. ${\tau }_1>0,{\tau }_2\in [0,{\tau }_2^{\left(1\right)}).$

Selecting parameters $r=0.5487,$ $a=0.0155,$ $m=0.0947,$ $b=0.9525,$ $n=0.5148,$ $c=0.8431,$ $k=0.9350,$ $\beta =0.9734,$ $\alpha =0.3725,$ and $d=0.3209,$ system (5) becomes

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{0.5487x}{1+0.6395y(t-{\tau }_1)}}-(1+0.9525y(t-{\tau }_1))0.3725x-0.0155x^2-0.8431xy,\hfill \\ {\displaystyle \frac{dy}{dt}}=0.9734e^{-0.3209{\tau }_2}x(t-{\tau }_2)y(t-{\tau }_2)-0.0947y-0.5148y^2.\hfill \end{array}\right.$$

(38)

Calculations confirm that system (38) possesses an internal equilibrium at $E(0.1535,0.1044)$, with conditions $(H_0),(H_1),(H_2),(H_6)$, and $(H_7)$ satisfied. We set ${\tau }_2=0.0212$ within the stability interval. Treating ${\tau }_1$ as the bifurcation parameter, and the critical delay is ${\tau }_1^{(0)}=4.5$. According to Theorem 5, when ${\tau }_1=1<{\tau }_1^{(0)}$, the internal equilibrium $E(0.1535,0.1044)$ is asymptotically stable, as shown in Figure 7. When ${\tau }_1=9>{\tau }_1^{(0)}$, the internal equilibrium $E(0.1535,0.1044)$ becomes unstable, and a Hopf bifurcation occurs, as illustrated in Figure 8.

• Case 5. ${\tau }_1\in [0,{\tau }_1^{\left(0\right)}),{\tau }_2>0.$

Calculations confirm that the parameters of Case 3 satisfy conditions $(H_0),(H_1),(H_2),(H_8)$, and $(H_9)$ for Case 5. For analytical continuity, we select Case 3’s parameters $r=0.8714,$ $a=0.1113,$ $m=0.1939,$ $b=0.7968,$ $n=0.0043,$ $c=0.0418,$ $k=0.2079,$ $\beta =0.9323,$ $\alpha =0.2393,$ and $d=0.4149,$ and system (5) becomes

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{0.8714x}{1+0.2079y(t-{\tau }_1)}}-(1+0.7968y(t-{\tau }_1))0.2393x-0.1113x^2-0.0418xy,\hfill \\ {\displaystyle \frac{dy}{dt}}=0.9323e^{-0.4149{\tau }_2}x(t-{\tau }_2)y(t-{\tau }_2)-0.1939y-0.0043y^2.\hfill \end{array}\right.$$

(39)

Calculations confirm that system (39) possesses an internal equilibrium at $E(0.2156,1.6565)$, with conditions $(H_0),(H_1),(H_2),(H_8)$, and $(H_9)$ satisfied. Figure 9 depicts the variation curves of $S_0$ and $S_1$ versus ${\tau }_2$. We observe that $S_0$ intersects the x-axis at two points, calculated as ${\tau }_2^{(1)}=0.49$ and ${\tau }_2^{(2)}=4.16$. According to Theorem 7, when ${\tau }_2=0.1<{\tau }_2^{(1)}$, the internal equilibrium $E(0.2156,1.6565)$ is asymptotically stable, as shown in Figure 10. When ${\tau }_2=0.8>{\tau }_2^{(1)}$, the internal equilibrium $E(0.2156,1.6565)$ becomes unstable, and a Hopf bifurcation occurs, as illustrated in Figure 11.

5. Conclusions

This paper examines the dynamics of a predator–prey model characterized by double delays, encompassing a fear delay ${\tau }_1$ and a delay ${\tau }_2$ in the conversion of prey biomass into predator biomass after predation. Five delay cases (1) ${\tau }_1={\tau }_2=0,$ (2) ${\tau }_1>0,{\tau }_2=0,$ (3) ${\tau }_1=0,{\tau }_2>0,$ (4) ${\tau }_1>0,{\tau }_2\in (0,{\tau }_2^{\left(1\right)}),$ and (5) ${\tau }_1\in [0,{\tau }_1^{\left(0\right)}),{\tau }_2>0$ are analyzed for system stability and Hopf bifurcation conditions. Taking case (5) as an example, Hopf bifurcation properties are studied via normal form theory and the central manifold theorem, complemented by numerical simulations to illustrate delay impacts on system dynamics.

This paper reveals the impact mechanism through which fear delay and biomass conversion delay influence system stability. Fear-induced delayed responses may exacerbate prey vulnerability, while biomass conversion delay can constrain predator population overgrowth. These two factors collectively regulate system equilibrium, providing insights for controlling endangered species. Practically, this research can predict periodic population fluctuations driven by fear effects and energy transfer processes, thereby providing a scientific foundation for the formulation of ecological protection strategies and ecosystem management.