2. Preliminaries

2.1. Existence, uniqueness, and boundedness of solutions

Let $\mathbf{C}=C([-\tau ,0],\mathbb{R}_+)$ be a set of continuous functions mapping $[-\tau ,0]$ to $\mathbb{R}_+$ equipped with the supremum norm, where $\mathbb{R}_+$ is the set of positive real numbers. Consider the initial data $\phi =(\phi _1(t),\phi _2(t))\in \mathbf{C\times C}$ for system (1.2). The right-hand side of system (1.2) is continuously differentiable with respect to the arguments $x(t)$ , $y(t)$ , $x(t-\tau )$ , and $y(t-\tau )$ . By the existence and uniqueness theorem [Reference Hale and Lunel8, Reference Kuang14], there exists a unique solution $(x_1(t), x_2(t))$ for $t$ in a small neighbourhood of $t=0$ and satisfies the initial condition $(x_1(t), x_2(t))=(\phi _1(t),\phi _2(t))$ for $t\in [-\tau , 0]$ . Moreover, the solution is continuously differentiable.

Lemma 2.1. $x(t)$ and $y(t)$ are positive for $t\gt 0$ given the initial condition $\phi \in \mathbf{C\times C}$ .

Proof. It follows from the first equation of system (1.2) and the formula of variation of constants that

\begin{equation*}x(t)=x(0)e^{ \int _{0}^{t} r(1-\frac {x(s)}{K})-\frac {my(s)}{a+x(s)}\,ds }.\end{equation*}

Since $x(0)\gt 0$ , then $x(t)\gt 0$ for $t\gt 0$ on the interval of existence. For $y(t)$ , since $y(0)\gt 0$ and $y(t)$ is continuous, then either $y(t)\gt 0$ for $t\gt 0$ on the interval of existence, or $\exists$ $t_0\gt 0,$ such that $y(t_0)=0$ , but $y'(t_0)\leq 0$ . However, by substituting $t=t_0$ into the second equation of system (1.2), we obtain

\begin{equation*} y'(t_0)=e^{-d\tau }\frac {cmx(t_0-\tau )y(t_0-\tau )}{a+x(t_0-\tau )}\gt 0, \end{equation*}

Which is a contradiction. Hence, $y(t)$ is positive for $t\gt 0$ .

Lemma 2.2. The solution $(x(t), y(t))$ is eventually uniformly bounded from above provided that the initial condition $\phi \in \mathbf{C\times C}$ .

Proof. We will first show that $x(t)$ is eventually uniformly bounded from above. From the first equation of system (1.2), by Lemma 2.1, we have

\begin{equation*}\frac {dx}{dt}=rx(t)\left(1-\frac {x(t)}{K}\right)-\frac {mx(t)y(t)}{a+x(t)}\lt rx(t)\left(1-\frac {x(t)}{K}\right).\end{equation*}

Let $X(t)$ be the solution to $\frac {dX}{dt}=X(t)(1-\frac {X(t)}{K})$ with $X(0)=x(0)\gt 0$ . By the comparison theorem,

\begin{equation*}\limsup _{t\rightarrow \infty }x(t)\leq \limsup _{t\rightarrow \infty }X(t)=K.\end{equation*}

To show $y(t)$ is eventually uniformly bounded from above, let $U(t)=ce^{-d\tau }x(t-\tau )+y(t)$ , then

\begin{eqnarray*} \frac {dU}{dt}&=&ce^{-d\tau }\frac {d(x(t-\tau ))}{dt}+\frac {d(y(t))}{dt}\\ &=&ce^{-d\tau }\left ((r+d)x(t-\tau )-\frac {r(x(t-\tau ))^2}{K}\right )-dU(t). \end{eqnarray*}

Notice that

\begin{equation*}(r+d)x(t-\tau ) -\frac {r(x(t-\tau ))^2}{K}-\frac {(r+d)^2K}{4r}=-\left (\frac {\sqrt {r}x(t-\tau )}{\sqrt {K}}-\frac {(r+d)\sqrt {K}}{2\sqrt {r}}\right )^2\leq 0,\end{equation*}

and accordingly,

\begin{equation*}\frac {dU}{dt}\leq ce^{-d\tau }\frac {(r+d)^2K}{4r}-dU(t).\end{equation*}

Let $Y(t)$ be the solution to $\frac {dY}{dt}=ce^{-d\tau }\frac {(r+d)^2K}{4r}-dY(t)$ with $Y(0)=U(0)$ . Then,

\begin{equation*}Y(t)=\frac {ce^{-d\tau }(r+d)^2K}{4rd}+\left (U(0)-\frac {ce^{-d\tau }(r+d)^2K}{4rd}\right )e^{-dt}.\end{equation*}

By the comparison theorem,

\begin{equation*}\limsup _{t\rightarrow \infty }y(t)\leq \limsup _{t\rightarrow \infty }U(t)\leq \limsup _{t\rightarrow \infty }Y(t)=\frac {ce^{-d\tau }(r+d)^2K}{4rd}\leq cK.\end{equation*}

Hence, $y(t)$ is eventually uniformly bounded from above.

Theorem 2.3. The solution $(x(t), y(t))$ of system (1.2) exists on $[-\tau , \infty )$ for $\phi \in \mathbf{C\times C}$ .

Proof. By Lemmas 2.1 and 2.2, the solution $(x(t), y(t))$ is eventually uniformly bounded. By the Continuation Theorem [Reference Hale and Lunel8, Reference Kuang14], the solution exists for all $t\gt 0$ .

2.2. Dynamics of the classical model and equilibrium points of system (1.2)

To fully understand the dynamics of the delayed model (1.2), we recall the dynamics of the classical model below, i.e., the corresponding ODE system of model (1.2), which was studied by Jing and Chen in the 1980s [Reference Jing and Chen13].

(2.1) \begin{equation} \begin{cases} \displaystyle \frac {dx(t)}{dt}=rx\left(1-\frac {x}{K}\right)-\frac {mxy}{a+x},\\[12pt] \displaystyle \frac {dy(t)}{dt}=y\left(-d+\frac {cmx}{a+x}\right). \end{cases} \end{equation}

Lemma 2.4. Let $K_c=\frac {ad}{cm-d}$ , and $K_0=\frac {a(cm+d)}{cm-d}$ . System (2.1) has three equilibrium points $E_0(0,0)$ , $E_K(K,0)$ and $E^*(\frac {ad}{cm-d}, \frac {rca(Kcm-Kd-ad)}{K(cm-d)^2})$ , where $E_0$ is always a saddle.

• • $E_K$ is globally asymptotically stable for $K\lt K_c$ and becomes unstable for $K\gt K_c$ . As $K$ increases and crosses the threshold $K=K_c$ , a transcritical bifurcation takes place between $E_K$ and $E^\ast$ . $E_K$ loses stability and $E^\ast$ enters the first quadrant.

• • $K=K_0$ is the Hopf bifurcation threshold for $E^*$ . $E^\ast$ is globally asymptotically stable for $K_c\lt K\lt K_0$ and unstable for $K\gt K_0$ .

• • System (2.1) has no periodic solution for $K\leq K_0$ , and a unique periodic solution for $K\gt K_0$ .

For system (1.2), it is easy to see that the equilibrium points $E_0(0, 0)$ and $E_K(K, 0)$ exist for all $\tau \gt 0$ . To explore the existence and stability of interior equilibrium points, we define the following parameters for convenience:

\begin{equation*}K_1=\frac {ad}{cme^{-d\tau }-d}, K_2=\frac {ad(3cm+d)}{c^2m^2-d^2}, \tau _{max}=\frac {1}{d} \ln \left (\frac {Kcm}{ad+dK}\right ).\end{equation*}

Then, $K_c\lt K_2\lt K_0$ and $K_c\lt K_1$ . We introduce the condition below.

• • Condition $C_0\,:\, cm\gt d$ , and $K\gt K_1$ (i.e., $\tau \lt \tau _{max}$ ).

Through straightforward calculations, we derive the following lemma.

Lemma 2.5. The interior equilibrium $E^*(\frac {da}{ce^{-d\tau }m-d},\frac {rce^{-d\tau }a(Kce^{-d\tau }m-Kd-da)}{K(ce^{-d\tau }m-d)^2})$ of system (1.2) exists under the condition $C_0$ .

Consider the stability of equilibrium points. Let $(x_0, y_0)$ be any equilibrium point of system (1.2) The characteristic matrix at $(x_0, y_0)$ is given by

\begin{equation*} J(x_0,y_0)=\begin{bmatrix} \bigg (r-\frac {2rx_0}{K}-\frac {may_0}{(a+x_0)^2}\bigg ) -\lambda & \quad \frac {-mx_0}{a+x_0} \\ e^{-\lambda \tau }\frac {ce^{-d\tau }may_0}{(a+x_0)^2} & \quad \bigg (-d+e^{-\lambda \tau }\frac {ce^{-d\tau }mx_0}{(a+x_0))}\bigg )-\lambda \end{bmatrix}.\end{equation*}

By substituting $x_0$ and $y_0$ with 0 in $J(x_0,y_0)$ , we obtain two simple eigenvalues of $J(0, 0)$ : $\lambda _1=r\gt 0$ and $\lambda _2=-d\lt 0$ . Applying the Hartman–Grobman theorem [Reference Hartman9], we establish the following lemma.

Lemma 2.6. Equilibrium $E_0$ is a saddle point for all $\tau \geq 0$ .

3. Local dynamics near equilibrium points

3.1. Local stability of $E^*$

Substituting the coordinates of $E^*$ into $J(x_0, y_0)$ , we obtain

\begin{align*} J(E^\ast ) = \left[\begin{array}{c@{\quad}c} \frac {rd(K(ce^{-d\tau }m-d)-a(ce^{-d\tau }m+d))}{Kce^{-d\tau }m(ce^{-d\tau }m-d)} -\lambda & \frac {-d}{ce^{-d\tau }}\\[5pt] e^{-\lambda \tau }\frac {r(Kce^{-d\tau }m-Kd-da)}{Km} & -d+de^{-\lambda \tau }-\lambda \end{array}\right]. \end{align*}

The characteristic equation is derived by setting the determinant of $J(E^\ast )$ to zero, i.e.,

(3.1) \begin{equation} P(\lambda ,\tau )\,:\!=\,\lambda ^2+[d-H(\tau )]\lambda +e^{-\lambda \tau }[-d\lambda +L(\tau )]-dH(\tau )=0, \end{equation}

where $H(\tau )$ and $L(\tau )$ are defined as follows

\begin{equation*}H(\tau )=r\bigg [\frac {de^{d\tau }}{cm}-\frac {ad}{K(ce^{-d\tau }m-d)}-\frac {ad^2e^{d\tau }}{(ce^{-d\tau }m-d)Kcm}\bigg ],\end{equation*}

\begin{equation*}L(\tau )=dH(\tau )+A(\tau ), \ \text{and}\ \ A(\tau )=rd\left (\frac {Kce^{-d\tau }m-Kd-da}{Kmce^{-d\tau }}\right ),\end{equation*}

with $\tau \in [0, \tau _{max})$ . Under condition $C_0$ , $A(\tau )$ is strictly positive, and the functions $H(\tau )$ , $L(\tau )$ , and $A(\tau )$ are well defined and bounded. For simplicity, we will drop the $(\tau )$ and express $H(\tau )$ , $L(\tau )$ , and $A(\tau )$ as $H, L$ , and $A$ when necessary.

Lemma 3.1. The sum of the multiplicity of roots of $P(\lambda , \tau )=0$ on $\mathbb{C}^+=\{\lambda \in \mathbb{C}\,:\, Re(\lambda )\gt 0\}$ can change only if a pair of conjugate complex roots appear on or cross the imaginary axis as $\tau$ varies continuously on $[0, \tau _{max})$ .

Proof. To show that no eigenvalue emerges from infinity, we write $P(\lambda ,\tau )= \lambda ^2 +\eta (\lambda ,\tau )$ , where $\eta (\lambda ,\tau )=[d-H(\tau )]\lambda +e^{-\lambda \tau }[-d\lambda +L(\tau )]-dH(\tau )$ . Since $H(\tau )$ and $L(\tau )$ are bounded on $[0, \tau _{max})$ , we have

\begin{equation*}\limsup _{Re\lambda \gt 0, |\lambda | \to \infty }\big |\lambda ^{-2} \eta (\lambda ,\tau )\big |\leq \limsup _{Re\lambda \gt 0, |\lambda | \to \infty }\left (\frac {2d+|H(\tau )|}{|\lambda |}+\frac {|L(\tau )|+d|H(\tau )|}{|\lambda |^2}\right )=0\lt 1.\end{equation*}

Furthermore, $P(0, \tau )=A(\tau )\gt 0$ for $\tau \in [0, \tau _{max})$ . Hence, $\lambda =0$ is not a root for $\tau \in [0, \tau _{max})$ . By Theorem 1.4, Chapter 3 of [Reference Kuang14], the desired result is obtained.

To study the stability of $E^\ast$ and the possibility of Hopf bifurcation at $E^*$ as $\tau$ varies, by Lemma 3.1, we assume $\lambda =\pm iw$ is a pair of purely imaginary roots of $P(\lambda , \tau )=0$ where $w\gt 0$ . This leads to

\begin{equation*}-w^2+ [d-H]iw+e^{-i\tau w}[{-}diw+L]-dH=0.\end{equation*}

Using Euler’s formula and equating the real and imaginary parts to zero, we obtain

(3.2) \begin{equation} \begin{cases} -w^2 +L\cos (\tau w)-dw\sin (\tau w)-dH=0,\\ (d-H)w -dw\cos (\tau w)-L\sin (\tau w)=0. \end{cases} \end{equation}

Solving for $\cos (\tau w)$ and $\sin (\tau w)$ , and using the assumption $\tau \in [0,\tau _{max})$ , we obtain

(3.3) \begin{equation} \begin{cases} \sin (\tau w)=\frac {dwL-HwL-d^2wH-dw^3}{L^2+d^2w^2},\\[5pt] \cos (\tau w)=\frac {LdH+Lw^2+(dw)^2-dHw^2}{L^2+d^2w^2}. \end{cases} \end{equation}

Both $\sin (\tau w)$ and $\cos (\tau w)$ are well defined, since $L^2\geq 0$ and $d^2w^2\gt 0$ . Applying $\sin ^2(\tau w)+\cos ^2(\tau w)=1$ to (3.3), we obtain

(3.4) \begin{equation} (d^2w^2+L^2)(w^4+H^2w^2+d^2H^2-L^2)=0. \end{equation}

Theorem 3.2. Let $K\gt K_2$ , and define

\begin{equation*}\bar {\tau }= \frac {1}{d}\ln \left (\frac {cm\sqrt {9a^2+4aK+4K^2}-3acm}{2\left (ad+dK\right )}\right ). \end{equation*}

The following statements hold.

1. (1) $0\lt \bar {\tau }\lt \tau _{max}$ .

2. (2) $L(\tau )+dH(\tau )\gt 0$ if $\tau \in [0, \bar {\tau })$ , $L(\tau )+dH(\tau )=0$ if $ \tau =\bar {\tau }$ , and $L(\tau )+dH(\tau )\lt 0$ if $\tau \in (\bar {\tau }, \tau _{max})$ .

3. (3) The unique solution of the hexic equation ( 3.4 ) for $0\leq \tau \lt \bar {\tau }$ and $K\gt K_2$ is given by

   (3.5) \begin{equation} w(\tau )=\sqrt {\frac {\sqrt {((H(\tau ))^4 - 4d^2(H(\tau ))^2 + 4(L(\tau ))^2)}}{2} - \frac {(H(\tau ))^2}{2}}. \end{equation}

Proof. Statement (1) is obtained by direct calculation.

(2). $L(\tau )+dH(\tau )$ is continuous for $\tau \in [0, \tau _{max})$ . It is easy to check that $L(\tau )+dH(\tau )$ equals zero if and only if $\tau =\bar {\tau }$ , and

\begin{equation*}L'(\bar {\tau })+dH'(\bar {\tau })=-\frac {4ard^2(K+a)(\sqrt {4K^2+4Ka+9a^2}-3a)}{(2K+5a-\sqrt {4K^2+4Ka+9a^2})^2K}\lt 0.\end{equation*}

Hence, the sign of $L(\tau )+dH(\tau )$ is obtained for $\tau \in [0, \tau _{max})$ .

(3) Let $Z=w^2$ , and substitute $Z$ into the hexic equation (3.4). Then, three solutions appear

\begin{equation*} Z_{\pm }=\frac {\pm \sqrt {H^4 - 4d^2H^2 + 4L^2}}{2}-\frac {H^2}{2}, Z_3=\frac {-L^2}{d^2}.\end{equation*}

Since $Z=w^2\gt 0$ , $Z_{-}$ and $Z_3$ are discarded. To ensure $w=\sqrt {Z_+}$ is well defined, it suffices to show

\begin{equation*}(H(\tau ))^4 - 4d^2(H(\tau ))^2 + 4(L(\tau ))^2\gt (H(\tau ))^4,\end{equation*}

i.e., $(L(\tau ))^2\gt d^2(H(\tau ))^2$ . Since $A(\tau )$ is positive for $\tau \in [0, \tau _{max})$ , we have $L(\tau )=dH(\tau )+A(\tau )\gt dH(\tau )$ . Moreover, by statement (2), we have $L(\tau )\gt -dH(\tau )$ for $\tau \in [0, \bar {\tau })$ . In conclusion, $L(\tau )\gt |dH(\tau )|$ , and $(L(\tau ))^2\gt d^2(H(\tau ))^2$ .

Corollary 3.3. The equilibrium $E^\ast$ is locally asymptotically stable if $K\leq K_2$ and $\tau \in (0, \tau _{max})$ , or $K_2\lt K\lt K_0$ and $\tau \in [\bar {\tau }, \tau _{max})$ .

Proof. Based on Theorem3.2, $d^2(H(\tau ))^2-(L(\tau ))^2\gt 0$ if $K\leq K_2$ and $\tau \in (0, \tau _{max})$ , or $K_2\lt K\lt K_0$ and $\tau \in [\bar {\tau }, \tau _{max})$ . Then, (3.4) has no positive root and the desired result follows.

3.2. Local Hopf bifurcation at $E^*$

By statement (3) of Theorem3.2, we assume $0\leq \tau \leq \bar {\tau }\lt \tau _{max}$ and $K\gt K_2$ , which guarantees that the hexic equation (3.4) has a positive root $w=w(\tau )$ , a necessary condition for Hopf bifurcation.

We explore the critical values for $\tau \in [0, \bar {\tau })$ at which Hopf bifurcation occurs. From the second equation of system (3.3), it is reasonable to define the angle $\tau w(\tau )$ using the inverse cosine function, while also considering the sign of $\sin (\tau w(\tau ))$ . Here $\sin (\tau w(\tau ))$ is the composition of the functions $\sin (\tau w)$ and $w(\tau )$ , which are given by the first equation of system (3.3) and equation (3.5), respectively.

Lemma 3.4. Define $\breve {\tau }={\frac {1}{d}\ln \left ( {\frac {cm \left ( K-a \right ) }{d \left (K+a \right ) }} \right ) }$ .

1. (1) If $K\lt K_0$ , then $\breve {\tau }\lt 0$ , or $\breve {\tau }$ does not exist, and $\sin (\tau w(\tau ))\gt 0$ for all $\tau \in [0,\bar {\tau })$ .

2. (2) If $K=K_0$ , then $\breve {\tau }=0$ , $\sin (\breve {\tau } w(\breve {\tau }))= 0$ . Moreover, $\sin (\tau w(\tau ))\gt 0$ for all $\tau \in (0,\bar {\tau })$ .

3. (3) If $K\gt K_0$ , $\breve {\tau }$ is well defined with $0\lt \breve {\tau }\lt \bar {\tau }$ . Then, $\sin (\tau w(\tau ))$ changes sign as follows: $\sin (\tau w(\tau ))\lt 0$ for $0\lt \tau \lt \breve {\tau }$ ; $\sin (\tau w(\tau ))=0$ for $\tau =\breve {\tau }$ ; $\sin (\tau w(\tau ))\gt 0$ for $\breve {\tau }\lt \tau \lt \bar {\tau }$ .

Consequently, the function $\theta _n(\tau ), \tau \in [0, \bar {\tau })$ is well defined as follows.

• • If $\sin (\tau w(\tau ))\lt 0$ : $\theta _n(\tau )=-\arccos \left (\frac {LdH+Lw^2+d^2w^2-dHw^2}{L^2+d^2w^2}\right )+2n\pi .$

• • If $\sin (\tau w(\tau ))\geq 0$ : $\theta _n(\tau )=\arccos \left (\frac {LdH+Lw^2+d^2w^2-dHw^2}{L^2+d^2w^2}\right )+2n\pi .$

Here, $n\in \mathbb{N}$ , and $\mathbb{N}$ is the set of all nonnegative integers.

Proof. Solving $\sin (\tau w(\tau ))=0$ for $H(\tau )$ , we obtain three solutions

\begin{equation*}H(\tau )=0, H(\tau )=\frac {L(\tau )}{d}, H(\tau )=2d.\end{equation*}

The solution $H(\tau )=\frac {L(\tau )}{d}$ is discarded since it is valid only for $\tau =\tau _{max}$ . The third solution $H(\tau )=2d$ is also discarded. Indeed, substituting $H(\tau )=2d$ into $\sin (\tau w(\tau ))=0$ yields $|L(\tau )|+L(\tau )=0$ . Since $A(\tau )\gt 0$ , we have

\begin{equation*}0\lt 2d^2+A(\tau )=dH(\tau )+A(\tau )=L(\tau )=-|L(\tau )|\leq 0,\end{equation*}

which leads to a contradiction. Therefore, the remaining solution $H(\tau )=0$ holds when $\tau =\breve {\tau }$ .

Case (1). If $K\lt K_0$ , then $\breve {\tau }\lt 0$ , or $\breve {\tau }$ does not exist. Then, $H(\tau )\lt 0$ for all $\tau \in [0, \bar {\tau })$ . By continuity, $\sin (\tau w(\tau ))$ does not change sign as $\tau$ varies in $[0,\bar {\tau })$ . We are now investigating the sign of $\sin (\tau w(\tau ))$ .

Since $L(\tau )=dH(\tau )+A(\tau )$ and $w(\tau )$ is given by (3.5), we obtain

\begin{equation*} \textrm {sign}\{\sin (\tau w(\tau )\}= \textrm {sign}\{dL(\tau )-H(\tau )L(\tau )-d^2H(\tau )-d(w(\tau ))^2\}= \textrm {sign}\{\xi _1-\xi _2\}, \end{equation*}

where $\xi _1=2dA(\tau )-d(H(\tau ))^2-2A(\tau )H(\tau )$ and $ \xi _2=d\sqrt {(H(\tau ))^4+8dA(\tau )H(\tau )+4(A(\tau ))^2}.$

Since $ L(\tau )\gt -dH(\tau )$ for $\tau \in [0, \bar {\tau })$ , it follows that $A(\tau )=L(\tau )-dH(\tau )\gt -2dH(\tau )$ . Noting that $H(\tau )\lt 0$ , we get $\xi _1\gt 3d(H(\tau ))^2-4d^2H(\tau )=dH(\tau )(3H(\tau )-4d)\gt 0$ . It is apparent that $\xi _2\geq 0$ .

Next, we analyse the sign of $\xi _1^2-\xi _2^2$ . Since $H(\tau )\lt 0$ , we obtain

\begin{eqnarray*} \textrm {sign}\{\xi _1^2-\xi _2^2\}&=&\textrm {sign}\{ A(\tau ) H(\tau ) \left (H(\tau ) -2 d \right ) \left (dH(\tau )+A(\tau )+d^{2}\right )\}\\ &=&\textrm {sign}\{dH(\tau )+A(\tau )+d^{2}\}\gt 0. \end{eqnarray*}

Since $A(\tau )\gt -2dH(\tau )$ , it follows that $dH(\tau )+A(\tau )+d^{2}\gt -dH(\tau )+d^2\gt 0$ . Thus, $\xi _1^2-\xi _2^2\gt 0,$ implying $\xi _1\gt \xi _2\gt 0$ . Hence,

\begin{equation*}\textrm {sign}\{\sin (\tau w(\tau )\}=\textrm {sign}\{\xi _1-\xi _2\}= \frac {\textrm {sign}\{\xi ^2_1-\xi ^2_2\}}{\textrm {sign}\{\xi _1+\xi _2\}} =1, \ \forall \tau \in [0,\bar {\tau }). \end{equation*}

Cases (2) and (3). Since $K_0\gt a$ , if $K\geq K_0$ , then $\breve {\tau }$ is well defined. In this case, to study the sign of $\sin (\tau w(\tau ))$ , it suffices to check the sign of its derivative at $\breve {\tau }$ . We have

\begin{equation*}\frac {d}{d\tau }(\sin (\tau w(\tau )))\big |_{\tau =\breve {\tau }}=\frac {\sqrt {\frac {a r d}{K}}\, \left (a -K \right )^{2}}{2 a^{2}}\gt 0,\end{equation*}

and which implies that $\sin (\tau w(\tau ))$ changes sign from negative to positive at $\breve {\tau }$ . The additional condition $K\gt K_0$ ensures that $\breve {\tau }$ is positive, and the desired results follow.

By Theorem3.2 and Lemma 3.4, we obtain two functions

\begin{equation*}\theta =\tau w(\tau ), \ \theta =\theta _n(\tau ), \tau \in [0, \bar {\tau }].\end{equation*}

See Figure 1. These two functions are defined at $\tau =\bar {\tau }$ by continuous extension. The following lemma is obvious.

Figure 1.

Intersection of $\tau w(\tau )$ and $\theta _n(\tau )$ . Here, the graphs of $\tau w(\tau )$ and> $\theta _n(\tau )$ are sketched in a relatively simple way, but ensuring that the properties of Lemma 3.5 are all satisfied.

Lemma 3.5. Consider two functions $\theta =\tau w(\tau )$ and $\theta =\theta _n(\tau )$ , where $n\in \mathbb{N}$ .

1. (1) $\theta =\tau w(\tau )\geq 0$ on $\tau \in [0, \bar {\tau }]$ and vanishes only at $\tau =0$ and $\tau =\bar {\tau }$ .

2. (2) $\theta _0(0)\gt 0 (=0, \lt 0, resp.)$ if $K\lt K_0 (=K_0, \gt K_0, resp)$ .

3. (3) $\theta _0(\bar {\tau })=\pi$ regardless of $K$ .

4. (4) If $K\lt K_0$ , then there exists $\zeta \gt 0$ such that $0\lt \zeta \leq \theta _0(\tau )\leq \pi$ for $\tau \in [0,\bar {\tau }]$ .

Proof. Statements (1) and (2) follow from Theorem3.2 and Lemma 3.4.

For statements (3) and (4), a straightforward calculation gives $\sin (\bar {\tau } w(\bar {\tau }))=0$ and $\cos (\bar {\tau } w(\bar {\tau }))=-1$ , which implies that $\theta _0(\bar {\tau })=\pi$ . Since $\theta _0(\tau )$ is continuous on $[0, \bar {\tau }]$ , it attains the maximum and minimum values on $[0, \bar {\tau }]$ . By statement (1) of Lemma 3.4, $\sin (\tau w(\tau ))\gt 0$ for all $\tau \in [0,\bar {\tau })$ , so $\zeta =\displaystyle \min _{\tau \in [0,\bar {\tau }]}\theta _0(\tau )\gt 0$ and $\displaystyle \max _{\tau \in [0,\bar {\tau }]} \theta _0(\tau )=\pi$ .

Theorem 3.6. Assume that there exists $\tau ^\star \in (0, \bar {\tau })$ such that

\begin{equation*}\tau w(\tau )_{|\tau ^\star }=\theta _n(\tau )_{|\tau ^\star }, \ \text{and}\ \ (\tau w(\tau ))'_{|\tau ^\star }\neq (\theta _n(\tau ))'_{|\tau ^\star }\end{equation*}

for some $n\in \mathbb{N}$ . Then system (1.2) undergoes a Hopf bifurcation at $\tau =\tau ^\star$ , and a family of non-constant periodic solutions exists in a neighbourhood of $E^\ast$ for $\tau$ in a small neighbourhood of $\tau =\tau ^\star$ .

Proof. Let $\lambda (\tau )=\alpha (\tau )\pm i\beta (\tau )$ be a pair of conjugate complex eigenvalues of $P(\lambda , \tau )=0$ . The condition $\tau w(\tau )_{|\tau ^\star }=\theta _n(\tau )_{|\tau ^\star }$ implies that $\alpha (\tau ^\star )=0$ and $\beta (\tau ^\star )=w(\tau ^\star )\gt 0$ , meaning that $P(\lambda , \tau )=0$ has a pair of purely imaginary eigenvalues $\pm iw(\tau ^\star )$ at $\tau =\tau ^\star$ . Next, we check the transversality condition by defining the function

\begin{equation*}S_n(\tau )=\tau -\frac {\theta _n(\tau )}{w(\tau )}, n\in \mathbb{N}.\end{equation*}

By Theorem 2.2 of [Reference Beretta and Kuang3] (the geometric stability switch criterion), we obtain

\begin{equation*} \delta (\tau ^\star )\,:\!=\,\textrm {sign} \bigg \{\frac {d \alpha (\tau )}{d\tau }\bigg |_{\tau =\tau ^\star } \bigg \}=\textrm {sign} \bigg \{\frac {d F(w(\tau ))}{dw}\bigg |_{\tau =\tau ^\star } \bigg \}\cdot \textrm {sign}\bigg \{\frac {d S_n(\tau )}{d\tau }\bigg |_{\tau =\tau ^\star } \bigg \}, \end{equation*}

where $F(w(t))=(w(\tau ))^4+(H(\tau ))^2(w(\tau ))^2+d^2(H(\tau ))^2-(L(\tau ))^2$ . Thus,

\begin{equation*}F_w(w(\tau ))=4(w(\tau ))^3+2(H(\tau ))^2w(\tau )\gt 0, \, \forall \tau \in (0,\bar {\tau }).\end{equation*}

By the definition of $S_n(\tau )$ , we have

\begin{equation*} \frac {dS_n(\tau )}{d\tau }=1-\frac {\theta '_n(\tau )w(\tau )-w'(\tau )\theta _n(\tau )}{(w(\tau ))^2}=\frac {\tau (w(\tau ))^2-[\theta _n'(\tau )\tau w(\tau )- \tau w'(\tau )\theta _n(\tau )]}{\tau (w(\tau ))^2}.\end{equation*}

Since $(\tau w(\tau ))'=\tau w'(\tau ) +w(\tau )$ , we get $\tau w'(\tau )=(\tau w(\tau ))'-w(\tau )$ . Noting that $ \theta _n(\tau ^\star )=\tau ^\star w(\tau ^\star )$ , we obtain

\begin{eqnarray*} \frac {dS_n(\tau )}{d\tau }\bigg |_{\tau =\tau ^\star }&=&\dfrac {\tau ^\star (w(\tau ^\star ))^2-\theta _n'(\tau ^\star )\tau ^\star w(\tau ^\star )+\big[(\tau w(\tau ))'_{|\tau ^\star }-w(\tau ^\star )\big]\tau ^\star w(\tau ^\star )}{\tau ^\star (w(\tau ^\star ))^2}\\ &=&\dfrac {\tau ^\star w(\tau ^\star )\big[(\tau w(\tau ))'_{|\tau ^\star }-\theta _n'(\tau ^\star )\big]}{\tau ^\star (w(\tau ^\star ))^2}=\dfrac {(\tau w(\tau ))'_{|\tau ^\star }-\theta _n'(\tau ^\star )}{w(\tau ^\star )}. \end{eqnarray*}

Since $w(\tau ^\star )\gt 0$ , we obtain

\begin{equation*}\delta (\tau ^\star )=\textrm {sign}\{ (\tau w)'-\theta '_n(\tau )_{|\tau =\tau ^\star }\}\neq 0,\end{equation*}

which completes the proof.

Remark 3.7. $\delta (\tau ^\star )$ reveals the direction along which the complex conjugate eigenvalues $\lambda =\alpha (\tau )\pm i\beta (\tau )$ cross the imaginary axis as $\tau$ increases through $\tau =\tau ^\star$ . If $\delta (\tau ^\star )=1$ (resp. $\delta (\tau ^\star )=-1$ ), $\lambda$ crosses the imaginary axis from left to right (resp. from right to left). If $\delta (\tau ^\star )\neq 0$ , the graphs of two functions $\theta =\tau w(\tau )$ and $\theta =\theta _n(\tau )$ intersect transversely at $\tau =\tau ^\star$ . Otherwise, the two graphs are tangent.

It is technically challenging to characterize the number of intersections between $\theta =\tau w(\tau )$ and $\theta =\theta _n(\tau )$ or the roots of $S_n(\tau )=0$ , $n\in \mathbb{N}$ . There may be no roots or several roots.

Definition 3.8. Let $\chi (n)$ be the number of roots of $S_n(\tau )=0$ on $(0, \bar {\tau })$ , where $n\in \mathbb{N}$ .

The following hypothesis is necessary for the occurrence of Hopf bifurcation.

1. (H1) Suppose that $S_n(\tau )=0$ has finitely many roots on $(0, \bar {\tau })$ , i.e., $0\leq \chi (n)\lt \infty , \forall n\in \mathbb{N}$ , and these $\chi (n)$ roots are given and arranged in the following order

   \begin{equation*}\tau _n^{(1)}\lt \tau _n^{(2)}\lt \cdots \lt \tau _n^{(i)}\lt \cdots \tau _n^{(\chi (n))},\end{equation*}
   with $\delta (\tau _n^{(i)})=\pm 1$ $(i=1, 2, \cdots , \chi (n))$ .

Remark 3.9. Let $\Lambda =\{j\in \mathbb{N}|\chi (j)\neq 0\}$ . If $\chi (n)=2$ for all $n\in \Lambda$ , then we denote $\tau _n^{(1)}$ and $\tau _n^{(2)}$ as $\tau _n^{-}$ and $\tau _n^{+}$ , respectively.

Example 1. Two sets of parameters are chosen to show the distribution of roots of $S_n(\tau )=0$ .

1. (a) Choose $K= 1, r=30, m =1, c=4, d=0.1, a=1$ . See Figure 2(a). Here, $K\lt K_0=\frac {41}{39}$ , and we have $\chi (0)=\chi (1)=\chi (2)=\chi (3)=2$ , and $\chi (n)=0$ for $n\geq 4$ . The critical Hopf bifurcation values are

   \begin{align*} \left .\begin{array}{llll} \tau _0^-=0.013562, & \tau _0^+=23.67336, & \tau _1^-=3.8062, & \tau _1^+=21.49988,\\ \tau _2^-=7.81234, & \tau _2^+=19.55209, & \tau _3^-=12.7067, & \tau _3^+=17.10348. \end{array}\right . \end{align*}

2. (b) Choose $K= 20, r=10, m =1, c=4, d=0.1, a=5$ . See Figure 2(b). Here, $K\gt K_0=\frac {205}{39}$ , and we have $\chi (0)=1$ , $\chi (1)=2$ , $\chi (2)=4$ , and $\chi (n)=0$ for $n\geq 3$ . The critical Hopf bifurcation values are

   \begin{align*} \left .\begin{array}{llll} \tau _0^{(1)}=32.42808, & \tau _1^{(1)}=6.2049, & \tau _1^{(2)}=32.22374, & \\ \tau _2^{(1)}=13.96008, & \tau _2^{(2)}=24.85518, & \tau _2^{(3)}=30.21286, & \tau _2^{(4)}=32.00694. \end{array}\right . \end{align*}

Figure 2.

Left panel: Roots of $S_n(\tau )=0$ for the parameters given in Example 1 (a), where $K\lt K_0$ . Right panel: Roots of $S_n(\tau )=0$ for the parameters given in Example 1 (b), where $K\gt K_0$ .

Corollary 3.10. Assume (H1), then Hopf bifurcation occurs at a sequence of critical values $\tau =\tau _n^{(i)}$ and there exists a family of non-constant periodic solutions in a neighbourhood of $E^\ast$ for $\tau$ in a small neighbourhood of $\tau =\tau _n^{(i)}$ .

The following is an attempt to study the roots of $S_n(\tau )=0$ for certain special cases.

Proposition 3.11. Let $M=\displaystyle \max _{\tau \in [0, \bar {\tau }]}\{\tau w(\tau )\}$ . Assume that

\begin{equation*}K\lt K_0, \ \text{and}\ \ (2N+1)\pi \lt M\lt (2N+3)\pi \end{equation*}

for some $N \in \mathbb{N}$ . Then, there are three cases.

1. (1) If $n\leq N$ , $\theta _n(\tau )$ intersects $\tau w(\tau )$ at least twice.

2. (2) If $n=N+1$ , $\theta _n(\tau )$ and $\tau w(\tau )$ intersect either at least twice, or once tangentially, or they never intersect.

3. (3) If $n\geq N+2$ , $\theta _n(\tau )$ and $\tau w(\tau )$ never intersect in $(0,\bar {\tau })$ .

Proof. By Lemma 3.5, $\theta _n(\tau )$ ranges between $[\zeta +2n\pi , (2n+1)\pi ]$ . From statement (1) of Lemma 3.5, we have $\tau w(\tau )\big |_{\tau =0}=0$ and $ \tau w(\tau )\big |_{\tau =\bar {\tau }}=0$ and $M\gt 0$ . For the first case, from the intermediate value theorem, it follows that $\theta _n(\tau )$ must intersect $\tau w(\tau )$ at least twice since $\theta _n(0)\geq \zeta +2n\pi \gt 0$ , $M\gt (2N+1)\pi \gt \max _{\tau \in [0, \bar {\tau }]}\theta _n(\tau )$ , and $\theta _n(\bar {\tau })\gt 0$ . For the third case, we have $n\geq N+2$ , hence $\min _{\tau \in [0, \bar {\tau }]}\theta _n(\tau )\gt 2n\pi \geq 2(N+2)\pi \gt M$ and the result follows. The second case is trivial, as all possibilities might hold here.

3.3. Local dynamics and bifurcations involving $E_K$

Theorem 3.12.

1. (1) $E_K$ is asymptotically stable for $\tau \gt \tau _{max}$ and unstable for $0\lt \tau \lt \tau _{max}$ .

2. (2) System (1.2) undergoes a transcritical bifurcation involving $E_K$ and $E^\ast$ at $\tau =\tau _{max}.$

Proof. (1) The characteristics equation at $E_K$ has the following form

\begin{equation*} (\lambda +r)\left(\lambda +d-\frac {ce^{-d\tau }mK}{a+K}e^{-\lambda \tau }\right)=0. \end{equation*}

Therefore, $\lambda =-r$ is an eigenvalue, and the other eigenvalues are the roots of the equation

(3.6) \begin{equation} Q(\lambda )\,:\!=\,\lambda +d-\frac {ce^{-d\tau }mK}{a+K}e^{-\lambda \tau }=0. \end{equation}

As one did for $E^\ast$ , an analogous analysis can be performed to investigate the stability of $E_K$ . Here, using Theorem4.7(a) in Chapter 4 of [Reference Smith22] we conclude that $E_K$ is asymptotically stable if $\tau \gt \tau _{max}$ . If $0\lt \tau \lt \tau _{max}$ , then $Q(0)\lt 0$ . Note that $\displaystyle \lim _{\lambda \rightarrow \infty , \lambda \in \mathbb{R}}Q(\lambda )=\infty ,$ by continuity of $Q(\lambda )$ , there exists a positive real root. Hence, $E_K$ is always unstable.

(2) If $\tau =\tau _{max}$ , then $E_K=E^\ast$ and $Q(0)=0, Q'(0)=1+d\tau _{max}\neq 0$ . Hence, $\lambda =0$ is a simple eigenvalue.

Let $\hat {\lambda }(\tau )$ be a root of $Q(\lambda )=0$ such that $\hat {\lambda }(\tau _{max})=0$ . To verify that $E_K$ changes stability at $\tau = \tau _{max}$ , we compute

\begin{equation*}\frac {d}{d\tau }\textrm {Re}(\hat {\lambda })\bigg |_{\tau =\tau _{max}}=-\frac { cmdKe^{-d\tau }}{a+K+\tau cmK e^{-d\tau }}\bigg |_{\tau =\tau _{max}} \lt 0.\end{equation*}

As a consequence, $E_K$ changes stability from unstable to stable. To investigate the stability switch of $E^*$ at $\tau =\tau _{max}$ , let $\tilde {\lambda }(\tau )$ be a root of $P(\lambda , \tau )=0$ such that $\tilde {\lambda }(\tau _{max})=0$ . By the chain rule, we have

\begin{equation*} \frac {dP(\lambda ,\tau )}{d\tau } = \frac {\partial P(\lambda ,\tau )}{\partial \lambda }\frac {d\lambda }{d\tau } + \frac {\partial P(\lambda ,\tau )}{\partial \tau } =0. \end{equation*}

Substituting $\tau =\tau _{\max }$ and using $\tilde {\lambda }(\tau _{\max })=0$ , we obtain

\begin{equation*} \left .\frac {dP(\lambda , \tau )}{d\tau }\right |_{\tau =\tau _{\max }} = -dH'(\tau )+L'(\tau ) - \frac {d\tilde {\lambda }}{d\tau }\bigl (H(\tau )+\tau L(\tau )\bigr ) \Big |_{\tau =\tau _{\max }}. \end{equation*}

Solving for $\frac {d\tilde {\lambda }}{d\tau }$ and taking the real part yields

\begin{equation*}\textrm {Re}\left (\frac {d}{d\tau }\tilde {\lambda }\right )\bigg |_{\tau =\tau _{max}}=\frac {d}{d\tau }\textrm {Re}(\tilde {\lambda })\bigg |_{\tau =\tau _{max}}=\frac {L'(\tau )-dH'(\tau )}{H(\tau )+\tau L(\tau )}\bigg |_{\tau =\tau _{max}}=\frac {d^2}{\ln \left (\frac {cmK}{d\,\left (a+K\right )}\right )+1}\gt 0. \end{equation*}

Hence, the real part of $\tilde {\lambda }$ changes sign at $\tau =\tau _{max}$ from negative to positive. This implies that $E^*$ moves from the first quadrant to the fourth quadrant and changes stability from stable to unstable. Thus, $E_K$ and $E^\ast$ exchange stability at $\tau =\tau _{max}$ , where they merge in a transcritical bifurcation.

4. Global Hopf bifurcation and continuity of branches

By the local Hopf bifurcation Theorem3.6 and Corollary 3.10, the periodic solutions bifurcating from $E^\ast$ exist in a small neighbourhood of a sequence of critical values $\tau =\tau _n^{(i)}$ where $i=1, 2, \cdots , \chi (n)$ and $n\in \mathbb{N}$ . In this section, we study the evolution of periodic solutions as $\tau$ varies within $(0, \tau _{max})$ . The global Hopf bifurcation theory developed by Wu is implemented (see Theorem 3.3 of [Reference Wu25]), and connected components of non-trivial periodic solutions in Fuller’s space are considered. By estimating the period of the period solution and studying the $\tau$ -interval in which the period solution exists, we can describe the onset and termination of Hopf bifurcation branches under certain conditions.

Let $\tilde {x}(t)=x(t\tau )$ , $\tilde {y}(t)=y(t\tau )$ , then system (1.2) can be rewritten as

(4.1) \begin{equation} \dot {z}=F(z_t,\tau ,T), \ (t, \tau , T)\in \mathbb{R}_+\times (0, \tau _{max})\times \mathbb{R}_+, \end{equation}

where

\begin{equation*}z(t)=\begin{bmatrix} \tilde {x}(t)\\ \tilde {y}(t) \end{bmatrix}, \, F(z_t,\tau ,T)= \begin{bmatrix} \tau r\tilde {x}(t)\left(1-\frac {\tilde {x}(t)}{K}\right)-\tau \frac {m\tilde {x}(t)\tilde {y}(t)}{a+\tilde {x}(t)}\\[5pt] -\tau d \tilde {y}(t)+\tau e^{-d\tau }\frac {cm\tilde {x}(t-1)\tilde {y}(t-1)}{a+\tilde {x}(t-1)} \end{bmatrix}. \end{equation*}

Let $z_t(s)=z(t+s)$ for $s\in [-1,0]$ , so that $z_t\in C([-1, 0], \mathbb{R}^2_+)$ . Hence, the domain of $F(z_t,\tau , T)$ is $ C([-1, 0], \mathbb{R}^2_+) \times (0,\tau _{max}) \times \mathbb{R}_+$ . By identifying the subspace of $C([-1, 0], \mathbb{R}^2_+)$ consisting of all constant mapping with $\mathbb{R}^2_+$ , we obtain a restricted mapping

\begin{equation*}\tilde {F}(z,\tau ,T)\,:\!=\,F(z_t,\tau ,T)_{|\mathbb{R}_+^2\times (0,\tau _{max})\times \mathbb{R}_+}= \begin{bmatrix} \tau r\tilde {x}\left(1-\frac {\tilde {x}}{K}\right)-\tau \frac {m\tilde {x}\tilde {y}}{a+\tilde {x}}\\[5pt] -\tau d\tilde {y}+\tau e^{-d\tau }\frac {cm\tilde {x}\tilde {y}}{a+\tilde {x}} \end{bmatrix}. \end{equation*}

Then, $\tilde {F}$ is twice continuously differentiable, and condition (A1) of Theorem 3.3 of [Reference Wu25] is fulfilled.

Let $\mathscr{N}\,(F)=\{(z,\tau ,T)| \tilde {F}(z,\tau ,T)=0\}$ be the set of stationary solutions of system (4.1). From our analysis in Section 3, we know

\begin{equation*}\mathscr{N}\,(F)=\{(z, \tau , T)| z\in \{E_0, E_K, E^\ast \}, (\tau , T)\in (0, \tau _{max})\times \mathbb{R}_+\}.\end{equation*}

It is apparent that the function $F(z_t, \tau , T)$ is continuously differentiable with respect to its first argument $z_t$ and the characteristics matrix function at the stationary solution $(z, \tau , T)\in \mathscr{N}\,(F)$ is

\begin{equation*}\Delta _{(z,\tau ,T)}(\lambda )=\lambda Id-DF(z, \tau , T)(e^{\lambda \cdot } Id).\end{equation*}

Here, $DF(z, \tau , T)$ is the complexification of the derivative of $F(z_t, \tau , T)$ with respect to $z_t$ evaluated at $(z, \tau , T)\in \mathscr{N}\,(F)$ . Since $\Delta _{(z,\tau ,T)}$ is continuous, condition (A3) of Theorem 3.3 of [Reference Wu25] is fulfilled.

Furthermore, by Lemma 2.6 and Theorem3.12, we establish that $\lambda =0$ is not an eigenvalue of $E_0$ , $E_K$ , or $E^\ast$ for $\tau \in (0, \tau _{max})$ , meaning $\text{det}(DF(z, \tau , T))\neq 0$ for $(z, \tau , T)\in \mathscr{N}\,(F)$ . Thus, $DF(z, \tau , T)$ is isomorphic to $\mathbb{R}^2$ at each stationary solution, and condition (A2) of Theorem 3.3 of [Reference Wu25] is fulfilled.

For the global Hopf bifurcation, by Lemma 2.1 and Theorem3.6, $(E^*, \tau , T)\in \mathscr{N}\,(F)$ is the only stationary solution under consideration. Assume condition (H1) is satisfied, i.e., $S_n(\tau )=0$ has $\chi (n)$ roots on $(0, \bar {\tau })$ , where $\chi (n)\lt \infty , \forall n\in \mathbb{N}$ , and these $\chi (n)$ roots are given and arranged in the following order

\begin{equation*}\tau _n^{(1)}\lt \tau _n^{(2)}\lt \cdots \lt \tau _n^{(i)}\lt \cdots \tau _n^{(\chi (n))}.\end{equation*}

Furthermore, $\delta (\tau _n^{(i)})=\pm 1$ $(i=1, 2, \cdots , \chi (n))$ .

By the definition of a centre for a stationary solution [Reference Wu25], the points

(4.2) \begin{equation} \left(E^\ast , \tau _n^{(i)}, \frac {2\pi }{\tau _n^{(i)}w(\tau _n^{(i)})}\right)=\left(E^\ast , \tau _n^{(i)}, \frac {2\pi }{\theta _n(\tau _n^{(i)})}\right), \ n\in \mathbb{N}, \ i=1, 2,\cdots , \chi (n) \end{equation}

are centres of system (4.1). For each centre, it is the only centre in its small neighbourhood, and it only has finitely many purely imaginary characteristic values, whose imaginary part takes the form

\begin{equation*}\tau _n^{(i)}w\left(\tau _n^{(i)}=i\frac {2\pi }{2\pi /(\tau _n^{(i)}w(\tau _n^{(i)}))}=\frac {2\pi }{2\pi /(\theta _n(\tau _n^{(i)}))}.\right.\end{equation*}

Here, $T_n^{(i)}\,:\!=\,2\pi /(\tau _n^{(i)}w(\tau _n^{(i)}))=2\pi /(\theta _n(\tau _n^{(i)}))$ is the minimal period of periodic solution bifurcating at $\tau _n^{(i)}$ for the scaled model (4.1). Therefore, $(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ given in (4.2) are isolated centres. Moreover, the first crossing number of each isolated centre $\gamma _1(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ is well defined, and

\begin{equation*}\gamma _1\left(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})}\right)=-\textrm {sign}\left (\frac {d}{d\tau }Re(\lambda )\big |_{\tau =\tau _n^{(i)}}\right )=-\textrm {sign}(S'(\tau _n^{(i)}))=-\delta (\tau _n^{(i)})=\pm 1.\end{equation*}

Hence, condition (A4) of Theorem 3.3 of [Reference Wu25] is fulfilled.

Define the closed set $\Sigma (F)$ in $C([-1, 0], \mathbb{R}^2_+)\times (0, \tau _{max})\times \mathbb{R}_+$ as follows:

\begin{equation*}\Sigma (F)=Cl\{(z,\tau ,T) \in C([-1, 0], \mathbb{R}^2_+)\times (0, \tau _{max})\times \mathbb{R}_+|z\ \text{is a $T$-periodic solution of system (4.1)}\}.\end{equation*}

Let $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ be the connected component of $(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ in $\Sigma (F).$

By ensuring that conditions (A1)-(A4) of Theorem 3.3 in [Reference Wu25] hold, we can now apply the global Hopf bifurcation theorem to system (4.1).

In the following, we analyse the boundedness and connections of $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ .

Proof. By Lemma 2.1, $\xi _1$ and $\xi _2$ are positive for all $t\geq 0$ . By Lemma 2.2, we have $\displaystyle \limsup _{t\rightarrow \infty } x(t)\leq K$ . We assert that $\xi _1(t)\leq K$ for all $t\geq 0$ . If not, then there exists $t_1\gt 0$ such that $\xi _1(t_1)\gt K$ . Then,

\begin{equation*}K\lt \xi _1(t_1)=\lim _{n\rightarrow \infty }\xi _1(t_1+nT)\leq \limsup _{t\rightarrow \infty } \xi _1(t)\leq K,\end{equation*}

which is a contradiction. By a similar argument, we conclude that $\xi _2(t)\leq cK$ for all $t\geq 0$ .

Proof. At a periodic solution with period 1, we have $\tilde {x}(t-1)=\tilde {x}(t)$ and $\tilde {y}(t-1)=\tilde {y}(t)$ . To show that system (4.1) has no periodic solution of period 1, we replace $\tilde {x}(t-1)$ and $\tilde {y}(t-1)$ with $\tilde {x}(t)$ and $\tilde {y}(t)$ , respectively, reducing system (4.1) to the ODE system

(4.3) \begin{equation} \begin{cases} \frac {d\tilde {x}}{dt}=\tau r\tilde {x}\left(1-\frac {\tilde {x}}{K}\right)-\tau \frac {m\tilde {x}\tilde {y}}{a+\tilde {x}},\\[5pt] \frac {d\tilde {y}}{dt}=\tau \tilde {y}\left(-d+ce^{-d\tau }\frac {m\tilde {x}}{a+\tilde {x}}\right). \end{cases} \end{equation}

By Lemma 2.4 and substituting $c$ with $ce^{-d\tau }$ , we obtain that system (4.3) does not have a periodic solution when $K\lt \frac {a(cme^{-d\tau }+d)}{cme^{-d\tau }-d}$ . Notice that $K_0=\frac {a(cm+d)}{cm-d}\lt \frac {a(cme^{-d\tau }+d)}{cme^{-d\tau }-d}$ , and therefore system (4.3) has no periodic solutions if $K\lt K_0$ . Consequently, system (4.1) has no periodic solutions of period 1.

We have the estimate of $T_n^{(i)}$ , the minimal period of periodic solution bifurcating at $\tau _n^{(i)}$ for the scaled model (4.1) below.

Proof. By Lemma 3.5, we have $2n\pi +\zeta \leq \theta _n(\tau _n^{(i)})\leq (2n+1)\pi$ for all $\tau \in [0,\bar {\tau }]$ . Then

\begin{equation*}\frac {1}{n+1}=\frac {2\pi }{(2n+2)\pi }\lt \frac {2\pi }{(2n+1)\pi }\leq T_n^{(i)}\leq \frac {2\pi }{2n\pi +\zeta }\lt \frac {1}{n}.\end{equation*}

Proof. Consider the Lyapunov functional

\begin{equation*} V(\tilde {x}, \tilde {y})=\tilde {x}-K-K\ln {\frac {\tilde {x}}{K}}+\frac {mK}{da}\tilde {y}+\tau \frac {cm^2Ke^{-d\tau }}{da}\int _{t-1}^{t}\frac {\tilde {x}(s)\tilde {y}(s)}{a+\tilde {x}(s)}ds. \end{equation*}

By Lemma 2.1, $V(\tilde {x}, \tilde {y})$ is well-defined. Furthermore, $V(\tilde {x}, \tilde {y})\geq 0$ for all $\tilde {x}\gt 0, \tilde {y}\geq 0$ , and $V(\tilde {x}, \tilde {y})=0$ if and only if $(\tilde {x}, \tilde {y})=(K, 0)$ , we compute the total derivative along system (4.1) with respect to $t$ :

\begin{eqnarray*} \dot {V}(\tilde {x}, \tilde {y})&=&r\tau \tilde {x}(t)\left(1-\frac {\tilde {x}(t)}{K}\right)-\tau \frac {m\tilde {x}(t)\tilde {y}(t)}{a+\tilde {x}(t)}-K\tau \left (r\left(1-\frac {\tilde {x}(t)}{K}\right)-\frac {m\tilde {y}(t)}{a+\tilde {x}(t)}\right )\\ & &+\,\tau \frac {cm^2Ke^{-d\tau }}{da}\cdot \frac {\tilde {x}(t-1)\tilde {y}(t-1)}{a+\tilde {x}(t-1)}-\tau \frac {mK}{a}\tilde {y}(t)\\ & &+\,\tau \frac {cm^2Ke^{-d\tau }}{da}\cdot \frac {\tilde {x}(t)\tilde {y}(t)}{a+\tilde {x}(t)}-\tau \frac {cm^2Ke^{-d\tau }}{da}\cdot \frac {\tilde {x}(t-1)\tilde {y}(t-1)}{a+\tilde {x}(t-1)}\\ &=&\tau \left [-\frac {r}{K}(K-\tilde {x})^2 +\tilde {y}\left (\frac {-m\tilde {x}+Km}{a+\tilde {x}}-\frac {mK}{a}+\frac {cm^2Ke^{-d\tau }\tilde {x}}{da(a+\tilde {x})}\right )\right ]. \end{eqnarray*}

For $\tau \geq \tau _{max}$ , we obtain

\begin{eqnarray*} \dot {V}(\tilde {x}, \tilde {y})&\leq & \tau \left [-\frac {r}{K}(K-\tilde {x})^2 +\tilde {y}\left (\frac {-m\tilde {x}+Km}{a+\tilde {x}}-\frac {mK}{a}+\frac {cm^2Ke^{-d\tau _{max}}\tilde {x}}{da(a+\tilde {x})}\right )\right ]\\ &=&-\frac {\tau r}{K}(K-\tilde {x})^2\leq 0. \end{eqnarray*}

The last equality holds if and only if $\tilde {x}=K$ and $\tilde {y}\geq 0$ . From the first equation of system (4.1), the largest invariant set of $\{(K, \tilde {y})|\tilde {y}\geq 0\}$ is the singleton $\{(K, 0)\}$ . By Lyapunov–LaSalle’s invariance principle [Reference Kuang14], $(K,0)$ is globally asymptotically stable for $\tau \geq \tau _{max}$ .

Proof. (1). By Lemma 4.2, all periodic solutions of system (4.1) are uniformly bounded. Therefore, the projection of $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ onto $C([-1, 0], \mathbb{R}^2_+)$ is bounded.

By Lemmas 2.4 and 4.5, no periodic solution exists for $\tau =0$ or $\tau \geq \tau _{max}$ . Consequently, the connected components do not extend beyond the finite interval $(0,\tau _{max})$ . Hence, the projection of $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ onto $\tau$ -space is bounded.

Furthermore, by Lemma 4.3, there are no periodic solutions of period 1, and consequently, none with period $\frac {1}{n}, n=1,2,\cdots$ . By the continuity of the connected component, the projection of $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ onto the $T$ -space is a subset of the finite interval $(\frac {1}{n+1}, \frac {1}{n})$ . Hence, all connected components $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ are bounded in Fuller’s space, where $n=1, 2, \cdots$ .

(2). By Lemma 4.4, the minimal period of the periodic solution bifurcating at $\tau _n^{\pm }$ is $T_n^{\pm }=\frac {2\pi }{\theta (\tau _n^{\pm })}\in (\frac {1}{n+1}, \frac {1}{n})$ . By Lemma 4.3, the projection of $\mathscr{C}\,(E^*,\tau _n^{\pm }, \frac {2\pi }{\theta (\tau _n^{\pm })})$ onto the $T$ -space is a subset of the finite interval $(\frac {1}{n+1}, \frac {1}{n})$ . Moreover, $\mathscr{C}\,(E^*,\tau _n^{-}, \frac {2\pi }{\theta (\tau _n^{-})})$ and $\mathscr{C}\,(E^*,\tau _n^{+}, \frac {2\pi }{\theta (\tau _n^{+})})$ are the only two connected components whose third arguments belong to $(\frac {1}{n+1}, \frac {1}{n})$ .

By Theorem4.1, the first crossing number of $(E^*,\tau _n^{\pm }, \frac {2\pi }{\theta (\tau _n^{\pm })})$ on $\mathscr{C}\,(E^*,\tau _n^{\pm }, \frac {2\pi }{\theta (\tau _n^{\pm })})$ satisfies

\begin{equation*}\sum _{(z,\tau ,T)\in \mathscr{C}\,(E^*,\tau _n^{\pm }, \frac {2\pi }{\theta (\tau _n^{\pm })})\cap \mathscr{N}\,(F)}\gamma _1(z,\tau ,T)=0.\end{equation*}

Note that

\begin{equation*}\gamma _1\left(E^*,\tau _n^{-}, \frac {2\pi }{\theta (\tau _n^-)}\right)=-1, \ \text{and}\ \ \gamma _1\left(E^*,\tau _n^{+}, \frac {2\pi }{\theta (\tau _n^+)}\right)=1.\end{equation*}

Hence, $\mathscr{C}\,(E^*,\tau _n^{-}, \frac {2\pi }{\theta (\tau _n^{-})})$ and $\mathscr{C}\,(E^*,\tau _n^{+}, \frac {2\pi }{\theta (\tau _n^{+})})$ must coincide, initiating at $\tau =\tau _n^-$ and terminating at $\tau _n^+$ in the $\tau$ -space (or vice versa). Since $\tau _j^-\lt \tau _i^-\lt \tau _i^+\lt \tau _j^+$ for $j\lt i$ , the components $\mathscr{C}\,(E^*,\tau _n^{\pm }, \frac {2\pi }{\theta (\tau _n^{\pm })})$ are nested, where $n=1, 2, \cdots$ .

If $\chi (0)=2$ and assume the projection of $\mathscr{C}\,(E^*,\tau _0^{-}, \frac {2\pi }{\theta (\tau _0^{-})})$ and $\mathscr{C}\,(E^*,\tau _0^{+}, \frac {2\pi }{\theta (\tau _0^{+})})$ onto the $T$ -space are bounded, by Theorem4.1 and a similar argument as that of Theorem4.6, we obtain the following result.

Proof. By Lemma 4.2, all periodic solutions of system (4.1) are uniformly bounded. Therefore, the projection of $\mathscr{C}\,(E^*,\tau _0^{\pm }, \frac {2\pi }{\theta (\tau _0^{\pm })})$ onto $C([-1, 0], \mathbb{R}^2_+)$ is bounded. By Lemmas 2.4 and 4.5, no periodic solution extends beyond the interval $(0,\tau _{\max })$ , so the projection of $\mathscr{C}\,(E^*,\tau _0^{\pm }, \frac {2\pi }{\theta (\tau _0^{\pm })})$ onto $\tau$ -space is bounded. It follows from the assumption that $\mathscr{C}\,(E^*,\tau _0^{-}, \frac {2\pi }{\theta (\tau _0^{-})})$ and $\mathscr{C}\,(E^*,\tau _0^{+}, \frac {2\pi }{\theta (\tau _0^{+})})$ are the only two connected components whose third components lie in a bounded subset of $(1, \infty )$ , and the product of their first crossing numbers satisfies

\begin{equation*} \gamma _1\left (E^*,\tau _0^{-}, \frac {2\pi }{\theta (\tau _0^{-})}\right )\cdot \gamma _1\left (E^*,\tau _0^{+}, \frac {2\pi }{\theta (\tau _0^{+})}\right ) = -1. \end{equation*}

Therefore, by Theorem4.1, the two components $\mathscr{C}\big(E^*,\tau _0^{-}, \frac {2\pi }{\theta (\tau _0^{-})}\big)$ and $\mathscr{C}\big(E^*,\tau _0^{+}, \frac {2\pi }{\theta (\tau _0^{+})}\big)$ coincide and connect two bifurcation points $\tau _0^-$ and $\tau _0^+$ .

It is important to note that the nonexistence of periodic solutions in the ODE system (2.1) (i.e., system (1.2) plays a crucial role in establishing the boundedness and connectedness of the connected components, as demonstrated in the proof of Theorem4.6. However, when $K\gt K_0$ , system (1.2) admits a periodic solution at $\tau =0$ . Consequently, system (4.1) also possesses a periodic solution of period 1. As a result, we lose the ability to estimate the period of the connected components in the $T$ -space, and whether $\mathscr{C}\,(E^*,\tau _n^{-}, \frac {2\pi }{\theta (\tau _n^{-})})$ and $\mathscr{C}\,(E^*,\tau _n^{+}, \frac {2\pi }{\theta (\tau _n^{+})})$ coincide remains unclear if $\chi (n)=2$ (see Figure 3 for these two connected components. It is not clear whether they are connected due to the computational challenge).

Figure 3.

Global Hopf bifurcation and connected components for $K\lt K_0$ . The parameters and values of $\tau _n^\pm (n=0, 1, 2, 3)$ are provided in Example 1 (a).

Nevertheless, we will explore the relations between the connected components for $K\gt K_0$ through numerical analysis using DDE-BifTool. Instead of analysing $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ for system (4.1), we consider the connected components $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{w(\tau _n^{(i)})})$ for system (1.2), since a periodic solution exists at $\tau =0$ .

• • A unique connected component and the limit cycle of the ODE model (2.1).

  If $\chi (0)=1$ and $\chi (n)=0$ for $n\geq 1$ , $S_0(\tau )=0$ has a unique root $\tau =\tau _0^{(1)}$ on $[0, \bar {\tau })$ and system (1.2) has a unique connected component $\mathscr{C}\,(E^*,\tau _0^{(1)}, \frac {2\pi }{w(\tau _0^{(1)})})$ . When $\tau =0$ , $J(E^\ast )$ has a pair of conjugate complex eigenvalues with positive real part, meaning $E^\ast$ is unstable and a stable limit cycle $\Gamma _0$ (i.e, an isolated periodic solution) exists around $E^\ast$ . As $\tau$ increases from $0$ to $\tau _0^{(1)}$ , and passes by $\tau _0^{(1)}$ , Lemma 3.1 implies that this pair of complex eigenvalues vary continuously, crossing the imaginary axis from right to left at $\tau =\tau _0^{(1)}$ so that $E^\ast$ becomes stable. Meanwhile, the limit cycle $\Gamma _0$ evolves continuously. However, $\Gamma _0$ does not necessarily persist over the entire interval $[0, \tau _0^{(1)})$ , as it may vanish due to a bifurcation of periodic solutions at some $\tau \lt \tau _0^{(1)}$ . If no such bifurcation occurs for $\tau \in (0, \tau _0^{(1)})$ , then $\Gamma _0$ must disappear through a local Hopf bifurcation at $\tau =\tau _0^{(1)}$ , i.e., $(\Gamma _0, 0, \frac {2\pi }{\bar {w}})\in \mathscr{C}\,(E^*,\tau _0^{(1)}, \frac {2\pi }{w(\tau _0^{(1)})})$ , where $\bar {w}$ is the frequency of the limit cycle $\Gamma _0$ .Thus, the limit cycle of the ODE model (2.1) belongs to a connected component of the Hopf bifurcation of the DDE model (1.2) in Fuller’s space. As illustrated in Figure 4, $\Gamma _0$ varies continuously on $[0, \tau _0^{(1)}]$ and terminates $\tau =\tau _0^{(1)}$ through a Hopf bifurcation.

Figure 4.

Global Hopf bifurcation and a unique connected component $\mathscr{C}\,(E^*,\tau _0^{(1)}, 2\pi /w(\tau _0^{(1)}))$ for $K\gt K_0$ . The left panel shows the amplitude and the right panel displays the period of the periodic solution for $\tau \in [0, \tau _0^{(1)}]$ . The parameter are $K=7, r=1, m=1, c=1, d=0.1, a=5$ , and $K\gt K_0=6.11$ .

• • Multiple connected components.

• If $\chi (n)\gt 2$ for some $n\in \mathbb{N}$ , then $S_n(\tau )=0$ has $\chi (n)$ critical values $\tau =\tau _n^{{(i)}}$ , leading to $\chi (n)$ connected components $\mathscr{C}\,(E^*,\tau _n^{(i)}, \frac {2\pi }{\theta (\tau _n^{(i)})})$ , where $i=1, 2, \cdots , \chi (n)$ . Studying the relationship between these connected components is technically challenging, regardless of whether $K\lt K_0$ or $K\gt K_0$ . To illustrate this complexity, we consider the parameter values from Example1(b), where $\chi (0)=1, \chi (1)=2$ , $\chi (2)=4$ , and $\chi (n)=0$ for $n\geq 3$ . Based on Figure 5 we have the following observations:

  Figure 5.

  Global Hopf bifurcation and multiple connected components for $K\gt K_0$ . The right panel is a magnified view of the left panel near $\tau =\tau _0^{(1)}$ . The parameter values and $\tau _n^{(i)} (n=0, 1, 2)$ are specified in Example 1 (b).

  

1. – For $n=0$ : $\mathscr{C}\,(E^*,\tau _0^{(1)}, \frac {2\pi }{w(\tau _0^{(1)})})$ initials at $\tau _0^{(1)}$ , and experiences a sequence of saddle-node bifurcations of limit cycles occurs in the interval $[32.33, 32.4]$ . The curve does not extend any further due to computational challenges.

2. – For $n=1$ : $\mathscr{C}\,(E^*,\tau _1^{(1)}, \frac {2\pi }{w(\tau _1^{(1)})})$ and $\mathscr{C}\,(E^*,\tau _1^{(2)}, \frac {2\pi }{w(\tau _1^{(2)})})$ coincide, i.e., the periodic solution initials at $\tau _1^{(1)}$ and terminates at $\tau _1^{(2)}$ .

3. – For $n=2$ : $\mathscr{C}\,(E^*,\tau _2^{(1)}, \frac {2\pi }{w(\tau _2^{(1)})})$ and $\mathscr{C}\,(E^*,\tau _2^{(2)}, \frac {2\pi }{w(\tau _2^{(2)})})$ coincide, $\mathscr{C}\,(E^*,\tau _2^{(3)}, \frac {2\pi }{w(\tau _2^{(3)})})$ and $\mathscr{C}\,(E^*,\tau _2^{(4)}, \frac {2\pi }{w(\tau _2^{(4)})})$ coincide. That is to say, we have the formation of a periodic solution at $\tau _{2}^{(1)}$ terminating at $\tau _{2}^{(2)}$ , and a formation of another periodic solution at $\tau _{2}^{(3)}$ terminating at $\tau _{2}^{(4)}$ .

5. Bifurcation diagram

In this section, we examine the bifurcation diagram of system (1.2) in the $(\tau , K)$ -plane. By statement (2) of Lemma3.12, a transcritical bifurcation involving $E_K$ and $E^\ast$ occurs when $\tau =\tau _{max}$ , i.e., $(\tau , K)$ lies on the curve $\mathscr{L}_c$ , defined as

\begin{equation*}\mathscr{L}_c\,:\!=\, \left\{(\tau , K)\in \overline {\mathbb{R}}_+\times \mathbb{R}_+| K=\frac {ad}{cme^{-d\tau }-d}\right\} .\end{equation*}

By Corollary 3.10, there exists a sequence of bifurcation curves through which Hopf bifurcation occurs at $E^\ast$ . Let $\ell _n(\tau , K)\,:\!=\,\tau w(\tau , K)-\theta _n(\tau . K)$ . The Hopf bifurcation curve is given by

\begin{equation*}\mathscr{L}_n= \{(\tau , K)\in \overline {\mathbb{R}}_+\times \mathbb{R}_+|\ell _n(\tau , K)=0, 0\leq \tau \lt \bar {\tau }\}, \ n=0, 1, 2, \cdots .\end{equation*}

Proof. Statement (1) follows directly from Lemma 2.5.

(2) Since $\ell _0(0, K_0)=0-\arccos (1)=0$ , the point $(0, K_0)$ serves as an endpoint of $\mathscr{L}_0$ . By Lemma 3.4, for any $K\geq K_0$ , $\tau w(\tau )$ intersects $\theta _0(\tau )$ at least once. Hence, the graph of $\ell _0(\tau , K)=0$ exists for all $K\geq K_0$ .

(3) By direct calculation, we obtain

\begin{equation*}\ell _0(0, K_0)=0, \text{and} \ \frac {\partial \ell _0(\tau , K)}{\partial K}_{|(0, K_0)}=-\frac {(cm-d)\sqrt {rd}}{mca\sqrt {c^2m^2-d^2}}\neq 0.\end{equation*}

By the implicit function theorem, there exists a function $K=K(\tau )$ for $\tau \in [0, \varepsilon )$ , where $\varepsilon \gt 0$ is small such that $K_0=K(0)$ and $\ell _0(K(\tau ), \tau )=0$ . Furthermore, a straightforward calculation yields

\begin{equation*}K'(\tau )_{|\tau =0}=-\frac {mca(c^2m^2-2cmd-d^2)}{(cm-d)^2}.\end{equation*}

If $K'(\tau )_{|\tau =0}=0$ (i.e., $cm=(1+\sqrt {2})d$ ), then by implicit differentiation, $K''(\tau )_{|\tau =0}=3\sqrt {2}ac^2m^2$ . This establishes the desired result.

Figure 6.

Bifurcation diagram in the $(\tau , K)$ -plane. The curve $\mathscr{L}_c$ represents the transcritical bifurcation, while $\mathscr{L}_n (n=0, 1, 2, 3)$ are curves of Hopf bifurcation. Dash lines highlight critical parameter values that are not bifurcation curves.

In what follows, we will sketch the bifurcation diagram of the system (1.2) in the first quadrant of the $(\tau , K)$ plane. There are a curve of transcritical bifurcation $\mathscr{L}_c$ and a curve of Hopf bifurcation $\mathscr{L}_0$ . By statement (3) of Lemma 5.1, in a neighbourhood of $(0, K_0)$ , the curve $\mathscr{L}_0$ exhibits different monotonicity behaviours depending on whether $cm\leq (1+\sqrt {2})d$ or $cm\gt (1+\sqrt {2})d$ . The simplest sketch of $\mathscr{L}_0$ is presented in Figure 6, ensuring minimal changes in monotonicity for $\tau \in [0, \check {\tau })$ .

The two curves $\mathscr{L}_c$ and $\mathscr{L}_0$ divide the first quadrant of $(\tau , K)$ -plane into three regions $V_a$ , $V_b$ , and $V_c$ , defined as follows:

\begin{eqnarray*} &&V_a=\{(\tau , K)\in \overline {\mathbb{R}}_+\times \mathbb{R}_+| \ell _0(\tau , K)\gt , 0\leq \tau \lt \bar {\tau }\},\\ &&V_b=\{(\tau , K)\in \overline {\mathbb{R}}_+\times \mathbb{R}_+|\ell _0(\tau , K)\lt 0, 0\leq \tau \lt \bar {\tau }\}, V_c=\{(\tau , K)\in \overline {\mathbb{R}}_+\times \mathbb{R}_+|\tau \gt \tau _{max}\}. \end{eqnarray*}

By Theorem4.5, $E_K$ is globally asymptotically stable for $(K, \tau )\in V_c$ , and system (1.2) undergoes a transcritical bifurcation for $(K, \tau )\in \mathscr{L}_c$ . $E^\ast$ is locally asymptotically stable for $(K, \tau )\in V_b$ , then system (1.2) experiences a Hopf bifurcation for $(K, \tau )\in \mathscr{L}_0$ . $E^\ast$ is unstable for $(K, \tau )\in V_a$ , and a periodic solution exists for $(K, \tau )$ in a tubular neighbourhood of $\mathscr{L}_0$ .

The existence of additional Hopf bifurcation curves $\mathscr{L}_n (n=1, 2, {\cdots}\,)$ depends on the choice of parameters. If such curves exist, they are positioned above $\mathscr{L}_0$ and do not intersect any other Hopf bifurcation curve. Numerical studies suggest two possible topological structures for $\mathscr{L}_n$ : $\mathscr{L}_n\simeq \mathbb{R}$ , or $\mathscr{L}_n\simeq \mathbb{S}^1$ , where $\simeq$ means “is topologically isomorphic”. Figure 6 illustrates an example where $\mathscr{L}_1\simeq \mathbb{R}$ , $\mathscr{L}_2\simeq \mathbb{R}$ , $\mathscr{L}_3\simeq \mathbb{S}^1$ .

The region $V_a$ is divided into several subregions by the Hopf bifurcation curves $\mathscr{L}_n (n=1, 2, {\cdots}\,)$ . Whenever $(K, \tau )$ crosses $\mathscr{L}_n$ , the Hopf bifurcation occurs, and a periodic solution emerges in a tubular neighbourhood of $\mathscr{L}_n$ . For $K\lt K_0$ , if $\mathscr{L}_n (n=2, 3, {\cdots}\,)$ transversally intersects the horizontal line exactly twice, by Theorem4.6, the connected components $\mathscr{C}\,(E^*,\tau _n^{-}, \frac {2\pi }{w(\tau _n^{-})})$ and $\mathscr{C}\,(E^*,\tau _n^{+}, \frac {2\pi }{w(\tau _n^{+})})$ coincide. That is, a periodic solution emerging at $(\tau _n^-, K)$ will terminate at $(\tau _n^+, K)$ . Determining the number of periodic solutions within the subregions between successive Hopf bifurcation curves is challenging. This is due to the potential presence of other bifurcations affecting periodic solutions. Even in the absence of such bifurcations, tracing the periodic solutions as $(\tau , K)$ crosses $\mathscr{L}_n (n=0, 1, 2 {\cdots}\,)$ remains difficult, as there is no definitive criterion for predicting the onset and termination of connected components in global Hopf bifurcation when $K\gt K_0$ or $\chi (n)\gt 2$ .

Remark 5.3. If $cm\gt (1+\sqrt {2})d$ , define

\begin{equation*}\tau ^\ast =\arg \min _{\tau \in (0, \bar {\tau })}\{\ell _0(\tau , K_0)=0\}.\end{equation*}

Then, for any point $(\tau , K)\in \mathscr{L}_0$ with $0\lt \tau \lt \tau ^\ast$ , we have $K\lt K_0$ (see Figure 6 (b)).

Biological significance of $\tau ^\ast$ . Predator–prey coexistence via oscillatory patterns can occur at lower environmental carrying capacities when delay ( $\tau \gt 0$ ) is introduced, compared to scenarios without delay ( $\tau = 0$ ) under the condition that $(1+\sqrt {2})d\lt cm$ , and $0\lt \tau \lt \tau ^\ast$ . The significance of $\tau ^\ast$ will be further illustrated in the next section.