*3.1. Existence Of Equilibrium Point*

When the parameters of system (3) meet the following assumptions

$$(H\_1) \; k - a\_1 > 0,\tag{4}$$

system (3) has a boundary equilibrium *E*<sup>1</sup> = (*x*(1), 0), where

$$\mathbf{x}^{(1)} = \frac{km}{k - a\_1}$$

.

When the parameters of system (3) meet the following assumptions

$$\gamma\_1(H\_2) \begin{cases} \sqrt{(c\_1a\_2 + (k - a\_1)c\_2)^2 - 4c\_1c\_2b\_2km} > 0, \\ c\_1a\_2 + (k - a\_1)c\_2 + \sqrt{(c\_1a\_2 + (k - a\_1)c\_2)^2 - 4c\_1c\_2b\_2km} > 0, \\ c\_1a\_2 - (k - a\_1)c\_2 - \sqrt{(c\_1a\_2 + (k - a\_1)c\_2)^2 - 4c\_1c\_2b\_2km} > 0, \end{cases} \tag{5}$$

system (3) has a positive equilibrium *E*<sup>2</sup> = (*x*(2), *y*(2)), where

$$x^{(2)} = \frac{c\_1 a\_2 + (k - a\_1)c\_2 + \sqrt{(c\_1 a\_2 + (k - a\_1)c\_2)^2 - 4c\_1 c\_2 b\_2 km}}{2c\_1 b\_2},$$

$$y^{(2)} = \frac{c\_1 a\_2 - (k - a\_1)c\_2 - \sqrt{(c\_1 a\_2 + (k - a\_1)c\_2)^2 - 4c\_1 c\_2 b\_2 km}}{2c\_1 c\_2}.$$

When the parameters of system (3) meet the following assumptions

$$\begin{aligned} \chi(H\_3) \begin{cases} \sqrt{(c\_1a\_2 + (k-a\_1)c\_2)^2 - 4c\_1c\_2b\_2km} > 0, \\ c\_1a\_2 + (k-a\_1)c\_2 - \sqrt{(c\_1a\_2 + (k-a\_1)c\_2)^2 - 4c\_1c\_2b\_2km} > 0, \\ c\_1a\_2 - (k-a\_1)c\_2 + \sqrt{(c\_1a\_2 + (k-a\_1)c\_2)^2 - 4c\_1c\_2b\_2km} > 0, \end{cases} \end{aligned} \tag{6}$$

system (3) has a positive equilibrium *E*<sup>3</sup> = (*x*(3), *y*(3)), where

$$x^{(3)} = \frac{c\_1 a\_2 + (k - a\_1)c\_2 - \sqrt{(c\_1 a\_2 + (k - a\_1)c\_2)^2 - 4c\_1 c\_2 b\_2 km}}{2c\_1 b\_2},$$

$$y^{(3)} = \frac{c\_1 a\_2 - (k - a\_1)c\_2 + \sqrt{(c\_1 a\_2 + (k - a\_1)c\_2)^2 - 4c\_1 c\_2 b\_2 km}}{2c\_1 c\_2}.$$

*3.2. Stability and Existence of Hopf Bifurcation for E*<sup>1</sup> = (*x*(1), 0)

When (*H*1) holds, system (3) has equilibrium *E*1, similar to the calculation method in [19–22], we calculate the stability of the equilibrium *E*<sup>1</sup> and the existence of Hopf bifurcation. The characteristic equation of system (3), evaluated at *E*1, is given as follows:

$$
\lambda (\lambda - a\_1 + e^{-\lambda \tau} k) = 0. \tag{7}
$$

Note that *λ* = 0 is always the root of the Equation (7). Next, we only need to consider the following equation,

$$
\lambda - a\_1 + e^{-\lambda \tau} k = 0. \tag{8}
$$

When *τ* = 0, Equation (8) becomes

$$
\lambda + k - a\_1 = 0,\tag{9}
$$

it leads to *λ*<sup>1</sup> = −(*k* − *a*1) < 0, due to (*H*1) holds.

When *τ* > 0, we try to discuss the existence of Hopf bifurcation. We assume that *λ* = *iω*(*ω* > 0) is a pure imaginary root of Equation (8). Substituting it into Equation (8) and separating the real and imaginary parts, we obtain:

$$\begin{cases} k\sin(\omega \tau) = \omega\_\prime\\ k\cos(\omega \tau) = a\_1. \end{cases} \tag{10}$$

Equation (10) derives the following results,

$$Q\_1 \stackrel{\triangle}{=} \sin(\omega\_1 \tau\_1) = \frac{\omega\_1}{k}, \ R\_1 \stackrel{\triangle}{=} \cos(\omega\_1 \tau\_1) = \frac{a\_1}{k}.\tag{11}$$

Adding the square of the two equations, we obtain

$$
\omega\_1^2 - a\_1^2 + k^2 = 0,\tag{12}
$$

then it gives *ω*<sup>1</sup> = - *<sup>k</sup>*<sup>2</sup> − *<sup>a</sup>*<sup>2</sup> <sup>1</sup>, which makes sense due to the assumptions (*H*1). We obtain the expression of *<sup>τ</sup>*(*j*) <sup>1</sup> as follows:

$$\tau\_1^{(j)} = \begin{cases} \frac{\arccos(R\_1) + 2j\pi}{\omega\_1}, Q\_1 > 0, \\\frac{-\arccos(R\_1) + 2(j+1)\pi}{\omega\_1}, Q\_1 < 0, \, j = 0, \, 1, \, 2, \, 3, \, 4, \, \cdots \, \, . \end{cases} \tag{13}$$

Let *λ* = *λ*(*τ*) be the root of Equation (8), satisfying *λ*(*τ*(*j*) <sup>1</sup> ) = i*ω*1. Differentiating both sides of (8) with respective to *τ* gives that

$$\left. \mathrm{Re} (\frac{d\lambda}{d\tau})^{-1} \right|\_{\tau = \tau\_1^{(j)}} = \frac{k^2 - a\_1^2}{k\omega\_1^2} > 0. \tag{14}$$

**Theorem 2.** *When the parameters of system* (3) *meet the assumptions* (*H*1)*, for any of τ* 0*, characteristic Equation* (7) *has a zero root. When <sup>τ</sup>* = *<sup>τ</sup>*(*j*) <sup>1</sup> *, characteristic Equation* (7) *has a zero root and a pair of pure imaginary roots, and when <sup>τ</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*(0) <sup>1</sup> )*, Equation* (7) *has a zero root, and other roots have negative real parts, when τ* > *τ*(0) <sup>1</sup> *, the equilibrium E*<sup>1</sup> *of system* (3) *is unstable.*
