*3.3. Stability and Existence of Hopf Bifurcation for E*2,3

Next, we analyze the stability of system (3) for *Ei* = (*x*(*i*), *y*(*i*)) (*i* = 2, 3), and the characteristic equation of system (3), evaluated at *Ei*, is given by

$$
\lambda^2 + (-a\_1 + c\_1 y^{(i)} + c\_2 y^{(i)}) \lambda + c\_2 y^{(i)} (-a\_1 + c\_1 y^{(i)}) - c\_1 b\_2 x^{(i)} y^{(i)} + (k \lambda + k c\_2 y^{(i)}) e^{-\lambda \tau} = 0. \tag{15}
$$

When *τ* = 0, Equation (15) becomes

$$
\lambda^2 + T\_2^{(i)}\lambda + D\_2^{(i)} = 0,\tag{16}
$$

where

$$T\_2^{(i)} = (k - a\_1 + c\_1 y^{(i)} + c\_2 y^{(i)}), \\ D\_2^{(i)} = c\_2 y^{(i)} (k - a\_1 + c\_1 y^{(i)}) - c\_1 b\_2 x^{(i)} y^{(i)}.$$

where (*x*(*i*), *y*(*i*))=(*x*(2), *y*(2)), the parameters of system (3) meet the assumptions (*H*2), we can prove that

$$T\_2^{(2)} > 0, \ D\_2^{(2)} = -y^{(2)} \sqrt{(c\_1 a\_2 + (k - a\_1)c\_2)^2 - 4c\_1 c\_2 b\_2 k m} < 0,\tag{17}$$

thus, *E*<sup>2</sup> is unstable when *τ* = 0.

When (*x*(*i*), *y*(*i*))=(*x*(3), *y*(3)) and the parameters of system (3) satisfy the assumptions (*H*3), we can prove that

$$T\_2^{(3)} > 0, \; D\_2^{(3)} = y^{(3)} \sqrt{(c\_1 a\_2 + (k - a\_1) c\_2)^2 - 4c\_1 c\_2 b\_2 k m} > 0,\tag{18}$$

thus, *E*<sup>3</sup> is locally asymptotically stable when *τ* = 0. 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 (15). Substituting it into Equation (15) and separating the real and imaginary parts, we obtain:

$$\begin{cases} \omega^2 + c\_2 y^{(i)} (a\_1 - c\_1 y^{(i)}) + c\_1 b\_2 x^{(i)} y^{(i)} = k \omega \sin(\omega \tau) + k c\_2 y^{(i)} \cos(\omega \tau), \\ \omega (a\_1 - c\_1 y^{(i)} - c\_2 y^{(i)}) = k \omega \cos(\omega \tau) - k c\_2 y^{(i)} \sin(\omega \tau), \end{cases} \tag{19}$$

Equation (19) derives the following results,

$$\begin{aligned} Q\_2^{(i)} &\triangleq \sin(\omega \tau) = \frac{k\omega(\omega^2 + c\_2 y^{(i)}(a\_1 - c\_1 y^{(i)}) + c\_1 b\_2 \mathbf{x}^{(i)} y^{(i)}) - k c\_2 y^{(i)} \omega(a\_1 - c\_1 y^{(i)} - c\_2 y^{(i)})}{k^2 \omega^2 + k^2 c\_2^2 y^{(i)2}}, \\ R\_2^{(i)} &\triangleq \cos(\omega \tau) = \\ &\omega \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{aligned} \tag{20}$$

*<sup>k</sup>ω*2(*a*<sup>1</sup> <sup>−</sup> *<sup>c</sup>*1*y*(*i*)) + *<sup>c</sup>*1*b*2*x*(*i*)*y*(*i*)) + *kc*2*y*(*i*)(*ω*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*2*y*(*i*)(*a*<sup>1</sup> <sup>−</sup> *<sup>c</sup>*1*y*(*i*)) + *<sup>c</sup>*1*b*2*x*(*i*)*y*(*i*)) *k*2*ω*<sup>2</sup> + *k*2*c*<sup>2</sup> 2*y*(*i*)<sup>2</sup> .

Adding the square of the two equations, we obtain

*ω*<sup>4</sup> + *T*(*i*) <sup>3</sup> *<sup>ω</sup>*<sup>2</sup> <sup>+</sup> *<sup>D</sup>*(*i*) <sup>3</sup> = 0, (21)

where

$$\begin{aligned} T\_3^{(i)} &= 2c\_2 y^{(i)} (a\_1 - c\_1 y^{(i)}) + 2c\_1 b\_2 x^{(i)} y^{(i)} + (a\_1 - c\_1 y^{(i)} - c\_2 y^{(i)})^2 - k^2, \\ D\_3^{(i)} &= 2c\_1 c\_2 b\_2 x^{(i)} y^{(i)2} (a\_1 - c\_1 y^{(i)}) + (c\_2 y^{(i)} (a\_1 - c\_1 y^{(i)}))^2 + (c\_1 b\_2 x^{(i)} y^{(i)})^2 - (k c\_2 y^{(i)})^2. \end{aligned}$$

For convenience, we let *ω*<sup>2</sup> = *z*, Equation (21) becomes

$$h(z) = z^2 + T\_3^{(i)} z + D\_3^{(i)} = 0. \tag{22}$$

When the parameters of system (3) meet the following assumptions—*D*(*i*) <sup>3</sup> < 0, Equation (22) has one positive root *<sup>z</sup>*2; If *<sup>T</sup>*(*i*) <sup>3</sup> <sup>&</sup>gt; 0, *<sup>D</sup>*(*i*) <sup>3</sup> > 0 hold, Equation (22) has no positive root; If *T*(*i*) <sup>3</sup> <sup>&</sup>lt; 0, *<sup>D</sup>*(*i*) <sup>3</sup> > 0 hold, Equation (22) has two positive roots *z*3, *z*4. We hypothesize that Equation (22) has positive roots *zn* (*<sup>n</sup>* <sup>=</sup> 2, 3, 4), then *<sup>ω</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>*zn* (*<sup>n</sup>* <sup>=</sup> 2, 3, 4). From (20), we can solve the critical value of time delay,

$$\tau\_n^{(j)} = \begin{cases} \frac{\arcsin(Q\_2^{(j)}) + 2j\pi}{\omega\_n}, R\_2 > 0, \\\\ \frac{-\arcsin(Q\_2^{(j)}) + 2(j+1)\pi}{\omega\_n}, R\_2 < 0, \ n = 2, 3, 4, \ j = 0, 1, 2, \cdots \end{cases} \tag{23}$$

Let *<sup>λ</sup>* = *<sup>λ</sup>*(*τ*) be the root of (15), satisfying *<sup>λ</sup>*(*τ*(*j*) *<sup>n</sup>* ) = *iω<sup>n</sup>* (*n* = 2, 3, 4). Differentiating both sides of Equation (15) with respective to *τ* gives that:

$$\left. \mathrm{Re} (\frac{d\lambda}{d\tau})^{-1} \right|\_{\tau = \tau\_n^{(j)}} = \frac{2\omega\_n^2 + 2c\_2 y^{(i)} (a\_1 - c\_1 y^{(i)}) + 2c\_1 b\_2 x^{(i)} y^{(i)} + (a\_1 - c\_1 y^{(i)} - c\_2 y^{(i)})^2 - k^2}{k^2 \omega\_n^2 + k^2 c\_2^2 y^{(i)2}} \tag{24}$$
 
$$= h'(z\_n) \neq 0.$$

**Theorem 3.** *Considering the stability of E*<sup>2</sup> *and E*<sup>3</sup> *for system* (3)*, we come to the following conclusions:*

*(1) When* (*H*2) *holds, the equilibrium E*<sup>2</sup> *is unstable for any τ* 0*;*

*(2) When* (*H*3) *holds, we discuss the stability of equilibrium E*<sup>3</sup> *of system* (3) *below.*

*(a) When <sup>T</sup>*(3) <sup>3</sup> <sup>&</sup>gt; 0, *<sup>D</sup>*(3) <sup>3</sup> > 0 *hold, Equation* (22) *has no positive root, the equilibrium E*<sup>3</sup> *is locally asymptotically stable for any τ* 0*;*

*(b) If <sup>D</sup>*(3) <sup>3</sup> <sup>&</sup>lt; <sup>0</sup> *holds, Equation* (22) *has one positive roots <sup>z</sup>*2*, then when <sup>τ</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*(0) <sup>2</sup> )*, the equilibrium E*<sup>3</sup> *is locally asymptotically stable, and unstable when <sup>τ</sup>* <sup>&</sup>gt; *<sup>τ</sup>*(0) <sup>2</sup> *;*

*(c) If <sup>T</sup>*(3) <sup>3</sup> <sup>&</sup>lt; 0, *<sup>D</sup>*(3) <sup>3</sup> > 0 *hold, system* (3) *undergoes a Hopf bifurcation at the trivial equilibrium <sup>E</sup>*<sup>3</sup> *when <sup>τ</sup>* <sup>=</sup> *<sup>τ</sup>*(*j*) *<sup>n</sup>* (*<sup>n</sup>* <sup>=</sup> 3, 4; *<sup>j</sup>* <sup>=</sup> 0, 1, 2, ···)*. Then,* <sup>∃</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>N</sup> makes* <sup>0</sup> <sup>&</sup>lt; *<sup>τ</sup>*(0) <sup>4</sup> < *τ*(0) <sup>3</sup> <sup>&</sup>lt; *<sup>τ</sup>*(1) <sup>4</sup> <sup>&</sup>lt; *<sup>τ</sup>*(1) <sup>4</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>τ</sup>*(*m*−1) <sup>3</sup> <sup>&</sup>lt; *<sup>τ</sup>*(*m*) <sup>4</sup> <sup>&</sup>lt; *<sup>τ</sup>*(*m*+1) <sup>4</sup> *. When <sup>τ</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*(0) <sup>4</sup> ) <sup>∪</sup> '*<sup>m</sup> l*=1 (*τ*(*l*−1) <sup>3</sup> , *<sup>τ</sup>*(*l*) <sup>4</sup> )*, the equilibrium <sup>E</sup>*<sup>3</sup> *of the system* (3) *is locally asymptotically stable, and when <sup>τ</sup>* <sup>∈</sup> *<sup>m</sup>*'−<sup>1</sup> *l*=0 (*τ*(*l*) <sup>4</sup> , *<sup>τ</sup>*(*l*) <sup>3</sup> ) ∪

(*τ*(*m*) <sup>4</sup> , +∞)*, the equilibrium E*<sup>3</sup> *is unstable.*
