L. Zhao et al.

The characteristic equation of the system (4) is
系统（4）的特征方程为

Equation (5) is equivalent to
等式（5）等价于

The following two cases are discussed according to Eq. (6):
根据式（6）讨论以下两种情况：

(i) . （一） .

(ii) . （二） .

For (i), it is only necessary to consider , suppose that is one of its roots. Separating the real and imaginary parts, then it follows that
对于（i），只需要考虑 ，假设这是 它的根之一。将真实部分和虚部分分开，然后得出

Combined with the expression of
结合表达

Accordingly 因此

and

.

It is found from Eq. (8) that
从式（8）中可以看出，

where 哪里

If the values of in system (3) are given, substitute Eq. (10) of in Eq. (8) and the value of can be calculated from by Maple software. Finally, the value of can be obtained by Eq. (11).
如果给出了系统（3）中的值 ，则将方程（10） 代入方程（8），则可以通过Maple软件计算 出的值 。最后，可以通过方程（11）得到的 值。

Define 定义

For (ii), take into account 1) . Let be one of its roots, thus
对于（ii），考虑 1）。 让我们 成为它的根源之一，因此

In view of Eq. (12)
根据式（12）

Hence 因此

and

From Eq. (13) 由式 （13）

where 哪里

If the values of in system (3) are given, use the same method as (i), the value of can be obtained by Eq. (13).
如果给出了系统（3）中的值 ，则使用与（i）相同的方法，可以通过方程（13）获得的 值。

Let

Next, define the first bifurcation point as
接下来，将第一个分岔点定义为

The following assumption is given hereinafter:
以下给出以下假设：

(H1) or .
（H1） 或 .

Lemma 1. Suppose that is the root of Eq. (6) near meeting , , then the following transversality condition holds
引理 1.假设这是 方程（6）的根 ， 则 以下横向条件成立

Proof. For (i), differentiating both sides of with respect to , and bring in the expression of , it can be deduced that
证明。对于（i），将 的两侧 相 差，并引入 的 表达式，可以推导出

where 哪里

From Eq. (23), after direct calculation, the following result can be obtained
从式（23）直接计算后，可得到以下结果

where 哪里

For (ii), differentiating both sides of with respect to , instead of (i), becomes in the transversal condition, and the rest of the proof is the same as (i).
对于（ii），将两 边相对于 ，而不是（i）进行微分， 则 处于横向条件，其余证明与（i）相同。

Apparently, the hypothesis hints that transversality condition is satisfied.
显然，该假设暗示满足横向条件。

Based on the above discussion, the following theorem can be obtained.
基于上述讨论，可以得到以下定理。

Theorem 1. Assume that (H1) holds, then
定理 1.假设 （H1） 成立，则

(1) The trivial equilibrium of system (3) is asymptotically stable when and becomes unstable when .
（1）系统（3）的平凡平衡在时 渐近稳定，在 时变为不稳定。

(2) System (3) undergoes a Hopf bifurcation at .
（2） 系统 （3） 在 处发生 Hopf 分岔。

Remark 4.1. The relationship between the roots of Eqs. (11) and (16) with respect to can be solved as follows. If the values of parameters are given, is expressed by Eq. (10), and then its expression is substituted into Eq. (8)). Using Maple software, the value of can be solved by
备注 4.1.方程根之间的关系。（11）和（16）可以 按如下方式解决。如果给出参数 的值， 则用方程（10）表示，然后将其表达式代入方程（8））。使用Maple软件，可以通过 以下方式解决

Note that at this time, the values of , are all known. Finally, use Maple software to calculate the value of . In addition, it can also be substituted back into Eq. (10) to verify whether the calculated value is correct. The calculation of Eq. (16) is similar.
请注意，此时 ，的 值都是已知的。最后，使用Maple软件计算出的 值。此外，还可以将其代入方程（10）中，以验证计算 值是否正确。方程（16）的计算与此类似。

For (i), use the method in Sec. 4.1, from Eq. (12) we have,
对于（i），使用方程（12）中4.1中的方法，

From Eq. (8) that
从式（8）可以看出

where 哪里

.

If the values of in system (3) are given, the value of can be obtained by Eq. (20), which is applied using the same method as in the previous section.
如果给出了系统（3）中的 值，则可以通过方程（20）获得其值，该方程使用与上一节相同的方法应用。

Define 定义

For (ii), we can get
对于（ii），我们可以得到

As can be seen from Eq. (13)
从式（13）可以看出

where 哪里

If the values of in system (3) are given, the value of can be obtained by Eq. (22).
如果给出系统（3）中的 值，则可以通过方程（22）得到的 值。

Let

Next, define the first bifurcation point as
接下来，将第一个分岔点定义为

The following assumption is given hereinafter:
以下给出以下假设：

Lemma 2. Let be the root of Eq. (6) near meeting , , then the following transversality condition holds
引理 2.设 为方程（6）近 相交 的根， 则以下横向条件成立

Proof. For (i), differentiating both sides of with respect to , it can be deduced that
证明。对于（i），区分两边 相对于 ，可以推导出

where 哪里

From Eq. (23), the following result can be obtained
从式（23）可以得到以下结果

where 哪里

For (ii), differentiating both sides of with respect to , at this time, becomes 1) in the transversal condition, the remaining proof is similar to (i).
对于（ii），此时将 的两边 相差 化 为 1） 在横向条件下，剩余的证明类似于 （i）。

Apparently, the hypothesis hints that transversality condition is satisfied.
显然，该假设暗示满足横向条件。

Based on the above discussion, the following theorem can be obtained.
基于上述讨论，可以得到以下定理。

Theorem 2. Assume that (H2) holds, then
定理 2.假设 （H2） 成立，则

(1) The trivial equilibrium of system (3) is unstable when and becomes asymptotically stable when .
（1）系统（3）的平凡平衡在时 不稳定，在 时渐近稳定。

(2) System (3) undergoes a Hopf bifurcation at .
（2） 系统 （3） 在 处发生 Hopf 分岔。
Remark 4.2. The fractional delay competitive website system based on Lotka-Volterra competitive model has aroused the interest of scholars [Zhao et al., 2017; Xu et al., 2019]. Unfortunately, in all kinds of competition models proposed before, time delay is taken as the bifurcation parameter. In this paper, the system parameters are analyzed as the bifurcation parameters of the competitive website model, and the analysis method adopted is a very rich supplement to the existing technical means.
备注 4.2.基于Lotka-Volterra竞争模型的分数延迟竞争网站系统引起了学者们的兴趣[Zhao et al.， 2017;Xu 等人，2019 年]。不幸的是，在之前提出的各种竞争模型中，时间延迟都以分岔参数为分岔参数。本文将系统参数作为竞争性网站模型的分岔参数进行分析，所采用的分析方法对现有技术手段进行了非常丰富的补充。

Remark 4.3. So far, fractional-order systems with system parameters as bifurcation parameters have little research on bifurcation, especially dimensional systems Huang et al., 2018]. The conclusion of this paper is a rich supplement and an innovative strategy to the existing theoretical results of high-dimensional systems.
备注 4.3.到目前为止，以系统参数为分岔参数的分数阶系统对分岔的研究很少，尤其是 维度系统Huang et al.， 2018]。本文的结论是对现有高维系统理论成果的丰富补充和创新策略。

Remark 4.4. It should be noted that select system parameters or are used as bifurcation parameter, the equilibrium point of the system may change. But this does not affect the main results. In addition, in the numerical simulation of the fifth part, when is taken as the bifurcation parameter, the state of the system (3) changes from stable state to unstable state at the critical value. However, the state of the system (3) changes from unstable state to stable state if is taken as the bifurcation parameter, which is just opposite to that when is taken as the parameter.
备注 4.4.需要注意的是，选择系统参数 或 用作分岔参数时，系统的平衡点可能会发生变化。但这不会影响主要结果。此外，在第五部分的数值模拟中，当以分岔参数为 分岔参数时，系统（3）的状态在临界值处由稳定状态变为不稳定状态。然而，系统（3）的状态从不稳定状态变为稳定状态，如果 作为分岔参数，这与当作为参数的 状态正好相反。