It is not difficult to verify that if , then system (3) has the unique positive equilibrium , where
The Jacobian matrix of system (3) about the positive equilibrium is
Thus, the characteristic equation of system (3) at is
Multiplying on both sides of Eq. (4), it is easy to obtain
When , Eq. (5) reduces to
Obviously, . By the Routh-Hurwitz criterion, sufficient conditions for all roots of Eq. (6) to have a negative real part are given in the following form:
Thus, if condition (H1) Eq. (7)-Eq. (10) holds, is locally asymptotically stable in the absence of delay.
For , let () be the root of Eq. (5). Then we can get
Then we can get
According to , we consider the following two cases.
Case 1. , then Eq. (11) becomes
which is equivalent to
Let and denote
Let . Then Eq. (14) becomes
Then we can get the expression of , and we denote . Substitute into Eq. (11), we can get the expression of , and we denote . Thus, a function with respect to ω can be established by
If all the parameters of system (3) are given, we can calculate the roots of Eq. (15) by Matlab software package. Therefore, we make the following assumption in order to give the main results in this paper.
(H2) Eq. (15) has finite positive roots which are denoted by , respectively. For every fixed (), the corresponding critical value of time delay is
Case 2. , then Eq. (11) becomes
Similar as in Case 1, we can get the expression of denoted as and the expression of denoted by , and further we get a function with respect to ω that can be established by
We assume that Eq. (17) has finite positive roots denoted by , respectively. Then we can get the critical value of time delay corresponding to every fixed positive root of Eq. (17):
Then, when , Eq. (5) has a pair of purely imaginary roots .
Next, we verify the transversality condition. Taking the derivative of λ with respect to τ in Eq. (5), it is easy to obtain
Obviously, if condition (H3) holds, then . Therefore, by the Hopf bifurcation theorem in , we have the following results.
Theorem 1 For system (3), if conditions (H1)-(H3) hold, then the positive equilibrium of system (3) is asymptotically stable for , and system (3) undergoes a Hopf bifurcation at the positive equilibrium when .