It is easy to verify that if the condition (H1): holds, then system (2) has a unique positive equilibrium , where , and , where , , .
Let , . Dropping the bars, system (2) gets the following form:
(3)
where
(4)
(5)
(6)
(7)
(8)
(9)
with
(10)
(11)
(12)
(13)
(14)
The linearized system of system (3) is
(15)
The characteristic equation of system (15) is
(16)
where
Case 1. .
When , Eq. (16) becomes
(17)
where
It is easy to obtain , . From the expressions of and , we get . Then we can get . Therefore, if the condition (H1) holds, then the positive equilibrium of system (2) without delay is locally asymptotically stable.
Case 2. , .
When , , Eq. (16) becomes
(18)
where
Let () be the root of Eq. (18). Then we have
from which one can obtain
(19)
Since , we can conclude that if we have the condition (H2): , then , and further Eq. (19) has a unique positive root
and the corresponding critical value of the delay is
Define
When , then Eq. (18) has a pair of purely imaginary roots . Differentiating the two sides of Eq. (18), one can obtain
Thus,
Based on the analysis above and according to the Hopf bifurcation theorem in [21], we have the following results.
Theorem 1 Suppose that the conditions (H1) and (H2) hold. The positive equilibrium of system (2) is asymptotically stable for . System (2) undergoes a Hopf bifurcation at the positive equilibrium of system (2) when and a family of periodic solutions bifurcate from the positive equilibrium of system (2).
Case 3. , .
Substitute into Eq. (16), then Eq. (16) becomes
(20)
where
from which it follows that
(21)
Obviously, . Thus, we can conclude that Eq. (21) has a unique positive root
and the corresponding critical value of the delay is
Define
When , then Eq. (20) has a pair of purely imaginary roots and similar to Case 2, we can obtain
Based on the analysis above and according to the Hopf bifurcation theorem in [21], we have the following results.
Theorem 2 Suppose that the condition (H1) holds. The positive equilibrium of system (2) is asymptotically stable for . System (2) undergoes a Hopf bifurcation at the positive equilibrium of system (2) when and a family of periodic solutions bifurcate from the positive equilibrium of system (2).
Case 4. .
When , then Eq. (16) becomes
(22)
where
Multiplying by , Eq. (22) becomes
(23)
Let () be the root of Eq. (23). Then
It follows that
Then we have
(24)
where
(25)
(26)
(27)
Let , then Eq. (24) becomes
(28)
The discussion of the roots of Eq. (28) is similar to that in [22]. Denote
(29)
From Eq. (29), we have
Set
(30)
Let . Then Eq. (29) becomes
(31)
where
Define
(32)
(33)
(34)
(35)
(36)
Based on the conditions (H1) and (H2), we can conclude that . Then we have the following results according to Lemma 2.2 in [22].
Lemma 1 For Eq. (29),
-
(i)
if , then Eq. (29) has positive root if and only if and ;
-
(ii)
if , then Eq. (29) has positive root if and only if there exists at least one , such that and .
In what follows, we assume that (H41): the coefficients satisfy one of the following conditions in -: () , , and ; () and there exists at least one , such that and .
If the condition (H41) holds, Eq. (28) has at least one positive root . Thus, Eq. (24) has at least one positive root and the corresponding critical value of the delay is
Let
When , Eq. (23) has a pair of purely imaginary roots . Differentiating both sides of Eq. (23) with respect to τ, we get
Thus,
where
(37)
(38)
(39)
(40)
Obviously, if the condition (H42): holds, the transversality condition is satisfied. Based on the discussion above and according to the Hopf bifurcation theorem in [21] we have the following results.
Theorem 3 Suppose that the conditions (H1), (H2), (H41), and (H42) hold. The positive equilibrium of system (2) is asymptotically stable for . System (2) undergoes a Hopf bifurcation at the positive equilibrium of system (2) when and a family of periodic solutions bifurcate from the positive equilibrium of system (2).
Case 5. , .
We consider Eq. (16) with in its stable interval and is considered as the bifurcation parameter. Let () be the root of Eq. (16). Then
with
(41)
(42)
(43)
Then one can obtain
(44)
where
(45)
(46)
(47)
In order to give the main results in this paper, we make the following assumption. (H51): Eq. (44) has at least finite positive roots. If the condition (H51) holds, we denote the roots of Eq. (44) as . For every fixed (), the corresponding critical value of the delay is
Let . When , Eq. (16) has a pair of purely imaginary roots for . Differentiating Eq. (16) with respect to , one can obtain
Thus,
where
(48)
(49)
(50)
(51)
(52)
(53)
Obviously, if the condition (H52): holds, then . Thus, according to the Hopf bifurcation theorem in [21] we have the following result.
Theorem 4 Suppose that the conditions (H1), (H51), and (H52) hold and . The positive equilibrium of system (2) is asymptotically stable for . System (2) undergoes a Hopf bifurcation at the positive equilibrium of system (2) when and a family of periodic solutions bifurcate from the positive equilibrium of system (2).
Case 6. , .
We consider Eq. (16) with in its stable interval and is considered as the bifurcation parameter. Let () be the root of Eq. (16). Then
with
(54)
(55)
It follows that
(56)
where
(57)
(58)
(59)
Similar to Case 5, we make the following assumption. (H61): Eq. (56) has at least finite positive roots. We denote the roots of Eq. (56) as . For every fixed (), the corresponding critical value of the delay is
Let . When , Eq. (16) has a pair of purely imaginary roots for . Differentiating Eq. (16) with respect to , one can obtain
Thus,
where
(60)
(61)
(62)
(63)
(64)
(65)
Similar as in Case 5, we can conclude that if the condition (H62): holds, then . Thus, according to the Hopf bifurcation theorem in [21] we have the following results.
Theorem 5 Suppose that the conditions (H1), (H61), and (H62) hold and . The positive equilibrium of system (2) is asymptotically stable for . System (2) undergoes a Hopf bifurcation at the positive equilibrium of system (2) when and a family of periodic solutions bifurcate from the positive equilibrium of system (2).