In this section, we present some lemmas and definitions to prove our main results in Section 3.
Definition 2.1 Let be the unique positive equilibrium of model (1.2). If there exist constants , and such that every solution to the initial value problem (1.2) and (1.4) always satisfies
then is said to be globally exponentially stable.
Definition 2.2 All solutions are uniformly permanent if there exist positive constants m and M such that for any solution , we have
Lemma 2.1 There exists a unique positive global solution of model (1.2) and (1.4) on the interval .
Proof Because of , we have by using Theorem 5.2.1 in . Set for . From the variation of constants formula and initial condition , we obtain
for all .
It remains to show .
For the sake of contradiction, assume that is bounded. Note the fact that , we obtain from (1.2) that
This leads to
which excludes the possibility that . Hence it violates Theorem 2.3.1 in . So we obtain the existence of the unique global positive solution of (1.2) and (1.4) on . Therefore Lemma 2.1 is proved. □
Suppose that there exists a positive constant
then solutions of (1.2) and (1.4) are uniformly permanent with
Proof Let . By Lemma 2.1, for . From Theorem 1.6.1 in  and (2.1), we get that the solution to the initial value problem (1.2) and (1.4) is not greater than the solution to the initial value problem
In view of (2.2) and
We next show that . Otherwise, we assume that . For each , we define
It follows from that as and that
From the definition of , we know that or
which implies that
This yields that there exists such that
In view of (2.3), (2.4) and , for , we have
which obviously contradicts with (2.2). Thus, we have proved that .
Finally, we prove that . Again, by way of contradiction, we assume that . By the fluctuation lemma [, Lemma A.1], there exists a sequence such that
Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that
Then recall that , and it follows from
that (taking limits)
which contradicts with (2.2). This proves that . Hence the proof of Lemma 2.2 is completed. □