In view of ecology, a good situation occurs when all species co-exist. In this section, we will consider another stochastic persistence, that is, stochastic persistence in mean. Now, we present the definition of persistence in mean.
Definition 4 (see [7, 12])
The system (4) is said to be persistent in mean if
Firstly, we introduce a fundamental lemma which will be used.
Lemma 3 Consider the one-dimensional stochastic equation
where , , are positive, continuous and bounded functions, is a standard Brownian motion. Under the condition , for any initial value , the solution to (9) has the property
Proof We firstly show a.s. To define the Lyapunov function , using the Itô formula, we obtain
where , whose quadratic variation is
By virtue of the exponential martingale inequality, for any positive constants T, δ, β, we have
Choose , and , where , , and above. Hence
Obviously, we know . Applying the Borel-Cantalli lemma, we obtain that there exists some with such that for any , an integer such that for any , we get
for all . Then
Note that , , we have
For all with , we derive
Thus, for , we get . This implies
Letting , that is, , we can imply . By making , and , we get . Consequently,
Thus it remains to show that a.s. The quadratic variation of the stochastic integral is . So, the strong law of large numbers of local martingales yields that
Hence, for any , there exists some positive such that
For any , we have
Then, for any ,
That is, a.s. Then a.s. Thus a.s. Since ϵ is arbitrary, we conclude that
So, the proof is complete. □
Remark 1 Lemma 3 generalizes the works of  and .
To continue our analysis, let us impose the following hypothesis.
Assumption 2 , .
Theorem 1 tells us there is a unique global solution (which is positive for any initial value ) to the stochastic system (4). So, we conclude the following results by the comparison theorem. We can get
Denote that is the solution to the following stochastic equation:
with . And is the solution to the equation
with . It is obvious that , a.s. Moreover, we can have
We denote is the solution of the stochastic differential equation
with . And the stochastic equation
has the solution for initial value . Consequently, , a.s. To sum up, we have
Lemma 4 Under Assumption 2, for any initial value , the solution to (4) satisfies
Lemma 3, (10), (11) and (14) can straightforward imply the assertion.
Lemma 5 Under Assumption 2, for any initial value , the solution to (4) satisfies
Lemma 3, (12), (13) and (14) prove the result.
Theorem 7 Let Assumption 2 hold. Then, for any initial value , the system (4) is persistent in mean. That is, the system (4) has the properties
Proof Denote , by the Itô formula, we obtain
By virtue of the strong law of large numbers and Lemma 4, we get
On the other hand, denote , by the Itô formula, we obtain
So, we have
Dividing t on both sides yields
Letting , by virtue of the strong law of large numbers and Lemma 5, we have
The proof is complete. □