Lemma 2 The solutions of system (3) with initial value have the following properties:
Proof It follows from system (3) that
(7)
Set
(8)
where is a solution of system (8) with initial value and . By the comparison theorem for stochastic differential equations, it is easy to have
(9)
By Lemma 3.4 in [20], it is easy to get the following result:
here , and are all nonnegative functions defined on . If , then , a.s.
Note that (), then it follows from Eqs. (8) and (9),
In addition,
therefore, it leads to , a.s. Similarly, we can have , a.s. □
Lemma 3 [25]
Suppose that , where .
-
(I)
If there are positive constants , T and such that
for , where is a constant, , then , a.s. (i.e., almost surely).
-
(II)
If there are positive constants , T and such that
for , where is a constant, , then , a.s.
In the following, we give the result about weak persistence in the mean and extinction of the prey and predator population. Applying Itô’s formula to Eq. (3) leads to
Let , , then . For the prey population of system (3), we have
Theorem 3 (i) If , then the prey population will go to extinction a.s.
-
(ii)
If , then the prey population will be non-persistent in the mean a.s.
-
(iii)
If , then the prey population will be weakly persistent in the mean a.s.
-
(iv)
If , then the prey population will be strongly persistent in the mean a.s.
Proof (i) It follows from (10) that
(12)
, set , it is a martingale whose quadratic variation is . Making use of the strong law of large numbers for martingale yields
(13)
then
(14)
Taking superior limit on both sides of inequality (14) leads to , we can see that .
-
(ii)
By Eq. (12), we have
(15)
It follows from the property of superior limit and (13) that for arbitrary , there exists such that and for all . Substituting these inequalities into (15) yields
when , then
By and Lemma 3, we have , by virtue of the arbitrariness of ε, . Since the solution of system (3) is nonnegative, it is easy to have , that is to say, the prey population is non-persistent in the mean a.s.
-
(iii)
We only need to show that there exists a constant such that for any solution of system (3) with initial value , a.s. Otherwise, for arbitrary , there exists a solution with positive initial value and such that .
Let be sufficiently small so that
(16)
It follows from Eq. (11) that
(17)
here , it also has
(18)
By virtue of (17), it leads to , thus
(19)
On the other hand, it follows from Eq. (12) that
Taking the superior limit to the above inequality and making use of (13), (16) and (19), we have , that is to say, we have shown , this is a contradiction to Lemma 2. Therefore, , the prey population will be weakly persistent in the mean a.s.
-
(iv)
By Eq. (10), we get
It is easy to have
If , there exists sufficiently small such that . It follows from the property of superior limit, interior limit and (13) that for positive number ε, there exists a such that
for all . Then
By virtue of Lemma 3 and the arbitrariness of ε, we have
In other words, the prey population is strongly persistent in the mean a.s. □
Remark 1 The results of Theorem 3 illustrate that is the threshold between weak persistence in the mean and extinction. Note , if , then the prey population will be extinct, no matter whether there are predators. However, the prey population will survive when not considering environmental noise. This indicates that when the density of environmental noise is larger than the intrinsic growth rate of prey, it will cause the extinction of prey population. Therefore, it is more suitable to take into account stochastic perturbation in the systems. Here we can also find that the condition (iv) implies condition (iii), that is to say, the prey population must be weakly persistent in the mean when it is strongly persistent in the mean.
For the predator population, we have the following result.
Theorem 4 (i) If , then the predator population will go to extinction a.s.
-
(ii)
If , then the predator population will be non-persistent in the mean a.s.
-
(iii)
If , then the predator population will be weakly persistent in the mean a.s. where is the solution of Eq. (8) with initial value .
-
(iv)
If , then the predator population has a superior bound in time average, that is, .
Proof (i) If , then it follows from Theorem 3 that . By Eq. (11),
therefore, , then .
Now, if , it follows from the property of superior limit, interior limit and (13) that for sufficiently small ε, there exists a such that
for all . Applying Lemma 3 and the arbitrariness of ε yield
(20)
Substituting the above inequality into (11) gives
(21)
then
which means a.s.
-
(ii)
In the case (i), we have shown that if , then , therefore, . Now, we will prove that is still valid when . Otherwise, if , then it follows from Lemma 2 that . Making use of (21), one can see that
(22)
On the other hand, for arbitrary , there exists a such that
for all . Substituting these inequalities into (11) yields
Then an application of Lemma 3 and (22) results in . By virtue of (20) and the arbitrariness of ε,
which is a contradiction to our assumption, therefore, a.s.
-
(iii)
In the following, we need to show that a.s. Otherwise, for arbitrary , there exists a solution of system (3) with positive initial value such that . Let be sufficiently small so that
(23)
It follows from (11) that
Here is the solution of model (8) with initial value and , , a.s. for .
Because of
then
(24)
Consider the Lyapunov function , then is a positive function on , by Itô’s formula, (8) and (10), we have
Integrating from 0 to t and dividing by t on both sides of the inequality yield
Owing to , it leads to
here , then . Substituting the above inequality into (24), we have
Taking superior limit of the above inequality, we get
which contradicts Lemma 2, then a.s., that is to say, the predator population is weakly persistent in the mean a.s.
-
(iv)
It follows from Eq. (11) that
that is to say,
The following proof is similar to the proof of (iv) in Theorem 3, here we omit it. □
Remark 2 From the proof of Theorem 4, we can observe that if , then is straightforward. It shows that if the prey population goes to extinction, the predator population will also go to extinction, which is consistent with the reality. In the other case, , , which means the prey population will survive, but the predator population will go to extinction. Notice , this phenomenon may be caused by the large death rate of predator or the noise density .