In this section, we consider the stochastic differential equation with regime switching (1.4). If stochastic differential equation has a unique global (i.e., no explosion in a finite time) solution for any initial value, the coefficients of the equation are required to obey the linear growth condition and local Lipschitz condition. It is easy to see that the coefficients of (1.4) satisfy the local Lipschitz condition; therefore, there is a unique local solution on with initial value , , where is the explosion time.

And since our purpose is to reveal the effect of environmental noises, we impose the following hypothesis on intensities of environmental noises.

Assumption 4.1 ().

By virtue of comparison theorem, we will demonstrate that the local solution to (1.4) is global, which is motivated by [12]

Thus, is the unique solution of

with . By the comparison theorem, we get for a.s. It is easy to see that

with , has a unique solution

Obviously, , a.s.

Moreover,

So, we get , a.s., where

Besides,

Then,

where

To sum up, we obtain

It can be easily verified that , , , all exist on , hence

Theorem 4.2.

There is a unique positive solution of (1.4) for any initial value , , and the solution has the properties

where , , , are defined as (4.1),(4.4),(4.6), and (4.9).

Theorem 4.2 tells us that (1.4) has a unique global solution, which makes us to further discuss its properties.

Now, we will investigate certain asymptotic limits of the population model (1.4). Referred to [12], it is not difficult to imply that

Then, we give the following essential theorems which will be used.

Theorem 4.3.

Let Assumption 4.1 hold. Then, the solution has the property

Proof.

The proof is motivated by [12]. By (4.12) and (4.13), then

Thus, it remains to show that

Note the quadratic variation of is , thus the strong law of large numbers for local martingales yields that

Therefore, for any , there exists some positive constant such that for any

Then, for any , we have

Moreover, it follows from (4.13) that for the above and , we get

By virtue of (4.6), we can derive for

Using the generalized Itô Lemma, we can conclude that

Consequently,

Then, for

Denote and . It is obvious that . Hence,

So, we obtain

That is,

Therefore,

Note the fact that is arbitrary, we obtain that

By (4.12), we have a.s. Consequently,

So, we complete the proof.

Theorem 4.4.

Let Assumption 4.1 and hold. Then, has the property

Proof.

From (4.12) and (4.13), we have

Now, we only show that . By virtue of (4.12), it remains to demonstrate that . From the proof of Theorem 4.3, we know that for any , there exists some positive constant such that for any

It follows from (4.34) that

For above and , we get for
a.s. By the generalized Itô Lemma, then

Therefore,

Thus, it is easy to imply that

where and . By (4.9), we imply for

Consequently,

It follows from (4.40) that

Then,

For is arbitrary, we imply

Finally, we obtain

Theorem 4.5.

Let Assumption 4.1 and hold. Then, the solution to (1.4) obeys

Proof.

Using the generalized Itô Lemma, we have

Therefore,

Thus, let , by the ergodic properties of Markov chains, we have

Hence, by virtue of the strong law of large numbers of local martingales, we get

The proof is complete.