This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue of

The delay differential equation

In (

On the other hand, the population growth is often subject to environmental noise, and the system will change significantly, which may change the dynamical behavior of solutions significantly [

To our knowledge, much attention to environmental noise is paid on white noise ([

It should be pointed out that the stochastic logistic systems under regime switching have received much attention lately. For instance, the study of stochastic permanence and extinction of a logistic model under regime switching was considered in [

Since (

This paper is organized as follows. In the next section, we will show that there exists a positive global solution with any initial positive value under some conditions. In Sections

Throughout this paper, unless otherwise specified, let

Let

For convenience and simplicity in the following discussion, define

As

Assume that there are positive numbers

Since the coefficients of the equation are locally Lipschitz continuous, for any given initial data

Let

Noticing that

Now, for any

By the Gronwall inequality, we obtain that

Assume that there is a positive number

The following theorem is easy to verify in applications, which will be used in the sections below.

Assume that

The proof of this theorem is the same as that of the theorem above. Let

Note that condition (

Assume that there are positive numbers

Similarly, we can establish a corollary as follows.

Assume that there is a positive number

For convenience and simplicity in the following discussion, we list the following assumptions.

For each

For each

For each

For some

For each

For each

Equation (

The solutions of (

It is obvious that if a stochastic equation is stochastically permanent, its solutions must be stochastically ultimately bounded. So we will begin with the following theorem and make use of it to obtain the stochastically ultimate boundedness of (

Let

By Theorem

By the generalized Itô formula, Young's inequality and (

The inequality above implies

From (

Equation (

Solutions of (

This can be easily verified by Chebyshev's inequality and Theorem

Based on the results above, we will prove the other inequality in the definition of stochastic permanence. For convenience, define

If

All of the principal minors of

(i) Assumptions

(ii) Assumption (A4) implies that there exists a constant

If there exists a constant

Let

By Lemma

Define the function

Now, choose a constant

Under (A1′′), (A2), and (A3), (

The proof is a simple application of the Chebyshev inequality, Lemmas

Under (A1′′) and (A4), (

It is well-known that if

Assume that for some

In the previous sections we have shown that under certain conditions, the original (

Assume that (A1) holds. Then for any given initial data

By Theorem

It finally follows from (

Similarly, it is easy to prove the following conclusions.

Assume that (A1) and (A3′) hold. Then for any given initial data

Assume that (Al) and (A4′) hold. Then for any given initial data

If the noise intensities are sufficiently large in the sense that

Let A

Assume that for some

Assume that (A1′) holds. Then for any given initial data

By Theorem

By the well-known BDG's inequality [

From (

If there exists a constant

Applying the generalized Itô formula, for the fixed constant

Let

Let

Assume that (A1′′), (A2), and (A3) hold. Then for any given initial data

By Theorem

Let

Similarly, by using Lemmas

Assume that (A1′′) and (A4) hold. Then for any given initial data

If that for some

If

Consider a 2-dimensional stochastic differential equation with Markovian switching of the form

By Theorem

Let the generator of the Markov chain

Let the generator of the Markov chain

The authors are grateful to Editor Professor Elena Braverman and anonymous referees for their helpful comments and suggestions which have improved the quality of this paper. This work is supported by the National Natural Science Foundation of China (no. 10771001), Research Fund for Doctor Station of The Ministry of Education of China (no. 20113401110001 and no. 20103401120002), TIAN YUAN Series of Natural Science Foundation of China (no. 11126177), Key Natural Science Foundation (no. KJ2009A49), and Talent Foundation (no. 05025104) of Anhui Province Education Department, 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 090416237, no. 1208085QA15), Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).