A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.

In recent years, population models have merited a great deal of attention due to their theoretical and practical significance since the pioneering theoretical works by Lotka [

It is well known that time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time, capturing time, or other reasons. The effect of the past history on the stability of the system is an important problem in population biology. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. In 2002, Jin and Ma [

In fact, predator-prey system with time delays has also been investigated by lots of authors [

However, the research for cooperative systems with time delays is still relatively little because delays in mutualistic systems usually deprive the boundedness and persistence in [

In general, the delays appearing in different terms of an ecological system are not equal. Therefore, it is more realistic to consider dynamical system with different delays. Motivated by the references [

This paper is organized as follows. In Section

For convenience, letting

From the point view of biological meaning, we are interested in the positive equilibrium. It is obvious that system (

Next, we will consider the stability of the positive equilibrium and the existence of Hopf bifurcation.

Letting

The associated characteristic equation of system (

The characteristic equation of system (

When

We first introduce the following result which was proved by Ruan and Wei [

Consider the exponential polynomial

Let

The characteristic equation of system (

When

Let

From (

Define

the following transversality condition is satisfied:

This will show that there exists at least one eigenvalue with positive real part for

For simplifying, define

For system (

Suppose that

If

Suppose that (H1) holds.

If

If

If

The following transversality condition is satisfied:

Multiplying the

When

For system (

If (H1) (resp., (H1) and

If (H1) and

Denote

Define

In the following, differentiating Equation (

If the following assumption

For system (

The characteristic equation of system (

When

If the condition (H1) holds, then the following transversality condition is satisfied:

Differentiating (

For system (

Suppose that

Equation (

If (H3) holds, the roots of (

In the following, differentiating Equation (

Assume that

Therefore, by the general Hopf bifurcation theorem for FDEs in Hale [

For system (

If we only consider system (

We can obtain that the characteristic equation of system (

When

Multiplying

From (

Define

Since

For system (

Further, the corresponding characteristic equation with

When

Regarding

Suppose that

Equation (

Define

Differentiating (

Suppose that

For system (

In addition, if we consider system (

In this section, we will study the direction of the Hopf bifurcation and the stability of bifurcating periodic solution of system (

Without loss of generality, we only consider the case with

Turning to the linear problem

In fact, we can choose

Next, for

For

By some simple computation, we can obtain

In the remainder of this section, by using the same notations as in Hassard et al. [

However,

Furthermore, we can see that each

In (

The sign of

The sign of

The sign of

In this part, we will study the global continuation of periodic solutions bifurcating from the positive equilibrium

Let

It is easy to see that system (

Let

It follows from system (

In addition, we can easily observe that the following result is true:

The characteristic matrix of (

A stationary solution

From (

Note that the above equation is the same as (

For the above

Thus, the conditions (A1)–(A4) in [

In what follows, we define

From the above discussion, we have

All nonconstant periodic solutions of system (

Suppose that

If

On the other hand, we get, from the first equation of (

This shows that the nonconstant periodic solution of system (

System (

Let’s suppose that there exist a contradiction in system (

In the following, we state and prove our main result in this section.