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We investigate the dynamics of a differential-algebraic bioeconomic model with two time delays. Regarding time delay as a bifurcation parameter, we show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Using the theories of normal form and center manifold, we also give the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical tests are provided to verify our theoretical analysis.

Predator-prey systems with delay play an important role in population dynamics since Vito Volterra and James Lotka proposed the seminal models of predator-prey in the mid of 1920s. In recent years, predator-prey systems with delay have been studied extensively due to their theoretical and practical significance.

In 1973, May [

Recently, Song and Wei [

Considering the feedback time delay of predator species to the growth of species itself and also with the delay

They established the global existence results of periodic solutions bifurcating from Hopf bifurcations using a global Hopf bifurcation result due to Wu [

Noting that, in real situations, the feedback time delay of the prey to the growth of the species itself and the feedback time delay of the predator to the growth of the species itself are different. If the time delay of the predator to the growth of the species itself is zero, (

In 1954, Gordon [

This provides theoretical fundament for the establishment of bioeconomic systems by differential-algebraic equations (DAEs).

Let

Recently, a class of differential-algebraic bioeconomic models were proposed and analyzed in [

Based on the above economic theory and system (

The rest of this paper is organized as follows. In Section

In this section, we analyze the stability and the Hopf bifurcation of the differential-algebraic bioeconomic system (

For

In order to guarantee the existence of the positive equipment point

In order to analyze the local stability of the positive equilibrium point, we first use the linear transformation

Then the system (

According to the literature [

Thus the linearized system of parametric system (

The characteristic equation of the linearized system of parametric system (

The second-degree transcendental polynomial equation (

Let

either

either

From Lemma 2.1 in [

For system (

(i) If (H1)–(H3) hold, then the equilibrium

(ii) If (H1), (H2), and (H4) hold, then the equilibrium

(iii) If (H1), (H2), and (H5) hold, then there is a positive integer

Here,

Denote

In the previous section, we obtain the conditions for Hopf bifurcations at the positive equilibrium

Assuming the system (

Let

For

In fact, we can choose

For

Hence system (

For

Now, we calculate the coordinates to describe the center manifold

On the center manifold

Moreover, we have

Next, we will calculate

For

On the other hand,

It follows from (

Solving it, we have

Now we will seek appropriate

Noting that

It is easy to get

It is well known that, at the critical value

(i) The Hopf bifurcation is supercritical if

(ii) The bifurcated periodic solutions are unstable if

(iii) The period of bifurcated periodic solutions increases if

We now perform some simulations for better understanding of our analytical treatment.

Consider the following differential-algebraic system:

The only positive equilibrium point of the system (

When

When

By the aid of Mathematica, we can obtain the following values according to equalities in (

Figure

: Dynamic behavior of the differential-algebraic system (

Dynamic behavior of the differential-algebraic system (

Dynamic behavior of the differential-algebraic system (

The authors would like to thank the reviewers for their valuable comments which improved the paper. This work is supported by the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China (200720), the “333 Project” Foundation of Jiangsu Province, and the Foundation for Innovative Program of Jiangsu Province(CXZZ11_0870, CXZZ12_0383).