Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay

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.


Introduction
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 1 first proposed the delayed predator-prey system ẋ x r 1 − a 11 x t − τ − a 12 y , ẏ y −r 2 a 21 x − a 22 y , 1.1 where x and y can be interpreted as the population densities of prey and predator at time t, respectively.τ > 0 is the feedback time delay of prey species to the growth of species itself.r 1 > 0 denotes intrinsic growth rate of prey, and r 2 > 0 denotes the death rate of predator.The parameters a ij i, j 1, 2 are all positive constants.ẏ y −r 2 a 21 x − a 22 y t − τ .

1.2
They established the global existence results of periodic solutions bifurcating from Hopf bifurcations using a global Hopf bifurcation result due to Wu 5 , which was different from that used in Song and Wei 2 .In 6 , Yuan and Zhang also investigated the stability of the positive equilibrium and existence of Hopf bifurcation of the model 1.2 .
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, 1.2 becomes 1.1 .Meanwhile, if the time delay of the prey to the growth of the species itself is zero, 1.2 is simplified as ẋ x t r 1 − a 11 x t − a 12 y t , ẏ y t −r 2 a 21 x t − a 22 y t − τ .

1.3
In 1954, Gordon 7 studied the effect of harvest effort on ecosystem from an economic perspective and proposed the following economic theory: Net Economic Revenue NER Total Revenue TR − Total Cost TC .

1.4
This provides theoretical fundament for the establishment of bioeconomic systems by differential-algebraic equations DAEs .
Let E and y represent the harvest effort and the density of harvested population, respectively.Then TC cE and TR αEy, where α is unit price of harvested population and c is the cost of harvest effort.Considering the economic interest m of the harvest effort on the predator, we can establish the following algebraic equation: Recently, a class of differential-algebraic bioeconomic models were proposed and analyzed in 8-14 .For example, 8 discussed the problems of chaos and chaotic control for a differential-algebraic system.The existence and stability of equilibrium points, the sufficient conditions of existence for various bifurcations transcritical bifurcation, singular induced bifurcation, and Hopf bifurcation are invested in 9-14 .However, the stability and the direction of periodic solutions bifurcating from Hopf bifurcations have not been discussed there.Very recently, Zhang et al. 15 analyzed the stability and the Hopf bifurcation of a differential-algebraic bioeconomic system with a single time delay and Holling II type functional response and first investigated the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions.
Based on the above economic theory and system 1.3 , in this paper, we study the Hopf bifurcation of the following differential-algebraic bioeconomic system: The rest of this paper is organized as follows.In Section 2, regarding time delay τ as bifurcation parameter, first, we study the stability of the equilibrium point of system based on the new normal form approach proposed by Chen et al. 16 .Then, following the normal form approach theory and the center manifold theory introduced by Hassard et al. 3 , we compute the normal form for the Hopf bifurcation of system 1.6 and analyze the direction and stability of the bifurcating periodic orbits of the system.Numerical examples are given in Section 3 to illustrate our theoretical results.

Hopf Bifurcation
In this section, we analyze the stability and the Hopf bifurcation of the differential-algebraic bioeconomic system 1.6 .Furthermore, we will derive the formula for determining the properties of the Hopf bifurcation.

Existence of Hopf Bifurcation
For m > 0, the system 1. 6

2.1
In order to guarantee the existence of the positive equipment point P * , some restrictions must be satisfied also: In order to analyze the local stability of the positive equilibrium point, we first use the linear transformation x, y, E T Q • u, v, E T , where Then the system 1.6 yields

2.4
According to the literature 16 , considering the local parametrization of the third equation of the system 2.4 , and we can obtain the following parametric system of it: where

2.7
Thus the linearized system of parametric system 2.5 at 0,0 is given as below:

2.8
The characteristic equation of the linearized system of parametric system 2.8 at 0,0 is given by λ 2 pλ r sλ q e −λτ 0, 2.9 where

2.12
From Lemma 2.1 in 2 , we easily obtain the following results about the stability of the positive equilibrium and the Hopf bifurcation of 1.6 .Theorem 2.1.For system 1.6 .One has the following.
i If (H1)-(H3) hold, then the equilibrium E * of the system 1.6 is asymptotically stable for all τ ≥ 0.
iii If (H1), (H2), and (H5) hold, then there is a positive integer k, such that the equilibrium E * switches k times from stability to instability to stability; that is, E * is asymptotically stable when and unstable when Here,

Direction and Stability of the Hopf Bifurcation
In the previous section, we obtain the conditions for Hopf bifurcations at the positive equilibrium P * when τ τ ± n , n 0, 1, 2, . ... In this section, we will derive the formulaes determining the direction, stability, and period of these periodic solutions bifurcating from P * at τ.The basic techniques used here are the normal form and the center manifold theory developed by Hassard et al. 3 .
Assuming the system 1.6 undergoes Hopf bifurcations at the positive equilibrium P * at τ τ n , we let iω be the corresponding purely imaginary root of the characteristic equation at the positive equilibrium.In order to investigate the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions of system 1.6 , we consider the parametric system 2.5 of system 2.4 .
Let t tτ, τ τ n μ; then the parametric system 2.5 is equivalent to the following functional differential equation system:

2.19
In fact, we can choose

2.21
Hence system 2.17 can be rewritten as

2.22
For Ψ ∈ C −1, 0 , R 2 * , define and a bilinear inner product where η θ η θ, 0 .Then A * and A 0 are adjoint operators.By the discussion in Section 2.1, we know that ±iω are eigenvalues of A 0 .Thus they are also eigenvalues of A * .Suppose that q θ 1, q 1 T e iωτθ and q * s D q * 1 , 1 e iωτs are eigenvectors of A 0 and A * corresponding to iωτ and −iωτ, respectively.It is easy to obtain

2.25
Moreover, q * s , q θ 1, q * s , q θ 0. Now, we calculate the coordinates to describe the center manifold C 0 at μ 0. Using the same notations as in 3 , we define z q * , Z t , W t, θ Z t θ − 2 Re zq θ .

2.26
On the center manifold C 0 , we have W t, θ W z, z, θ , where and z and z are local coordinates for C 0 in the direction of q * and q * .Note that W is real if Z t is real; we only consider real solutions.For Z t ∈ C 0 , we have ż iωτz q * 0 f 0 z, z : iωτz g z, z , 2.28 where Moreover, we have

2.30
Next, we will calculate W 11 θ and W 20 θ .From 2.22 and 2.28 , we have

2.35
Now we will seek appropriate M and N. From 2.29 and 2.31 , we have where

2.39
It is easy to get M and N. Furthermore, we can get the following values:

2.40
It is well known that, at the critical value τ, the sign of μ 2 determines the direction of the Hopf bifurcation, β 2 determines the stability of bifurcated periodic solutions, while T 2 determines the period of bifurcated periodic solutions, respectively.In fact, we have the theorem as follows.Theorem 2.2.i The Hopf bifurcation is supercritical if μ 2 > 0 and subcritical if μ 2 < 0.
iii The period of bifurcated periodic solutions increases if T 2 > 0 and decreases if T 2 < 0.

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

3.1
The only positive equilibrium point of the system 3.1 is P 1.5, 2.0, 0.5 .It is easy to get p 2.3333, q 6, r 7, s 2. When τ 0, from 2.9 , the characteristic equation of the linearized system of parametric system of 3.1 at P is given by λ 2 p s λ r q 0.

3.2
By simple calculating, p s 4.3333 > 0, q r 13 > 0. According to Routh-Hurwitz stability criterion, P is stable.When τ > 0, we can further get r 2 − q 2 13 > 0, s 2 − p 2 2r 12.5556 > 0, and s 2 − p 2 2r 2 − 4 r 2 − q 2 105.6420 > 0 which mean H1 , H2 , and H5 hold.Furthermore, we have ω − we can conclude that the Hopf bifurcation of system 3.1 is subcritical, the bifurcating periodic solution exists when τ crosses τ 0 to the left, and the bifurcating periodic solution is unstable and decreases.Figure 1 shows the positive equilibrium point P of system 3.1 is locally asymptotically stable when τ 0.55 < τ 0 0.5638.The periodic solutions occur from P  14 Journal of Applied Mathematics when τ 0.563 < τ 0 0.5638 as is illustrated in Figure 2. When τ 0.57 > τ 0 0.5638, the positive equilibrium point P becomes unstable as shown in Figure 3.

Figure 1 :
Figure 1: : Dynamic behavior of the differential-algebraic system 3.1 with τ 0.55 < τ 0 , and the positive equilibrium point P is stable.

Figure 2 :
Figure 2: Dynamic behavior of the differential-algebraic system 3.1 with τ 0.563 < τ 0 , periodic solutions bifurcating from the positive equilibrium point P .

Figure 3 :
Figure 3: Dynamic behavior of the differential-algebraic system 3.1 with τ 0.57 > τ 0 , and the positive equilibrium point P is unstable.

has positive equipment point P
r 1 − a 12 y * /a 11 , E * m/ αy * − c and y * is the positive root of the following equation: α a 11 a 22 a 12 a 21 y 2 − ca 11 a 22 ca 12 a 21 αr 1 a 21 − αa 11 r 2 y ma 11 cr 1 a 21 − ca 11 r 2 0.