Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey

This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included.


Introduction
Predator-prey dynamics continues to draw interest from both applied mathematicians and ecologists due to its universal existence and importance.Many kinds of predator-prey models have been studied extensively 1-6 .It is well known that there are many species whose individual members have a life history that takes them through immature stage and mature stage.To analyze the effect of a stage structure for the predator or the prey on the dynamics of a predator-prey system, many scholars have investigated predator-prey systems with stage structure in the last two decades 7-15 .In 7 , Wang considered the following predator-prey system with stage structure for the predator and obtained the sufficient conditions for the global stability of a coexistence equilibrium of the system: where x t represents the density of the prey at time t.y 1 t and y 2 t represent the densities of the immature predator and the mature predator at time t, respectively.For the meanings of all the parameters in system 1.1 , one can refer to 7 .Considering the gestation time of the mature predator, Xu 8 incorporated the time delay due to the gestation of the mature predator into system 1.1 and considered the effect of the time delay on the dynamics of system 1.1 .
There has also been a significant body of work on the predator-prey system with stage structure for the prey.In 12 , Xu considered a delayed predator-prey system with a stage structure for the prey: where x 1 t and x 2 t denote the population densities of the immature prey and the mature prey at time t, respectively.y t denotes the population density of the predator at time t.All the parameters in system 1.2 are assumed positive.a is the birth rate of the immature prey.b is the transformation rate from immature individual to mature individuals.b 1 is the intraspecific competition coefficient of the mature prey.r 1 and r 2 are the death rates of the immature and the mature prey, respectively.r is the death rate of the predator.a 1 and a 2 are the interspecific interaction coefficients between the mature prey and the predator, respectively.a 1 x 2 / 1 mx 2 is the response function of the predator.And τ is a constant delay due to the gestation of the predator.In 12 , Xu investigated the persistence of system 1.2 by means of the persistence theory on infinite dimensional systems, and sufficient conditions are obtained for the global stability of nonnegative equilibrium of the model by constructing appropriate Lyapunov function.But studies on the predator-prey system not only involve the persistence and stability, but also involve many other behaviors such as periodic phenomenon, attractivity, and bifurcation 16-19 .In particular, the properties of periodic solutions are of great interest 20-24 .Therefore, F. Li and H. W. Li 14 considered the property of periodic solutions of the following system:

1.3
Motivated by the work of Xu 12 and F. Li and H. W. Li 14 and considering the intraspecific competition of the immature prey population, we consider the following system: where x 1 t and x 2 t denote the population densities of the immature prey and the mature prey at time t, respectively.y t denotes the population density of the predator at time t.The parameters a, a 1 , a 2 , b, b 1 , r, r 1 , r 2 , and m are defined as in system 1.3 .c is the intraspecific competition of the immature prey, τ 1 is the feedback delay of the mature prey, and τ 2 is the time delay due to the gestation of the predator.The organization of this paper is as follows.In Section 2, by analyzing the corresponding characteristic equations, the local stability of the positive equilibrium of system 1.4 is discussed, and the existence of Hopf bifurcation at the positive equilibrium is established.In Section 3, we determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem in 20 .And numerical simulations are carried out in Section 4 to illustrate the main theoretical results.Finally, main conclusions are included.

Local Stability and Hopf Bifurcation
From the viewpoint of biology, we are only interested in the positive equilibrium of system 1.4 .It is not difficult to verify that system 1.4 has a positive equilibrium E 0 x 0 1 , x 0 2 , y 0 , where if the following conditions hold: Let x 1 t z 1 t x 0 1 , x 2 t z 2 t x 0 2 , y t z 3 t y 0 , and we still denote z 1 t , z 2 t , and z 3 t by x 1 t , x 2 t , and y t .Then system 1.4 can be transformed to the following form: − ry t .

2.3
Then we can get the linearized system of system 2.

2.4
Therefore, the corresponding characteristic equation of system 2.4 is Next, we consider the local stability of the positive equilibrium E 0 x 0 1 , x 0 2 , y 0 and the Hopf bifurcation of system 1.4 for the different combination of τ 1 and τ 2 .where m 12 m 2 n 2 p 2 , m 11 m 1 n 1 p 1 q 1 , m 10 m 0 n 0 p 0 q 0 .It is not difficult to verify that m 12 > 0 and m 10 > 0. Thus, all the roots of 2.6 must have negative real parts, if the following condition holds: H 11 : m 12 m 11 > m 10 .Namely, the positive equilibrium E 0 x 0 1 , x 0 2 , y 0 is locally stable in the absence of time delay, if H 11 holds.

2.8
Squaring both sides and adding them up, we get the following sixth-degree polynomial equation: A 20

2.12
where To verify the transversality condition of Hopf bifurcation, differentiating the two sides of 2.7 with respect to τ 1 , and noticing that λ is a function of τ 1 , we can obtain

2.13
Thus, In conclusion, we have the following results.
It follows that

2.30
Let  Next, we verify the transversality condition.Differentiating 2.25 regarding τ and substituting τ τ 0 , we get Re dλ dτ where Thus, if the condition H 42 : AC BD / 0 holds, the transversality condition is satisfied.
Taking the derivative of λ with respect to τ 1 in 2.5 and substituting τ 1 τ 1 * , we get Re dλ dτ

2.38
Obviously, if the condition H 52 : P R Q R P I Q I / 0 holds, the transversality condition is satisfied.Through the above analysis, we have the following results.

Direction and Stability of Bifurcated Periodic Solutions
In Section 2, we have obtained the conditions under which a family of periodic solutions bifurcate from the positive equilibrium of system 1.4 when the delay crosses through the critical value.In this section, we will determine the direction of Hopf bifurcation and stability of bifurcating periodic solutions of system 1.4 with respect to τ 1 for τ 2 ∈ 0, τ 20 by using the normal form method and center manifold theorem introduced by Hassard et al. 20 .It is considered that system 1.4 undergoes Hopf bifurcation at τ 1 τ 1 * , τ 2 ∈ 0, τ 20 .Without loss of generality, we assume that τ 1 * > τ 2 * , where , y sτ 1 z 3 s .We still denote s by t.Then, system 1.4 can be transformed into the following system: where Hence, by the Riesz representation theorem, there exists a 3 × 3 matrix function η θ, μ : −1, 0 → R 3 whose elements are of bounded variation such that In fact, we choose

3.5
For φ ∈ C −1, 0 , R 3 , we define Then, system 3.1 can be transformed into the following operator equation:

3.19
Then, we can calculate the following values:

3.20
Based on the above discussion, we can obtain the following results.ii the stability of bifurcating periodic solutions is determined by the sign of σ: if σ < 0 σ > 0 , the bifurcating periodic solutions are stable (unstable); iii the period of the bifurcating periodic solution is determined by the sign of T : if T > 0 T < 0 , the bifurcating periodic solution increases (decreases).

Numerical Example
In this section, we give some numerical simulations to verify the theoretical analysis in Sections 2 and 3. Let a 8, a 1 4.25, a 2 3, b 5, b 1 1, c 0.5, m 2, r 1, r 1 1, and r 2 2.Then, we have the following particular case of system 1.4 :

4.1
It is not difficult to verify that a 2 > mr, bx 0 1 > r 2 − b 1 x 0 2 x 0 2 , namely, the conditions H 1 and H 2 hold.Therefore, system 4.1 has at least a positive equilibrium.By means of Matlab, we can get that the positive equilibrium of 4.1 is E 0 * 1.2111, 1.0000, 2.1568 .For τ 1 > 0, τ 2 0, we can get ω 10 1.3881, τ 10 0.9032.From Theorem 2.2, we know that the positive equilibrium E 0 * is asymptotically stable when τ 1 ∈ 0, τ 10 .The corresponding waveform and the phase plot are illustrated by Figure 1.When the time delay τ 1 passes through the critical value τ 10 , the positive equilibrium E 0 * will lose its stability and a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the positive equilibrium E 0 * .This property is illustrated by the numerical simulation in Figure 2. Similarly, we have ω 20 0.8497, τ 20 0.5124, when τ 2 > 0, τ 1 0. The corresponding waveform and the phase plots are shown in Figures 3 and 4.
For τ 1 τ 2 τ > 0, we can obtain ω 0 1.0000, τ 0 0.4178.From Theorem 2.2, we know that, when the time delay τ increases from zero to τ 0 , the positive equilibrium E 0 * is asymptotically stable.Once the time delay τ passes through the critical value τ 0 , the positive equilibrium E 0 * will lose its stability and a Hopf bifurcation occurs.This property is illustrated by the numerical simulation in Figures 5 and 6.

Conclusions
In this paper, a delayed predator-prey system with Holling type III functional response and stage structure for the prey population is investigated.Compared with literature 14 , we consider not only the time delay due to the gestation of the predator but also the negative feedback of the mature prey density and the intraspecific competition of the immature prey population.F. Li and H. W. Li 14 has obtained that the species in system 4.1 with only the time delay due to the gestation of the predator could coexist.However, we get that the species could also coexist with some available time delays of the mature prey and the predator.This is valuable from the view of ecology.
The sufficient conditions for the local stability of the positive equilibrium and the existence of local Hopf bifurcation for the possible combinations of two delays are obtained.The main results are given in Theorems 2. 1-2.5.By computation, we find that the time delay due to the gestation of the predator is marked because the critical value of τ 2 is smaller than that of τ 1 when we only consider them, respectively.Furthermore, the explicit formulae  which determines the direction of the bifurcation and the stability of the bifurcating periodic solutions is established when τ > 0 and τ 2 ∈ 0, τ 20 by using the normal form theory and center manifold theorem.The main results are given in Theorem 2.3.Finally, numerical simulations are carried out to support the obtained theoretical results.