Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing τ 1 and τ 2 as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.


Introduction
The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant topics, not only in ecology but also in mathematical ecology due to its universal existence and importance. In [1], Leslie introduced a predator-prey model in which the "carrying capacity" of the predator's environment is proportional to the number of prey:̇= where 1 , 2 , , and are positive constants and ( ) and ( ) denote the population of the prey and predator at time , respectively. The parameters 1 and 2 are the intrinsic growth rates of the prey and the predator. The value is the carrying capacity of the prey, and takes on the role of a prey-dependent carrying capacity for the predator; the parameter is a measure of the quality of the prey as food for the predator. However, this model has attracted the attention of some authors [2][3][4].
Time delays are often incorporated into population models for resource regeneration times, for example, maturing times and gestation periods [5,6]. Recently, great attention has been received and a lot of work has been carried out on the existence of the Hopf bifurcations in delayed population models (see [7][8][9] and references cited therein). The stability of positive equilibria and the existence and the direction of the Hopf bifurcations were discussed, respectively, in the references mentioned above. In [10], Yuan and Song considered the following delayed Leslie-Gower predator-prey system:̇( (2) They investigated the stability and the Hopf bifurcation of the above system without considering the effects of time delay on predator. Motivated by the above discussion, in this paper, by choosing the time delays 1 and 2 as bifurcation parameters, 2 Computational and Mathematical Methods in Medicine we investigate a modified Leslie-Gower predator-prey system with two delays described by the following system: where 1 and 2 are all positive constants. Due to crowding, the prey dynamics is delayed by 1 [11]. The negative feedback delay 2 is assumed in predator growth [12]. is a refuge protecting of the prey and ∈ [0, 1) is a constant. This leaves (1 − ) of the prey available to the predator.
The initial conditions for system (3) take the from where This paper is organized as follows. In Section 2, we investigate the effect of two delays 1 and 2 on the stability of the positive equilibrium of system (3). In Section 3, we derive the direction and stability of the Hopf bifurcation by using normal form and central manifold theory. In Section 4, numerical simulations are performed to support the stability results and chaos is observed. Finally, in Section 5, based on the global Hopf bifurcation theorem for general functional differential equations, we investigate the global existence of periodic solutions by using degree theory methods.

Local Stability Analysis and the Hopf Bifurcation
It is easy to see that system (3) has a unique positive equilibrium * ( * , * ), where * = Let̄= − * ,̄= − * and still denote bȳ= , = ; system (3) can be written aṡ where We then obtain the linearized systeṁ The corresponding characteristic equation is Since + < 0, − + > 0, we know that all roots have negative real parts.

Case 2. Consider
Theorem 2. For 1 = 0, the interior equilibrium point * = ( * , * ) is locally asymptotically stable for 0 < 2 < 2 0 and it undergoes the Hopf bifurcation at 2 = 2 0 given by Proof. On substituting 1 = 0, the characteristic equation (9) becomes Let ( > 0) be a purely imaginary root of (14); then it follows that Squaring both sides and adding them up, we get the following polynomial equation: It is easy to know that (16) has unique positive root 2 2 0 ; then the corresponding critical value of time delay 2 is  Let ( 2 ) = ± 2 0 be the root of (14); then the transversal condition can be obtained: Since we can obtain and then we can obtain (Re ) Case 3. Consider Theorem 3. If 2 = 0 holds, the interior equilibrium point * ( * , * ) is locally asymptotically stable for 0 < 1 < 1 0 and it undergoes the Hopf bifurcation at 1 = 1 0 given by where 1 0 is root of the corresponding characteristic equation.
Proof. The proof is similar to that in Case 2.
With going detailed analysis (26) it is assumed that there exists at least one real positive root * . Now (25) can be written as Equation (29) is simplified to give Computational and Mathematical Methods in Medicine and ± * are purely imaginary roots of (9) for 2 ∈ (0, 2 0 ]. Now verify the transversal condition of the Hopf bifurcation; differentiating equation (9) with respect to 1 , it is obtained that Then noting that To obtain the transversal condition, we also need the condition as follows: Case 5. 1 is fixed in the interval (0, 1 0 ) and 2 > 0.

Direction and Stability of the Hopf Bifurcation
In this section, we show that the system undergoes the Hopf bifurcation for different combinations of 1 and 2 satisfying sufficient conditions as described. Using the method based on the normal form theory and center manifold theory introduced by Hassard et al. in [13], we study the direction of bifurcations and the stability of bifurcating periodic solutions. Throughout this section, it is considered that the system undergoes the Hopf bifurcation at 2 = 2 0 , 1 ∈ (0, 1 0 ) at * . Let 2 = 2 0 + , ∈ , so that the Hopf bifurcation occurs at = 0. Without loss of generality, it is assumed that * 1 < 2 0 where * 1 ∈ (0, 1 0 ). Now we rescale the time by where ( ) = ( ( ) , ( )) , For convenience, ( ), ( ) are still as ( ), ( ), respectively; the nonlinear terms 1 and 2 are Define a family of operators as By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions ( , ) : where we choose , 0 ) , where ℎ ( , ) = ( 2 0 + ) ( ℎ 1 ℎ 2 ) , Hence, (3) can be rewritten aṡ where is the transpose of the matrix . For ∈ ([−1, 0], 2 ) and ∈ ([0, 1], ( 2 ) * ), in order to normalize the eigenvectors of operator and adjoint operator * , we define a bilinear inner product where ( ) = ( , 0).
Since ± 0 2 0 are eigenvalues of , they will also be the eigenvalues of * . The eigenvectors of and * are calculated corresponding to the eigenvalues + 0 2 0 and − 0 2 0 .

Numerical Simulations
To demonstrate the algorithm for determining the existence of the Hopf bifurcation in Section 2 and the direction and stability of the Hopf bifurcation in Section 3, we carry out numerical simulations on a particular case of (3) in the following form: ) , where 1 = 0.8, 2 = 1, = 1.3, = 0.7, = 1, and = 0.5. It is easy to show that system (55) has unique coexistence equilibrium * (0.545, 0.2725). By calculation, when 1 = 0, the critical delay for 2 is obtained as 2 0 = 1.3507 and 1 0 = 5.8228 when 2 = 0.
Further, under the condition of 1 = 1.28, when 2 = 1.32 < 2 0 = 1.9507, * is also stable (see Figure 3(a)), while, at 2 = 5.83, * loses stability and the Hopf bifurcation occurs from Figure 3(b); then using the algorithm derived in Section 3, we obtain that 2 = 312.8, 2 = −287.5, 2 = 106.56; we know the Hopf bifurcation is supercritical and bifurcating periodic solutions are stable and increase. When 2 = 7.9, system (55) becomes a chaotic solution in Figure 3(c). In Figure 3(d), the largest Lyapunov exponent diagram is plotted for variable 2 ; it is easy to know that when 2 > 7.55, the Lyapunov exponent is almost positive; then the chaos occurs.
Whereas, when 1 = 10.15 > 1 0 = 5.8228 and 2 = 1.2, system (55) becomes chaotic in Figure 4(a), in Figure 4(b), the largest Lyapunov exponent diagram is plotted for variable 1 ; it is easy to know that when 1 > 9.85, the Lyapunov exponent is almost positive; then the chaotic solutions occur.
However, * loses stability and the Hopf bifurcation occurs at 1 = 6.9, 2 = 2.1 in Figure 5(a). When 1 = 9.3, 2 = 2.7, a chaotic solution occurs in Figure 5(b). To explore the possibility of occurrence of chaos, the largest Lyapunov exponent diagrams are plotted with respect to key parameters 1 and 2 . In Figure 5(c), the largest Lyapunov exponent diagram is plotted for variable 1 when 2 = 2.7; it is easy to know that when 1 > 9.15, the Lyapunov exponent is almost positive; then the chaotic solution occurs. Similarly, in Figure 5(d), the largest Lyapunov exponent diagram is plotted for variable 2 when 1 = 9.3; it is easy to know that when 2 > 2.65, the Lyapunov exponent is almost positive; then the chaotic solution occurs.

Global Continuation of the Local Hopf Bifurcation
In this section, we will study the global continuation of periodic solutions bifurcating from the point * for 1 is fixed in the interval (0, 1 0 ). Further, the method we used here is based on the global Hopf bifurcating theorem for general functional differential equations introduced by Wu [14]. For convenience, we denote = 2 and write system (3)

Lemma 7.
Assume that (, , ) is an isolated center satisfying (1)(2)(3)(4) in [14]. Denote by (, , ) the connected component of (, , ) in Σ. Then either It is well known that if (ii) of the theorem is not true, then (, , ) is unbounded. However, when the projections of (, , ) onto -space and onto -space are bounded, then the projection of (, , ) onto -space is unbounded. Further, we show that the projection of (, , ) onto -space is away from zero; then the projection of -space must include [ , ∞). Following this idea, we can prove our results on the global continuation of the local Hopf bifurcation.
which implies that solutions of system (3) cannot cross the xaxes and y-axes. Thus, the nontrivial periodic orbits must be located in the interior of the first quadrant. Since ( ( ), ( )) is a nontrivial solution of (3) with ( ) > 0, ( ) > 0, then we havė