Abstract
A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing and 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.
1. 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 , and are positive constants and and denote the population of the prey and predator at time , respectively. The parameters and 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 [2–4].
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 [7–9] 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: 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 and as bifurcation parameters, we investigate a modified Leslie-Gower predator-prey system with two delays described by the following system: where and are all positive constants. Due to crowding, the prey dynamics is delayed by [11]. The negative feedback delay is assumed in predator growth [12]. is a refuge protecting of the prey and is a constant. This leaves 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 and 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.
2. Local Stability Analysis and the Hopf Bifurcation
It is easy to see that system (3) has a unique positive equilibrium , where
Let and still denote by , ; system (3) can be written as where We then obtain the linearized system The corresponding characteristic equation is where
Case 1. For, (9) becomes Since , we know that all roots have negative real parts.
Theorem 1. For , the interior equilibrium point is locally asymptotically stable.
Case 2. Consider
Theorem 2. For , the interior equilibrium point is locally asymptotically stable for and it undergoes the Hopf bifurcation at given by
Proof. On substituting , the characteristic equation (9) becomes Let 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 ; then the corresponding critical value of time delay is Let be the root of (14); then the transversal condition can be obtained: Since we can obtain and then we can obtain
Case 3. Consider
Theorem 3. If holds, the interior equilibrium point is locally asymptotically stable for and it undergoes the Hopf bifurcation at given by where is root of the corresponding characteristic equation.
Proof. The proof is similar to that in Case 2.
Case 4. is fixed in the interval () and .
Theorem 4. Assume that and ; then the equilibrium is asymptotically stable for ; moreover let hold; is defined below; then system (3) undergoes the Hopf bifurcation at when , where
Proof. We know in its stable interval and is considered as a parameter. Let be a root of (9). Separating real and imaginary parts, leads to
Let
where
We assumed that
Then and .
With going detailed analysis (26) it is assumed that there exists at least one real positive root . Now (25) can be written as
where
Equation (29) is simplified to give
and are purely imaginary roots of (9) for . Now verify the transversal condition of the Hopf bifurcation; differentiating equation (9) with respect to , it is obtained that
where
Then
noting that
To obtain the transversal condition, we also need the condition as follows:
Case 5. is fixed in the interval and .
Theorem 5. Assume that holds; let and ; then the equilibrium is asymptotically stable for , and system (3) undergoes the Hopf bifurcation at when , where and is root of the corresponding characteristic equation; moreover
Proof. The proof is similar to that in Case 4.
3. Direction and Stability of the Hopf Bifurcation
In this section, we show that the system undergoes the Hopf bifurcation for different combinations of and 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 at . Let , so that the Hopf bifurcation occurs at . Without loss of generality, it is assumed that where . Now we rescale the time by ; then system (3) can be written as where For convenience, are still as , respectively; the nonlinear terms and are Define a family of operators as By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions such that where we choose For , define where Hence, (3) can be rewritten as where and . For , define and the adjoint operator of as where is the transpose of the matrix .
For and , in order to normalize the eigenvectors of operator and adjoint operator , we define a bilinear inner product where .
Since are eigenvalues of , they will also be the eigenvalues of . The eigenvectors of and are calculated corresponding to the eigenvalues and .
Lemma 6. is the eigenvector of corresponding to ; is the eigenvector of corresponding to and where
Following the algorithms explained in Hassard et al. [13], we can obtain the properties of the Hopf bifurcation: where We know that and are constant vectors, computed as
As a result, we know and ; then is determined by the parameters and delays and . Thus, we can compute the following quantities: These expressions give a description of the bifurcating periodic solutions in the center manifold of system (3) at critical values and when which can be stated as follows: (i) gives the direction of the Hopf bifurcation: if , the Hopf bifurcation is supercritical (subcritical);(ii) determines the stability of bifurcating periodic solution: the periodic solutions are stable (unstable) if ;(iii) denotes the period of bifurcating period solutions: if , periodic solutions increase (decrease).
4. 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 , , , , , and . It is easy to show that system (55) has unique coexistence equilibrium . By calculation, when , the critical delay for is obtained as and when .
We can see from Figure 1(a) that is asymptotically stable at , while from Figure 1(b) loses stability and the Hopf bifurcation occurs at . From Figure 2(a), is asymptotically stable when , while from Figure 2(b) loses stability and the Hopf bifurcation occurs when .
(a)
(b)
(a)
(b)
Further, under the condition of , when , is also stable (see Figure 3(a)), while, at , loses stability and the Hopf bifurcation occurs from Figure 3(b); then using the algorithm derived in Section 3, we obtain that , , ; we know the Hopf bifurcation is supercritical and bifurcating periodic solutions are stable and increase. When , system (55) becomes a chaotic solution in Figure 3(c). In Figure 3(d), the largest Lyapunov exponent diagram is plotted for variable ; it is easy to know that when , the Lyapunov exponent is almost positive; then the chaos occurs.
(a)
(b)
(c)
(d)
Whereas, when and , system (55) becomes chaotic in Figure 4(a), in Figure 4(b), the largest Lyapunov exponent diagram is plotted for variable ; it is easy to know that when , the Lyapunov exponent is almost positive; then the chaotic solutions occur.
(a)
(b)
However, loses stability and the Hopf bifurcation occurs at , in Figure 5(a). When , , 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 and . In Figure 5(c), the largest Lyapunov exponent diagram is plotted for variable when ; it is easy to know that when , 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 when ; it is easy to know that when , the Lyapunov exponent is almost positive; then the chaotic solution occurs.
(a)
(b)
(c)
(d)
5. Global Continuation of the Local Hopf Bifurcation
In this section, we will study the global continuation of periodic solutions bifurcating from the point for is fixed in the interval . 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 and write system (3) in the following form: where . Following the work of Wu [14], we define , is a -periodic solution of (56)}, .
Lemma 7. Assume that is an isolated center satisfying in [14]. Denote by the connected component of in . Then either (i) is unbounded or(ii) is bounded, is finite, and for all , where is the crossing number of , if , or it is zero otherwise.
It is well known that if 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.
Lemma 8. If and hold, nontrivial periodic solutions of (3) are uniformly bounded.
Proof. Let be a nontrivial solution of system (3) through at with , . Then it follows from (3) that
which implies that solutions of system (3) cannot cross the x-axes 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 , then we have
It is easy to know for ; then if , we obtain
which implies that
Thus, for any , there exists a such that when , we have
It follows from the second equation of (3) that, for ,
Clearly, for sufficiently large . Thus, the nontrivial periodic solutions lying in the first quadrant of system (3) must be uniformly bounded.
Lemma 9. If and hold, system (3) has no nontrivial periodic solutions with period .
Proof. Assume that system (3) has a nontrivial periodic solution of period ; then the differential system
has periodic solution with period . Due to Lemma 7, we restrict our attention to , , respectively. System (63) also has the equilibrium ; we define
Obviously, is well defined and continuous for all . The function satisfies
Equation (65) shows that the positive equilibrium is the only extremum of the function in the first quadrant. It is easy to see that the point is a minimum, since
Clearly, the positive equilibrium is the global minimum; that is,
holds for all .
Calculate the derivative of along the solution of system (3). Use Razumikhin’s theorem (see [15]); when , we have
Thus, satisfies Lyapunov’s asymptotic stability theorem; we conclude that
which contradicts the fact that system (63) has periodic solutions. This ends the proof.
Theorem 10. If and hold, let and be defined in Case 5 in Section 2. Then, for each , system (3) has at least periodic solutions.
Proof. It is easy to know that the characteristic matrix of system (3) at the positive equilibrium is of the form
From the discussion of Section 2, it can be verified that , are isolated centers.
Let
Clearly, if and , then the necessary and sufficient conditions for are , and .
Defining
then we have the transversal number
By Theorem 3.2 of Wu [14], we conclude that the connected component through in is nonempty. Meanwhile, we have
and hence is unbounded.
From (36), we see that, for , . Then, we are in a position to prove that the projection of onto -space is , where . Clearly, it follows from the proof of Lemma 9 that system (3) with has no nontrivial periodic solution. Hence, the projection of onto -space is away from zero.
For a contradiction, we suppose that the projection of onto -space is bounded. This means that the projection of onto -space is included in an interval . Noting that and applying Lemma 9, we have for belonging to . This implies that the projection of the connected component onto -space is bounded. In addition, from Lemma 8, we obtain that the projection of onto -space is bounded if the projection of onto -space is bounded. Thus, the connected component crossing through is bounded, which is a contradiction. This implies that the projection of onto -space is for each , where . This is the end of the proof.
6. Conclusions
In this paper, we investigate the effect of the time delays and on the stability of the positive equilibrium of system (3) and derive the direction and stability of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical prediction and to explore the complex dynamics including chaos. Finally, we study the global continuation of periodic solutions bifurcating from the point for is fixed in the interval and show the global existence of the periodic solutions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the reviewers for their valuable comments and suggestions which have led to an improvement of this paper. This research is supported by the National Natural Science Foundation of China (Grant no. 11061016).