Abstract
A predator-prey system with two delays and stage-structure for both the predator and the prey is considered. Sufficient conditions for the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. Specially, the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are determined by applying the normal form theory and center manifold argument. Some numerical simulations for justifying the theoretical analysis are also provided.
1. Introduction
The predator-prey systems with stage-structure have been studied by many authors [1–5]. In [2], Aiello and Freedman built a stage-structured model of single species: where and denote the densities of the immature population and the mature population at time , respectively. However, it is well known that the predator-prey relationship often appears in population ecology. Multispecies predator-prey systems with stage structure are very important and have received much attention in recent years [6–16].
In [13], Xu investigated a predator-prey system with stage structure for the predator: where denotes the density of the prey at time . and denote the densities of the immature predator and the mature predator at time , respectively. Xu [13] proved that system (2) is permanent under certain conditions by means of the persistence theory on infinite-dimensional systems. And sufficient conditions were derived for the local and global stability of the coexistence equilibrium of the system. In [15], F. Li and H. Li investigated a predator-prey system with stage structure for the prey: where and denote the densities of the immature prey and the mature prey at time , respectively. denotes the density of the predator at time . F. Li and H. Li [15] studied the effect of the gestation time of the predator on the dynamics of system (3).
Since both the predator and the prey have a life history that takes them through immature stage and mature stage, it is reasonable to consider the predator-prey system with stage-structure for both the predator and the prey. Starting from this point, Wang [17] proposed the following delayed system: where and denote the densities of the immature prey and the mature prey at time , respectively. and denote the densities of the immature predator and the mature predator at time , respectively. is the intraspecific competition coefficient of the immature prey. and are the interspecific interaction coefficients between the immature prey and the mature predator. and are the death rates of the immature prey and the mature prey, respectively. and are the death rates of the immature predator and the mature predator, respectively. is the birth rate of the immature prey. and are the transformation rates from immature individuals to mature individuals for the prey and the predator, respectively. All the parameters in system (4) are assumed to be positive constants. And is a constant delay due to the gestation of the mature predator. Wang [17] considered the bifurcation phenomenon and the properties of periodic solutions of system (4).
The predator-prey systems with single delay have been investigated by many researchers. There are also many papers on the bifurcations of predator-prey systems with two or multiple delays [18–24]. Cui and Yan [20] investigated the stability and Hopf bifurcations of a three-species Lotka-Volterra food chain system by taking the sum of the two delays in the system as the bifurcation parameter. Gakkhar and Singh [23] investigated a modified Leslie-Gower predator-prey system with two delays and obtained the sufficient conditions for existence of Hopf bifurcation for possible combinations of the two delays. Motivated by the work above, we consider the following predator-prey system with two delays: where is feedback delay of the immature prey to the growth of the species itself and is the time delay due to the gestation of the mature predator.
The rest of this paper is organized as follows. In Section 2, sufficient conditions are established for the local stability of the positive equilibrium and the existence of Hopf bifurcation for possible combinations of the two delays in system (5). Section 3 is devoted to the properties of the Hopf bifurcation on the normal form theory and center manifold argument. Numerical simulations supporting the theoretical analysis are presented in Section 4. Finally, conclusions are given in Section 5.
2. Local Stability and Existence of Hopf Bifurcation
Considering the significance of ecology, we are interested only in positive equilibrium of system (5). It is not difficult to verify that if the condition : holds, then system (5) has a unique positive equilibrium , where Let , , , and . Dropping the bars, system (5) can be transformed to the following form: where The linear system of system (8) is The characteristic equation of system (10) is as follows: where
Case 1 (). For , (11) can be rewritten in the following form:
where
Clearly, . Therefore, by the Routh-Hurwitz theorem, if the conditions , and hold, then the positive equilibrium of system (5) without time delay is locally asymptotically stable.
Case 2 (). On substituting , (11) becomes where
Let be the root of (15). Then, we have
which follows that
with
Let , then (18) becomes
Define
Discussion about the roots of (20) is similar to that in [25]. Therefore, we have the following lemma.
Lemma 1. For (20), one has the following.(i)If , (20) has at least one positive root.(ii)If and , (20) has positive roots if and only if and .(iii)If and , (20) has positive roots if and only if there exists at least one such that and .
In what follows, one assumes that : the coefficients in satisfy one of the following conditions in :
(a) ;
(b) , , and ;
(c) , , and there exists at least one such that and .
If holds, one knows that (18) has at least a positive root such that (15) has a pair of purely imaginary roots . The corresponding critical value of the delay isDifferentiating both sides of (15) regarding , one gets
Thus,
where . Therefore, if the condition holds, then . Namely, if the condition holds, the transversality condition is satisfied. By the Hopf bifurcation theorem in [26], one has the following results.
Theorem 2. If the conditions hold, then
(i) the positive equilibrium of system (5) is asymptotically stable for ;
(ii) system (5) undergoes a Hopf bifurcation at the equilibrium when , and a family of periodic solutions bifurcate from near .
Case 3 (). On substituting , (11) becomes where Let be the root of (25). Then, we have It follows that where Let , then (28) becomes Define
According to Lemma 1, we know that if the coefficients in satisfy one of the following conditions in :(a′);(b′), , , and ;(c′), , and there exists at least one such that and .
Then, (28) has at least a positive roots such that (25) has a pair of purely imaginary roots . The corresponding critical value of the delay is Similarly as in Case 2, we can conclude that if the condition , then . That is, if the condition holds, the transversality condition is satisfied. According to the Hopf bifurcation theorem in [26], we obtain the following results.
Theorem 3. If the conditions hold, then(i)the positive equilibrium of system (5) is asymptotically stable for ;(ii)system (5) undergoes a Hopf bifurcation at the positive equilibrium when , and a family of periodic solutions bifurcate from near .
Case 4 (). For , (11) can be transformed into the following form: where Multiplying by , (33) becomes Let be the root of (35). Then, we can get Then, we obtain where It is well known that . Therefore, we have where Let , then (39) becomes
If all the parameters of system (5) are given, it is easy to calculate the roots of (41). Thus, we suppose the following:
Equation (41) has at least one positive real root.
Without loss of generality, we assume that (41) has eight positive real roots, which are denoted as , , , , . Then (39) has eight positive roots . And for every fixed , , the corresponding critical value of time delay is Let
Next, we verify the transversality condition. Differentiating both sides of (35) with respect to , we get Thus, where
Obviously, if the condition holds, then . Namely, if the condition holds, the transversality condition is satisfied. In conclusion, we have the following results.
Theorem 4. If the conditions hold, then(i)the positive equilibrium of system (5) is asymptotically stable for ;(ii)system (5) undergoes a Hopf bifurcation at the equilibrium when , and a family of periodic solutions bifurcate from near .
Case 5 5 ( and ). We consider system (5) under Case 2. That is, is in its stable interval and is considered as a parameter. Let be the root of (11). Then, we can get
where
It follows that
with
Suppose that : Equation (49) has at least finite positive real roots. If the condition holds, we denote the positive roots of (49) as , , , . Then, for every fixed , the corresponding critical value of time delay is Let Differentiating (11) regarding , we obtain with Hence, where
Clearly, if the condition holds, then . Thus, we have the following results.
Theorem 5. If the conditions hold and , then(i)the positive equilibrium of system (5) is asymptotically stable for ;(ii)system (5) undergoes a Hopf bifurcation at the equilibrium when , and a family of periodic solutions bifurcate from near .
3. Stability of the Bifurcating Periodic Solutions
In this section, we will determine the direction of Hopf bifurcation and stability of the bifurcated periodic solutions of system (5) with respect to for by using the normal form method and center manifold theorem introduced by Hassard et al. in [25]. Without loss of generality, we assume that , where .
Let , . Then, is the Hopf bifurcation value of system (5). Rescaling the time , then system (5) can be written as where with
Using the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation, such that In fact, we can choose For , we define Then system (57) can be transformed into the following operator equation: where .
The adjoint operator of A is defined by associated with a bilinear inner product: where .
By the discussion above, we know that are eigenvalues of and they are also eigenvalues of . We assume that are the eigenvectors of belonging to the eigenvalue and are the eigenvectors of belonging to the eigenvalue .
By a simple computation, we can get From (65), we can get Normalizing and by the condition and , one can get
Next, we can get the coefficients determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions by the algorithms given in [26]: with where and can be determined by the following equations, respectively:Therefore, we can calculate the following values: By the results of Hassard et al. [26], the following results are obtained.
Theorem 6. For system (5),(i)the direction of the Hopf bifurcation is determined by : if , the Hopf bifurcation is supercritical; if , the Hopf bifurcation is subcritical;(ii)the stability of the bifurcating periodic solutions is determined by : if , the bifurcating periodic solutions are stable; if , the bifurcating periodic solutions are unstable;(iii)the period of the bifurcating periodic solution is determined by : if , the period of the bifurcating periodic solutions increases; if , the period of the bifurcating periodic solutions decreases.
4. Numerical Simulation
In this section, we present some numerical simulations to support the theoretical analysis in Sections 2 and 3. We consider the following system by taking the same coefficients as in [17]: from which we can get , . Obviously, . Therefore, the condition is satisfied. Further, we get the unique positive equilibrium of system (73) by MATLAB software package. Then, we have , , , , and . Obviously, , , and . Namely, the conditions and hold.
For . From (18), we have Then, (20) becomes and it has two positive real roots: , . Thus, the condition is satisfied. Further, we have , . Then, we get . Therefore, the condition holds. From Theorem 2, we know that the positive equilibrium is asymptotically stable for , which can be shown in Figures 1, 2, and 3. The positive equilibrium is unstable when and system (73) undergoes a Hopf bifurcation at , and a family of periodic solutions bifurcate from . This property can be illustrated by Figures 4–6. Similarly, we have , for . For , the positive equilibrium is asymptotically stable from Theorem 3 and this property can be shown in Figures 7, 8, and 9. If , the positive equilibrium is unstable and a Hopf bifurcation occurs, and the corresponding waveforms and phase plots are shown in Figures 10, 11, and 12.
For , we can get . From Theorem 4, we know that the positive equilibrium is asymptotically stable for , which can be illustrated by Figures 13, 14, and 15. As can be seen from Figure 5 that when , the positive equilibrium is asymptotically stable. However, if , then the positive equilibrium becomes unstable and a family of bifurcating periodic solutions occur, which is illustrated by Figures 16, 17, and 18.
Lastly, regarding as a parameter and letting , we can obtain that . Further, we have and . Letting , we can know that the positive equilibrium is asymptotically stable from Theorem 5. When , then the positive equilibrium becomes unstable and a Hopf bifurcation occurs. The corresponding waveforms and phase plots are illustrated in Figures 19, 20, 21, 22, 23, and 24.
In addition, for system (73), we have by a simple computation. Further, from (72), we have , , and . Therefore, by Theorem 6, we can know that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the bifurcated periodic solutions decrease. Since the bifurcating periodic solutions of system (73) are stable, the predator and the prey species in system (73) can coexist in an oscillatory mode from the view of biology.
5. Conclusions
In this paper, a delayed predator-prey system with stage-structure for both the predator and the prey population is investigated. The sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation for the possible combinations of two delays are obtained. The main results are given in Theorems 2–5. The results show that the two time delays can play a complicated role in system (5). Furthermore, the explicit formulae determining the direction of the bifurcation and the stability of the bifurcating periodic solutions are established when and by using the normal form theory and center manifold theorem. The main results are given in Theorem 6. If the bifurcating periodic solutions are stable, the predator and the prey species can coexist in an oscillatory mode from the view of biology. Compared with literature [17], we consider not only the time delay due to the gestation of the mature predator but also the negative feedback delay of the immature prey. Wang [17] obtained that the species in system (5) with only the time delay due to the gestation of the mature predator could coexist under some conditions. However, we get that the species could also coexist with some available time delays of the immature prey and the mature predator under certain conditions. This is valuable from the view of ecology. Some numerical simulations are also included to support the obtained theoretical results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Acknowledgments
This work was supported by the National Nature Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2013A003, KJ2013B137).