Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.
Since the pioneering theoretical works by Lotka  and Volterra , there were a lot of authors who studied all kinds of predator-prey models modeled by ordinary differential equations (ODEs) [3–5]. To reflect that the dynamical behavior of the models depends on the past history of the system, it is often necessary to incorporate time delays into the models. Therefore, a more realistic predator-prey model should be described by delayed differential equations (DDEs) [6–26]. Some of them investigated discrete delays [6–20]; others were about distributed delays [21–24]; and both discrete and distributed delays were studied in . In general, delay differential equations exhibit more complicated dynamics on stability, periodic structure, bifurcation, and so on .
In the natural world, many individuals have a life story that takes them through two stages, immature and mature. The predator only catches the mature prey, as the immature preys are protected by their eggshells or refuge. Some predator-prey models with stage structure were investigated in [27–33]. Motivated by [25, 27, 31] and the references cited therein, in the present paper, we will consider the following stage-structured predator-prey model with discrete and distributed delay: where , , and can be interpreted as the population densities of the immature prey, mature prey, and predator at time , respectively. denotes the birth rate of the prey population; , , and denote the death rate of the immature prey, mature prey, and the predator; is the density-depended death rate of the predator; denotes the per capita per unit time predation rate of the predator; the term is the conversion rate from prey to predator, and the distributed delay may interpret as digest delay. The function is called the delayed kernel that is a nonnegative bounded function defined on . Following the ideas of Cushing et al. , we define as the following weak kernel function:
Next, we define a new variable: Then by use of linear chain trick technique, system (1) can be transformed into the following equivalent system:
The organization of this paper is as follows: In Section 2, we will get the conditions for the existence and stability of positive equilibrium of system (4). The occurring condition for Hopf bifurcation is also obtained. In Section 3, by use of normal form theory and central manifold argument, we illustrate the direction and stability of Hopf bifurcation. In Section 4, we give some brief discussion.
2. Stability of Positive Equilibrium and Existence of Hopf Bifurcation
In this section, we will firstly investigate the existence and stability of positive equilibrium of system (4) then study the effect of time delay on the system (4); that is, we will choose as bifurcating parameter to analyze Hopf bifurcation.
Theorem 1. There exists a unique positive equilibrium for system (4), if assumption holds. And , with
Note that when , (7) becomes It is easy to confirm that Thus, by the Routh-Hurwitz criterion we know that all the roots of (9) have negative real parts, which means that the positive equilibrium is locally asymptotically stable for .
Next, we will consider the case for . Suppose that there is a pure imaginary root , . Then we get
Separating the real and imaginary parts, we have Incorporating , we have where
In fact, if all the parameters of system (4) are given, it is easy to calculate the root of (15). Since , we conclude that if , then (15) has at least one positive real root. Without loss of generality, we assume that (15) has four positive root, defined by , , , , respectively. Then (13) has four positive roots as From (12), we obtain
Denote Then can be written as from which we can get
Thus, is a pair of purely imaginary root of (7). Define
In order to obtain the main result, it is necessary to make the following assumption:(H3).
Taking the derivative of with respect to in (7), it is easy to obtain and it may be rewritten as or equivalently, we have
Taking into the above equation, we get
Denote Then we have
Lemma 2 (see ). Consider the exponential polynomial where and are constants. As vary, the sum of the order of the zeros of on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
From Lemma 2, it is easy to obtain the following theorem.
Theorem 3. Suppose that (H1), (H2), and (H3) hold. Then the following results hold true.(i)The positive equilibrium of system (4) is asymptotically stable for .(ii)The positive equilibrium of system (4) undergoes a Hopf bifurcation when . That is system (4) has a periodic solution bifurcating from the positive equilibrium near .
3. Direction and Stability of the Hopf Bifurcation
In this section, we will derive the explicit formulae for determining the properties of the Hopf bifurcation at critical value of by using the normal form and the center manifold theory . Throughout this section, we always assume that system (4) undergoes Hopf bifurcation at the positive equilibrium for , and then is the corresponding purely imaginary roots of the characteristic equation at the positive equilibrium .
Let , , , , , and , and dropping the bars for simplification of notations, then system (4) is transformed into FDE defined in as where , , and where . By the Riesz representation theorem, there exists a function of bounded variation for , such that
In fact, we can choose where is the Dirac delta function. For , define Then system (32) is equivalent to where . For , define and a bilinear inner product where . Then and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of . Thus, they are also eigenvalues of . We need to compute the eigenvector of and corresponding to and , respectively.
Suppose that is the eigenvectors of corresponding to . Then . It follows from the definition of and that Because of , then we get from which we obtain
Similarly, let be the eigenvectors of corresponding to , and according to the definition of we have Note that . Then we get from which we can obtain By (40), we get Then we can choose such that , .
Nest, we will use the ideas in  to compute the coordinates describing center manifold at . Define
On the center manifold , we have , and where and are local coordinates for center manifold in the direction of and . Note that is real if is real, and we only consider real solutions. For the solution of (38), since and (38), we have Then, the above equation can be denoted as where From (48) and (49), we have and then we can obtain
From the definition of , we have
Comparing the coefficients with those of (52), we obtain
Note that on the center manifold near the origin and thus we obtain
By (58), we know that, for ,
Comparing the coefficients with those in (59), we get
Noting that , we get where is a constant vector.
By (34), we have
Thus, we obtain
Thus, we can determine and from (65) and (66). Furthermore, can be expressed by the parameters and delay. Thus, we can compute the following values: which determine the qualities of bifurcating periodic solution in the center manifold at critical value ; that is, determines the direction of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for determines the stability of the bifurcating periodic solution: the bifurcating periodic solution is stable (unstable) if ; and determines period of the bifurcating periodic solution: the period increases (decreases) if .
In this paper, we propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delay. Then, by introducing a new variable, the original system is transformed into an equivalent four-dimensional system (4). In Section 2, we analyze the existence and local stability of the positive equilibrium of the system (4). In fact, by use of the Routh-Hurwitz criterion, we know that the positive equilibrium is locally asymptotically stable for . Then, by some computation, we get a threshold in Theorem 3. The results indicate that if , the positive equilibrium of system (4) is asymptotically stable; if , the positive equilibrium of system (4) will undergo a Hopf bifurcation. In Section 3, by use of normal form theory and central manifold argument, we establish the formulae for the direction and the stability of the Hopf bifurcation. Our theoretical results show that stage structure and delay play an important role in the dynamics of the system, and delay may lead to complicated dynamic behaviors, such as Hopf bifurcation.
The first author is supported by Postdoctoral Science Foundation of China (no. 2011M501428) and Young Science Funds of Shanxi (no. 2013021002-2). The third author is supported by National Natural Science Foundation of China (no. 11171199). The authors would like to thank the referees of this paper for their helpful comments.
A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, Md, USA, 1925.
V. Volterra, “Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,” Memorie dell'accademia dei Lincei, vol. 2, pp. 31–113, 1926.View at: Google Scholar
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, Mass, USA, 1993.View at: MathSciNet
J. M. Cushing, Integro-Differential Equations and Delay Models in Population Dynamics, Springer, Berlin, Germany, 1977.View at: MathSciNet
B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.View at: MathSciNet