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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 628103, 19 pages
http://dx.doi.org/10.1155/2012/628103
Research Article

A Predator-Prey Model with Functional Response and Stage Structure for Prey

1School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China
2Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 24 September 2011; Accepted 25 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Xiao-Ke Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A predator-prey system with Holling type II functional response and stage structure for prey is presented. The local and global stability are studied by analyzing the associated characteristic transcendental equation and using comparison theorem. The existence of a Hopf bifurcation at the positive equilibrium is also studied. Some numerical simulations are also given to illustrate our results.

1. Introduction

Predator-prey system is a very classical and important mathematical model in population dynamics, there are many studies in progress nowadays (see [35]). In recent years, many scholars are giving their attention to predator-prey model with stage structure, and many results have been published. In [6, 7], the influences of prey or predator with stage structure to the state of the ecosystem are studied, respectively. A time-delay model of single-species growth incorporating stage structure as a reasonable generation of the logistic model was derived in [8]. This model assumes an average age to maturity which appears as a constant time delay reflecting a delayed birth of immatures and a reduced survival of immatures to their maturity. The model takes the form where and denote the immature and mature population densities, respectively; represents the birth rate of the immature population; and are the death rate of immature and mature population, respectively; denotes the length of time from the birth to maturity. It is shown in [8] that the introduction of stage structure does not affect the permanence of the species, but, under some hypotheses, it may maximize the total carrying capacity of the population.

Based on the previous model, several predator-prey models with stage structure are developed and studied (see [915]). Recently, Song et al. [16] investigated the following model with stage structure for prey: Their results showed that the positive equilibrium of system (1.2) was globally stable under some conditions.

On the other hand, the functional response plays an important role in all prey interactions. A functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. One significant component of the predator-prey relationship is the predator’s functional responses. When the predator spends some time searching for prey and some time, exclusive of searching, processing each captured prey item (i.e., handling time), Holling type II is a useful functional response (see [1721]). For example, Sun et al. [22] considered the following model with Holling type II functional response and stage structure for predator:

By analyzing the corresponding characteristic equations, the local stability of equilibria and existence of a Hopf bifurcation at the positive equilibrium are obtained in [22].

In this paper, we consider and study the following model with stage structure for prey: For continuity of the initial conditions, we further require

The remainder of the paper is organized as follows: in the next section, results on positivity and boundedness of solutions are presented. In Section 3, through using geometric stability switch criteria [1] to analyze the corresponding characteristic equations, we discuss the local stability of equilibria. Further, we study the existence of a Hopf bifurcation at the positive equilibria. The global stability of boundary equilibrium is obtained by using comparison theorem in Section 4. In Section 5, we try to interpret our results by numerical simulation. At last, we give some discussions.

2. Positive and Bounded

In this section, we discuss the positivity and boundedness of the solutions of system (1.4). Positivity means that the species is persistent and boundedness implies a natural restriction, which shows our model (1.4) is reasonable.

Theorem 2.1. The solutions of system (1.4) with initial conditions (1.5) are positive for all .

Proof. It is easy to see that and are positive for all .
We only show .
Since hence Considering the equation we have on , and from variation-of-constants formula, we get hence, Thus, on . By induction, we can show that for all . The proof is complete.

Theorem 2.2. Every solution of system (1.4) with initial conditions (1.5) are bounded for all , and all of these solutions are ultimately bounded.

Proof. Let , calculating the derivative of with respect to along the positive solution of system (1.4), we have For a small positive constant , Hence there exists a positive constant , such that thus we get Therefore, is ultimately bounded, that is, each solution of system (1.4) is ultimately bounded. The proof is complete.

3. Equilibria Analysis

Note that, in the system (1.4), the equation for the variable has a form If is bounded and as , then as ; that is, the asymptotic behavior of is dependent on that of . Therefore, in following sections we just need to study the asymptotic behavior of the following system:

By a straightforward calculation, we can get two nonnegative equilibria:

and , exists if and only if(): holds.

Further, if(): and () are satisfied, then system (3.2) has a unique positive equilibrium .

Remark 3.1. () is equivalent to , where . From (), we have and ; hence, if is a positive equilibrium, there must be In some of the subsequent sections, we always assume that (3.3) is valid; otherwise, there is no positive equilibrium of system (3.2).

3.1.

First, we analyze the stability of equilibrium , and the associated characteristic equation has the form

Theorem 3.2. If , then the equilibrium of system (3.2) is uniformly asymptotically stable.
If , then the equilibrium of system (3.2) is(i)unstable for all (see Remark 3.1),(ii)stable, but not asymptotically stable when ,(iii)asymptotically stable for all .

Proof. has always negative real part. Denote .
Let in , then we have that if , then that is, the trivial solution of system (3.2) is stable when there is no delay . If the increasing of leads to instability of system (3.2), then there is an such that for some . Note that Hence, But, from hypothesis , we know that (3.9) has no real root. Hence, the equilibrium of system (3.2) is uniformly asymptotic stable for any delay .
If , then the trivial solution of system (3.2) is unstable when there is no delay .
(i) When , we know that (3.9) has a positive root . For , let be defined by where, jointly with (3.9), we define the following maps:
If has no roots in when , then we know that the equilibrium of system (3.2) is unstable through using geometric stability switch criteria [1]. Otherwise, let , then which straightforwards implies (i).
(ii) When , we have , and is always a simple eigenvalue. Suppose that has a root , where , then a contradiction. Hence, (ii) is true.
(iii) When , (3.9) has no positive root, which means that the stability of does not change as increases. Let , where is a positive constant; we consider the following characteristic equation with delay-independent coefficients:
By Theorem [2, page 66] we know that (3.14) has only negative real-part eigenvalues; thus, (iii) is valid. The proof is complete.

Remark 3.3. From Theorem 3.2 we know that when the birth rate of prey species is less than death rate of itself, then the prey species tend to be extinct; therefore, the predator species reduces to zero since there is no enough foods, which accords with the basic evolution law of biology. We also see that time delay can stabilize an unstable system and keep this state to large enough. This phenomenon cannot appear in the case that the coefficients of a corresponding characteristic equation are independent of the delay.

3.2.

Now, we analyze the stability of . The characteristic equation of system (3.2) at takes the form

Denote and .

Obviously, the root of equation is positive when and negative when ; hence, we only need to consider . By the same method with the proof of Theorem 3.2, we know that has only negative real-part characteristic roots when .

Summarizing the discussion above, we get the following conclusion.

Theorem 3.4. If () is satisfied, then the equilibrium of system (3.2) is
unstable when (see Remark 3.1),
stable, but not asymptotically stable when ,
asymptotically stable when .

3.3.

Finally, we analyze the stability of positive equilibrium of system (3.2). The linearizing equation of system (3.2) at is hence, the characteristic equation takes the form where

Note that these are dependent on time delay since includes . In order to apply the geometric criterion of Beretta and Kuang [1], we rewrite into where

For convenience, let , then is strictly monotone increasing function in . If , hence, for all . Otherwise, that is, if , since , there is a unique such that .

Lemma 3.5. If () and (H3): hold, then and for all .
If (H2) and (H4): hold, then for all , when and when .

Proof. Obviously, . From (3.18), we have (1)If , then for all ; hence, for all .(2)If , then when and when ; hence, is true. The proof is complete.

Step 1. When , (3.17) becomes by Routh-Hurwitz Criterion we know that has only negative real-part eigenvalue if () holds and has positive real-part eigenvalue if () holds. Hence, we obtain the following theorem.

Theorem 3.6. If () and () are satisfied, then the equilibrium of system (3.2) is asymptotically stable when .
If () and () are satisfied, then the equilibrium of system (3.2) is unstable when .

Step 2. Suppose for some and , we get
Hence, where
Now, we take hypotheses () or () into account, respectively.

Case 1 (). In this case, since Thus, if , then (3.24) has no positive real root, and if , then and (3.24) has no real root, that is, the stability of equilibrium cannot change for all . Summarizing the discussion above, we obtain the following theorem.

Theorem 3.7. Assume that () and () are satisfied, then the equilibrium of system (3.2) is asymptotically stable for all .

Case 2 (). In this case, when , by the similar analysis we also obtain that the stability of equilibrium cannot change for all . Let , where is a positive constant; we consider the following characteristic equation with delay-independent coefficients: where and . By Theorem [2, page 66] we know that (3.27) has only negative real-part eigenvalues, then the equilibrium is asymptotically stable for all .
When , since , hence, , and ; we easily know that (3.24) has two positive real roots and , where
For , let be defined by where, jointly with (3.23), we define the following maps:
Obviously, , , and , which imply
Furthermore, we note that and (because of ); thus, equation has at least one root in ; let , then that is, the pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right. Summarizing the discussion above, we obtain the following result.

Theorem 3.8. Assume that () and () are satisfied; if equation has no root in , then the equilibrium of system (3.2) is unstable for all and becomes asymptotically stable for staying in some right neighborhood of ; hence, system (3.2) undergoes Hopf bifurcation when . The equilibrium of system (3.2) is asymptotically stable for .

4. Global Stability

In this section, we discuss the global stability of boundary equilibria. In some of the subsequent analysis, we will need the following result.

Lemma 4.1 (see [16]). Consider the following equation: for , then one has the following.(i)If , then .(ii)If , then .

Theorem 4.2. Assume that , then the equilibrium of system (3.2) is globally asymptotically stable.

Proof. We know that if , then is locally asymptotically stable. Now we show
From the first equation of system (3.2), we get
By Lemma 4.1 and comparing principle, we obtain that is, for the , there exists a such that which jointly with the second equation of system (3.2) leads to
Since the arbitrariness of , we have The proof is complete.

Theorem 4.3. Assume that () is satisfied, then the equilibrium of system (3.2) is globally asymptotically stable for .

Proof. We need to show attractivity. From the first equation of system (3.2), we get
By Lemma 4.1 and comparing principle, we obtain that is, for the , there exists a such that which jointly with the second equation of system (3.2) leads to
On the other hand, for , we have
Thus, for is small enough, there is which leads to that is, for the , there exists a such that hence, we have by hypothesis (), Lemma 4.1, and the arbitrariness of , there is
The proof is complete.

5. Numerical Simulation

In this section, we present some numerical simulations of system (3.2) to illustrate our theoretical analysis.

Example 5.1. Let , and , and we consider the following system: By calculating, we have that , and hypotheses () and () hold. By the discussion above, we obtain following corollary.

Corollary 5.2. For system (5.1), when is asymptotically stable, is globally stable when , and is globally stable when .

First, we take and initial value is that

Figure 1 illustrates that prey and predator converge asymptotically to zero, when and initial value is that .

628103.fig.001
Figure 1: The equilibrium is globally stable when .

Figure 2 illustrates that prey and predator converge asymptotically to , where .

628103.fig.002
Figure 2: The equilibrium is globally stable when .

Finally, we take ; system (5.1) has positive equilibrium .

Figure 3 shows that the positive solutions converge asymptotically to the steady-state .

628103.fig.003
Figure 3: The equilibrium is asymptotically stable when .

Example 5.3. In order to illustrate Theorem 3.8, we choose a set of parameters as , and , then system (3.2) reduces to By calculating, we have that , and hypotheses () and () hold.
Figure 4 shows that equation has no root and equation has a unique root , where , which leads to the following corollary.

628103.fig.004
Figure 4: The graphs of and .

Corollary 5.4. The equilibrium of system (5.3) is unstable for all and becomes asymptotically stable for staying in some right neighborhood of ; hence, system (5.3) undergoes Hopf bifurcation when .

By Corollary 5.4, we know that the positive equilibrium is unstable and positive solutions are oscillatory when , the bifurcating periodic solutions exist at least for the value of slightly larger than the critical value , the computer simulation is given in Figure 5. We continue to choose , by Corollary 5.4 we know that the positive equilibrium becomes asymptotically stable, and the computer simulation is depicted in Figure 6. In Figures 7 and 8, we plot the relationship between delay and ultimate amplitudes of and , which indicate the fact that the positive equilibrium is from instability to stability as increases.

628103.fig.005
Figure 5: The bifurcating periodic solutions from when .
628103.fig.006
Figure 6: The positive equilibrium is stable when .
628103.fig.007
Figure 7: The relationship between ultimate oscillation range of mature prey and delay .
628103.fig.008
Figure 8: The relationship between ultimate oscillation range of predator and delay .

6. Discussion

In this paper, a model which describes the Holling type predator-prey system with stage structure for prey is proposed. Sufficient conditions which ensure the stability of equilibria and the existence of Hopf bifurcation are obtained. Numerical results show the feasibility of our results.

By the analysis, we find the system (1.4) has complex dynamics behavior. Our results indicate that the stability switches of the positive equilibrium may occur as time delay increases and delay can stabilize system (1.4). Under condition (), the positive equilibrium is stable only if delay is larger, consequently, becomes smaller, that is, predator coexists with prey in lower population densities (with respect to delay , which is different from [22]. In [22], predator coexists with prey only if the population densities of prey are larger and those of predator are lower (with respect to delay . Moreover, Liu and Zhang [6] investigated system (1.2) when functional response is Beddington-DeAngelis type, by monotone dynamic theories, they obtained that the positive equilibrium is globally asymptotically stable provided that (describing the magnitude of interference among predators) is more than a positive constant; that is, their results cannot directly apply to study system (1.4). Furthermore, they did not consider , the death rate of mature prey population, which play an important role on stability of the equilibria and existence of Hopf bifurcation. Hypothesis () means that , where , and Hypothesis () means that . Hence, from Theorem 3.7 and Theorem 3.8, we know that can also stabilize system (1.4).

There are still many interesting and challenging mathematical questions that need to be studied for system (1.4). For example, we do not discuss the global stability of positive equilibrium. We will leave this for future work.

Acknowledgments

This work was partially supported by the NNSF of China (10961018), the NSF of Gansu Province of China (1107RJZA088), the NSF for Distinguished Young Scholars of Gansu Province of China (1111RJDA003), the Special Fund for the Basic Requirements in the Research of University of Gansu Province of China and the Development Program for Hong Liu Distinguished Young Scholars in Lanzhou University of Technology and the Scientific Research Foundation for the Young and Middle-Aged Teachers in Tianshui Normal University (TSA0937).

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