Discrete Dynamics in Nature and Society

Volume 2011, Article ID 380279, 19 pages

http://dx.doi.org/10.1155/2011/380279

## Dynamics of a Nonautonomous Leslie-Gower Type Food Chain Model with Delays

School of Mathematics and Quantitative Economics, Dongbei University of Finance & Economics, Dalian, Liaoning 116025, China

Received 11 October 2010; Accepted 26 January 2011

Academic Editor: M. De la Sen

Copyright © 2011 Hongying Lu and Weiguo Wang. 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 nonautonomous Leslie-Gower type food chain model with time delays is investigated. It is proved the general nonautonomous system is permanent and globally asymptotically stable under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness, and global asymptotic stability of a positive periodic solution of the system. The conditions for the permanence, global stability of system, and the existence, uniqueness of positive periodic solution depend on delays; so, time delays are profitless.

#### 1. Introduction

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. Food-chain predator-prey system, as one of the most important predator-prey system, has been extensively studied by many scholars, many excellent results concerned with the permanent property and positive periodic solution of the system, see [2–4] and the references cited therein. Recently, Nindjin and Aziz-Alaoui [5] proposed the following autonomous delayed predator-prey model with modified Leslie-Gower functional response system (1) represents an ecological situation where a prey population is the only food for a predator . This specialist predator , in turns, serves as the prey of a top-predator . The interaction between species and its prey has been modeled by the Volterra scheme. But, the interaction between species and its prey has been modeled by a modified version of Leslie-Gower scheme. About Leslie-Gower scheme one could refer to [6–13] and the references cited therein.

In [5], the authors showed that the system is uniformly persistent under some appropriate conditions and obtained sufficient conditions for global stability of the positive equilibrium of system (1).

We note that any biological or environment parameters are naturally subject to fluctuation in time, and if a model is to take into account such fluctuation then the model must be nonautonomous. On the other hand, time delays occur so often in nature, a number of models can be formulated as systems of differential equations with time delays (see, e.g., [2, 14, 15] and the references cited therein). Motivated by above considerations, in this paper, we consider the following general nonautonomous Leslie-Gower type food chain model with time delays of the form where , denote the time delays due to negative feedbacks of the prey, specialist predator and top-predator, respectively. is a time delay due to gestation, that is, mature adult predators can only contribute to the reproduction of predator biomass. can be regarded as a gestation period.

In this paper, for system (1.2) we always assume that for all are continuous and bounded above and below by positive constants on , and are continuous and differentiable bounded functions on , and is uniformly continuous with respect to on and .

Let , then we have . Let , then the function is the inverse function of the function , . Motivated by the application of system (1.2) to population dynamics, we assume that solutions of system (1.2) satisfies the following initial conditions

It is well known that by the fundamental theory of functional differential equation [16], one can prove the solution of system (1.2) with initial conditions (1.3) exists and remains positive for , we call such a solution the positive solution of system (1.2).

The organization of this paper is as follows. In Section 2, by using comparison theorem and further developing the analytical technique of [2, 14], we obtain a set of sufficient conditions, which ensure the permanence of the system (1.2). In Section 3, by constructing a suitable Lyapunov function, we establish a set of sufficient conditions, which ensure the global stability of the system (1.2). In Section 4, we will explore the existence and stability of the solutions of the periodic system (1.2). At last, the conclusion ends with brief remarks.

#### 2. Permanence

In this section, we establish a permanent result for system (1.2).

Here, for any bounded function

*Definition 2.1. *System (1.2) is said to be permanent, if there are positive constants and , such that each positive solution of system (1.2) satisfies

Theorem 2.2. *Assume that and hold assume further that**, **hold. Then system (1.2) is permanent.*

*Proof. *From the first equation of the system (1.2) it follows that

Let , by , then we have . Taking , where . Firstly, suppose is not oscillatory about . That is, there exists a , for such that
or

If (2.4) holds, then our aim is obtained. Suppose (2.5) holds, then for , we obtain
thus , as , which is contradiction with (2.5). Hence there must exist such that for . Secondly now assume that is oscillatory about for , that is, there exists a time sequence such that is a sequence of zeros of with and . Set a point where attends its maximum in . Thus we get . Then it follows from (2.3) that
which leads to
Integrating the both sides of (2.3) from to , it follows that
From (2.8) and (2.9) we have

Since is an arbitrary local maximum of , we can see that there exists a such that for all
Thus
For , from (2.11) and the second equation of the system (1.2) it follows that
Similar to the argument above, from (2.13) we obtain
Similarly, from the third equation of the system (1.2), we have
and so
Condition of Theorem 2.2 also implies that we could choose small enough such that
hence, for satisfies (2.17), from (2.12), (2.14), and (2.16), we know that there exists such that for and
From the first equation and (2.18) it follows that for ,
Note that implies that
Now we consider the following two cases.*Case 1. *Suppose , then for , from the positivity of the solution and (2.19) it follows that
Then from (2.12) and (2.21) it follows that , then there exists such that for
From the positivity of the solution, (2.19) and (2.22) it follows that
then we can see that as , which is contradiction with (2.12). Hence we have , which implies that there exists such that for .*Case 2. *Suppose , from (2.19), for , it follows that

Let
where .Firstly, suppose is not oscillatory about . That is, there exists a , for such that
or

If (2.26) holds, then our aim is obtained. Suppose (2.27) holds, then for , we obtain
thus there must exist such that for , which is a contradiction. Hence, (2.27) could not hold. Secondly now assume that is oscillatory about for , that is, there exists a time sequence such that is a sequence of zeros of with and . Set be a point where attends its minimum in . Thus, we get . Then it follows from (2.24) that
which implies that
Integrating (2.24) on the interval , we have
From (2.30) and (2.31), we get that

Since is an arbitrary local minimum of , we can have that there exists a such that for all
where
Taken , thus, we have
Thus, for satisfies (2.17), it follows from (2.35) that there exists large enough such that for

For , by using (2.36), from the second equation of system (1.2) it follows that
from (2.37), by a procedure similar to the discussion above, we can verify that
where
From (2.38), we see that there exists large enough such that for
Substituting (2.18) to the last equation of system (1.2), it follows that
from (2.41), similar to the argument of (2.35), we also have
where
Consequently, (2.12), (2.14), (2.16), (2.35), (2.38), and (2.42) show that under the assumption , for any positive solution of system (1.2), one has
where and , are independent of the solution of system (1.2), thus system (1.2) is permanent. This completes the proof of Theorem 2.2.

#### 3. Global Stability

Now we study the global stability of the positive solution of system (1.2). We say a positive solution of system (1.2) is globally asymptotically stable if it attracts all other positive solution of the system.

The following lemma is from [17], and will be employed in establishing the global stability of positive solution of system (1.2).

Lemma 3.1. *Let be a real number and be a nonnegative function defined on such that is integrable on and is uniformly continuous on , then .*

Theorem 3.2. *In addition to , assume further that**there exist constants such that **
where
**
Then for any positive solutions and of system (1.2), one has
*

*Proof. *For two arbitrary nontrivial solutions and of system (1.2), we have from Theorem 2.2 that there exist positive constants and , such that for all and
We define
Calculating the upper right derivative of along the solution of system (1.2), for , it follow that
On substituting (1.2) into (3.6), we derive that
It follows (3.4) and (3.7) that for

Let , by , we can obtain the inverse function of the function denoted by .

Define
We obtain from (3.8) and (3.9) for
We now define
where
It then follows from (3.10) that for
Similarly, we define
where
Calculating the upper right derivative of along solutions of (1.2), we derive that for
Similarly, we define
where
Calculating the upper right derivative of along solutions of (1.2), we derive that for
We now define a Lyapunov functional as
It then follows from (3.13), (3.16), (3.19), and (3.20) that for
By the hypothes is , there exist enough small positive constants , and a large enough constant , such that for all and
Integrating both sides of (3.21) on interval
It follows from (3.22) and (3.23) that
Therefore, is bounded on and also
By Theorem 2.2, we know that , are bounded on . On the other hand, it is easy to see that , are bounded for . Therefore, , are uniformly continuous on . By Lemma 3.1, one can conclude that
This completes the proof of Theorem 3.2.

*Remark 3.3. *In the proof of the stability of system (1.2), we construct the Lyapunov functional, which need the condition . Similar method and conditon can be found in the [2, 14].

#### 4. Existence and Stability of the Positive Periodic Solutions

In this section, we suppose that all the coefficients in system (1.2) are continuous and positive -periodic functions, then the system (1.2) is an -periodic system for this case.

We let the following denote the unique solution of periodic system (1.2) for initial value :

Now define Poincâre transformation is In this way, the existence of periodic solution of system (1.2) will be equal to the existence of the fixed point .

Theorem 4.1. *Assume that the conditions of hold, then system (1.2) with initial condition (1.3) has at least one positive -periodic solution.*

*Proof. *If assumption are satisfied, then from Theorem 2.2 we have that there exist positive constants and such that for all and
Let
then the compact region is a positive invariant set of system (1.2), and is also a close bounded convex set. So we have
also , thus . The operator is continuous because the solution is continuous about the initial value. Using the fixed point theorem of Brower, we can obtain that has at least one fixed point in , then there exists at least one strictly positive -periodic solution of system (1.2). This ends the proof of Theorem 4.1.

By constructing similar Lyapunov functional to those of Theorem 3.2, and using Theorem 4.1, we have the following theorem.

Theorem 4.2. *Assume that the conditions of hold, then system (1.2) has a unique positive -periodic solution which is globally asymptotically stable.*

#### 5. Concluding Remarks

In this paper, a nonautonomous Leslie-Gower type food chain model with time delays is investigated, which is based on the Holling type II and a Leslie-Gower modified functional response. By using comparison theorem, we prove the system is permanent under some appropriate conditions. Further, by constructing the suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. If the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness and global asymptotic stability of a positive periodic solution of the system. Our results have showed that the permanence, global stability of system and the existence, uniqueness of positive periodic solution depend on delays, and so time delays are profitless.

#### Acknowledgments

The authors are grateful to the Associate Editor, Manuel De la Sen, and two referees for a number of helpful suggestions that have greatly improved our original submission. This work is supported by the National Natural Science Foundation of China (no. 70901016) and Excellent Talents Program of Liaoning Educational Committee (no. 2008RC15).

#### References

- A. A. Berryman, “The origins and evolution of predator-prey theory,”
*Ecology*, vol. 73, no. 5, pp. 1530–1535, 1992. View at Google Scholar - X. Meng, J. Jiao, and L. Chen, “Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 68, no. 12, pp. 3633–3645, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Xu and M. A. J. Chaplain, “Persistence and global stability in a delayed predator-prey system with Michaelis-Menten type functional response,”
*Applied Mathematics and Computation*, vol. 130, no. 2-3, pp. 441–455, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments,”
*Nonlinear Analysis. Real World Applications*, vol. 5, no. 1, pp. 183–206, 2004. View at Publisher · View at Google Scholar - A. F. Nindjin and M. A. Aziz-Alaoui, “Persistence and global stability in a delayed Leslie-Gower type three species food chain,”
*Journal of Mathematical Analysis and Applications*, vol. 340, no. 1, pp. 340–357, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Aziz-Alaoui, “Study of a Leslie-Gower-type tritrophic population model,”
*Chaos, Solitons and Fractals*, vol. 14, no. 8, pp. 1275–1293, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Aziz-Alaoui and M. Daher Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,”
*Applied Mathematics Letters*, vol. 16, no. 7, pp. 1069–1075, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay,”
*Nonlinear Analysis. Real World Applications*, vol. 7, no. 5, pp. 1104–1118, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulation,”
*Memoirs of the Entomological Society of Canada*, vol. 45, pp. 1–60, 1965. View at Google Scholar - P. H. Leslie, “Some further notes on the use of matrices in population mathematics,”
*Biometrika*, vol. 35, pp. 213–245, 1948. View at Google Scholar · View at Zentralblatt MATH - P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator-prey type of interaction between two species,”
*Biometrika*, vol. 47, pp. 219–234, 1960. View at Google Scholar · View at Zentralblatt MATH - C. Letellier and M. A. Aziz-Alaoui, “Analysis of the dynamics of a realistic ecological model,”
*Chaos, Solitons and Fractals*, vol. 13, no. 1, pp. 95–107, 2002. View at Publisher · View at Google Scholar - R. K. Upadhyay and S. R. K. Iyengar, “Effect of seasonality on the dynamics of 2 and 3 species prey-predator systems,”
*Nonlinear Analysis. Real World Applications*, vol. 6, no. 3, pp. 509–530, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Meng and L. Chen, “Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay,”
*Journal of Mathematical Analysis and Applications*, vol. 339, no. 1, pp. 125–145, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. De la Sen, “Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 321, no. 2, pp. 621–650, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 2nd edition, 1977. - I. Barbălat, “Systems d'equations differential d'oscillations nonlinearities,”
*Revue Roumaine De Mathematiques Pures et Appliquees*, vol. 4, no. 2, pp. 267–270, 1959. View at Google Scholar