## Recent Developments and Applications on Discrete Fractional Equations and Related Topics

View this Special IssueResearch Article | Open Access

# Periodic Solutions of a Nonautonomous Plant-Hare Model with Impulses

**Academic Editor:**Thabet Abdeljawad

#### Abstract

A plant-hare model subjected by the effect of impulses is studied in this paper. Sufficient conditions are obtained for the existence of at least one positive periodic solution.

#### 1. Introduction

Classical predator-prey model has been well studied (e.g., see [1–8] and the references cited therein). To explore the impact of plant toxicity on the dynamics of plant-hare interactions, Gao and Xia [9] consider a nonautonomous plant-herbivore dynamical system with a toxin-determined functional response: where denotes the density of plant at time , denotes the herbivore biomass at time , is the plant intrinsic growth rate at time , is the per capita rate of herbivore death unrelated to plant toxicity at time , is the conversion rate at time , is the encounter rate per unit plant, is the fraction of food items encountered that the herbivore ingests, is the carrying capacity of plant, measures the toxicity level, and is the time for handing one unit of plant. To explore the impact of environmental factors (e.g., seasonal effects of weather, food supplies, mating habits, harvesting, etc.), the assumption of periodicity of parameters is more realistic and important. To this reason, they assumed that , , and are continuously positive periodic functions with period and , , , , are five positive real constants.

However, birth of many species is an annual birth pulse, for having more accurate description of the system, we need to consider using the impulsive differential equations. To see how impulses affect the differential equations, for examples, one can refer to [10–17]. Motivated by the above-mentioned works, in this paper, we consider the above system with impulses: where the assumptions on , , , , , , , and are the same as before, , is a strictly increasing sequence with , and . We further assume that there exists a such that and for .

Without loss of generality, we will assume for , and ; hence .

#### 2. Preliminaries

In this section, we cite some definitions and lemmas.

Let denote the space of -periodic functions which are continuous for , are continuous from the left for , and have possible discontinuities of the first kind at points ; that is, the limit from the right of exists but may be different from the value at . We also denote .

For the convenience, we list the following definitions and lemmas.

*Definition 1 (see [10]). *The set is said to be quasi-equicontinuous in if for any there exists a such that if ; ; and , then

Lemma 2 (see [10]). *The set is relatively compact if and only if *(1)* is bounded, that is, , for each , and some ;*(2)* is quasi-equicontinuous in .*

Lemma 3 (see [11]). *Assume that , then the following inequality holds:
*

*Before starting the main result, for the sake of convenience, one denotes*

#### 3. Existence of Positive Periodic Solutions

In order to obtain the existence of positive periodic solutions of (2), for convenience, we will summarize in the following a few concepts and results from [18] that will be basic for this section.

Let , be normed vector spaces, let be a linear mapping, and a continuous mapping. The mapping is called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, there exist continuous projectors and such that , . It follows that is invertible. We denote the inverse of that map by . If is an open bounded subset of , then the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .

Lemma 4 (see [18]). *Let be an open and bounded set. Let be a Fredholm mapping of index zero and let be -compact on . Assume*(a)* for each , , ;*(b)* for each , ;*(c)*.** Then has at least one solution in .*

If is a continuous -periodic function, then we set The following assumptions are valid throughout this paper:, , .

For convenience, we introduce two numbers as follows: where .

Theorem 5. *In addition to (), (), suppose that **. **Then system (2) has at least one positive -periodic solution.*

*Remark 6. *If the impulsive operators disappear, then . Then Theorem 5 reduces to the main results in Gao and Xia [9]. This implies that our result generalizes the previous one. It shows that the impulses do affect the system indeed.

*Proof. *Making the change of variables
Then, system (2) can be rewritten as
Take
and define
Both and are Banach spaces.

Define
,
, ; ,
It is not difficult to show that
Since is closed in , and are continuous projectors such that
It follows that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) exists, which is given by
Then and are defined by
Clearly, and are continuous. By using the Arzela-Ascoli theorem (see [10]), it is not difficult to prove that is compact for any open bounded set . Moreover, is bounded. Therefore, is -compact on with any open bounded set .

Now, we reach the position to search for an appropriate open, bounded subset for the application of the continuation theorem.

Corresponding to the operator equation , , we have
Suppose is a solution of (19) for a certain . Integrating the first equation of (19) over the interval , we obtain
Similarly, integrating the second equation of (19) over the interval , we obtain
It follows from the first equation of (19) and (20) and that
That is,
Similarly, it follows from the second equation of (19) and (21) and that
Since , there exists such that

From (20), we see that
which implies
So
This, combined with (23), gives
Similarly, it follows from (21) that
which implies
It follows from that
This, combined with (23), gives
It follows from (29) and (33) that

On the other hand, it follows from (21) and (34) that
which implies
It follows from () that
This, combined with (25), gives

Similarly, it follows from (21) and (34) that
which implies
So
This, combined with (25), gives
It follows from (38) and (42) that
Now, let us consider with . Note that
It follows from , , and that , which implies that the equation has only one solution
Choose such that
Set ; then . Let
It is clear that verifies the requirement (a) in Lemma 4. When , is a constant with . Then for . Simple computation shows that . Here, is taken as the identity mapping since .

By now, we have proved that verifies all the requirements in Lemma 4. Hence, (2) has at least one -periodic solution in .

#### Acknowledgment

This work is supported by NNSFC and the Natural Science Foundation of Fujiang Province (2013J01010).

#### References

- L. Chen,
*Mathematical Models and Methods in Ecology*, Science Press, Beijing, China, 1998, (in Chinese). - Y. Li, “Periodic solutions of a periodic delay predator-prey system,”
*Proceedings of the American Mathematical Society*, vol. 127, no. 5, pp. 1331–1335, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Song, Y. Peng, and J. Wei, “Bifurcations for a predator-prey system with two delays,”
*Journal of Mathematical Analysis and Applications*, vol. 337, no. 1, pp. 466–479, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y.-H. Xia, “Periodic solution of certain nonlinear differential equations: via topological degree theory and matrix spectral theory,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 22, no. 8, Article ID 940287, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Zhang, J. Liu, and Z. Teng, “Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 293–306, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhang and J. Luo, “Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 5, pp. 4109–4120, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Fan, Q. Wang, and X. Zou, “Dynamics of a non-autonomous ratio-dependent predator-prey system,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 133, no. 1, pp. 97–118, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-F. Huo, “Periodic solutions for a semi-ratio-dependent predator-prey system with functional responses,”
*Applied Mathematics Letters*, vol. 18, no. 3, pp. 313–320, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y.-F. Gao and Y.-H. Xia, “Periodic solutions of a nonautonomous plant-hare model,”
*Journal of Zhejiang University*, vol. 39, no. 5, pp. 507–511, 2012. View at: Google Scholar | MathSciNet - D. Baĭnov and P. Simeonov,
*Impulsive Differential Equations: Periodic Solutions and Applications*, vol. 66 of*Pitman Monographs and Surveys in Pure and Applied Mathematics*, Longman Scientific & Technical, Harlow, UK, 1993. View at: MathSciNet - Q. Wang, B. Dai, and Y. Chen, “Multiple periodic solutions of an impulsive predator-prey model with Holling-type IV functional response,”
*Mathematical and Computer Modelling*, vol. 49, no. 9-10, pp. 1829–1836, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Mahto, S. Abbas, and A. Favini, “Analysis of Caputo impulsive fractional order differential equations with applications,”
*International Journal of Differential Equations*, vol. 2013, Article ID 704547, 11 pages, 2013. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. M. A. El-Sayed, E. Ahmed, and H. A. A. El-Saka, “Dynamic properties of the fractional-order logistic equation of complex variables,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 251715, 12 pages, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. Hui and L. S. Chen, “Existence of positive periodic solution of periodic time-dependent predator-prey system with impulsive effects,”
*Acta Mathematica Sinica (English Series)*, vol. 20, no. 3, pp. 423–432, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Shao, P. Li, and G. Tang, “Dynamic analysis of an impulsive predator-prey model with disease in prey and Ivlev-type functional response,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 750530, 20 pages, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. O. Alzabut and T. Abdeljawad, “Exponential boundedness for solutions of linear impulsive differential equations with distributed delay,”
*International Journal of Pure and Applied Mathematics*, vol. 34, no. 2, pp. 203–217, 2007. View at: Google Scholar | MathSciNet - J. O. Alzabut and T. Abdeljawad, “On existence of a globally attractive periodic solution of impulsive delay logarithmic population model,”
*Applied Mathematics and Computation*, vol. 198, no. 1, pp. 463–469, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. E. Gaines and J. L. Mawhin,
*Coincidence Degree, and Nonlinear Differential Equations*, vol. 568 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1977. View at: MathSciNet

#### Copyright

Copyright © 2013 Haihui Wu and Yan Zhou. 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.