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Periodic Solutions of a Nonautonomous Plant-Hare Model with Impulses

DOI: 10.1155/2013/360182

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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


[1]  L. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing, China, 1998, (in Chinese).
[2]  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.
[3]  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.
[4]  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.
[5]  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.
[6]  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.
[7]  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.
[8]  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.
[9]  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.
[10]  D. Ba?nov 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.
[11]  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.
[12]  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.
[13]  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.
[14]  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.
[15]  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.
[16]  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.
[17]  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.
[18]  R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.


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