## Abstract and Applied Analysis

### Global Positive Periodic Solutions of Generalized $n$-Species Competition Systems with Multiple Delays and Impulses

#### Abstract

By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized $n$-species competition systems with multiple delays and impulses as follows: ${x}_{i}^{\prime }(t)={x}_{i}(t)[{a}_{i}(t)-{b}_{i}(t){x}_{i}(t)-{\sum }_{j=\mathrm{1}}^{n}{c}_{ij}(t){x}_{i}(t-{\tau }_{ij}(t))-{\sum }_{j=\mathrm{1}}^{n}{d}_{ij}(t){x}_{j}(t-{\gamma }_{ij}(t))-{\sum }_{j=\mathrm{1}}^{n}{e}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(s){x}_{j}(t+s)ds],\text{\hspace\{0.17em\}a}\text{.}\text{e}\text{.},\mathrm{ t}>\mathrm{0},t\ne {t}_{k},\mathrm{ k}\in {Z}_{+},\mathrm{ i}=\mathrm{1,2},\dots ,n;{\mathrm{ x}}_{i}({t}_{k}^{+})-{x}_{i}({t}_{k}^{-})={\theta }_{ik}{x}_{i}({t}_{k}),\mathrm{ i}=\mathrm{1,2},\dots ,n,\mathrm{ k}\in {Z}_{+};\text{\hspace\{0.17em\}\hspace\{0.17em\}and\hspace\{0.17em\}\hspace\{0.17em\}}{x}_{i}^{\prime }(t)={x}_{i}(t)[{a}_{i}(t)-{b}_{i}(t){x}_{i}(t)+{\sum }_{j=\mathrm{1}}^{n}{c}_{ij}(t){x}_{i}(t-{\tau }_{ij}(t))-{\sum }_{j=\mathrm{1}}^{n}{d}_{ij}(t){x}_{j}(t-{\gamma }_{ij}(t))-{\sum }_{j=\mathrm{1}}^{n}{e}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(s){x}_{j}(t+s)ds],\text{\hspace\{0.17em\}a}\text{.}\text{e}\text{.},\mathrm{ t}>\mathrm{0},\mathrm{ t}\ne {t}_{k},\mathrm{ k}\in {Z}_{+},\mathrm{ i}=\mathrm{1,2},\dots ,n;\mathrm{}{\mathrm{ x}}_{i}({t}_{k}^{+})-{x}_{i}({t}_{k}^{-})={\theta }_{ik}{x}_{i}({t}_{k}),\mathrm{ i}=\mathrm{1,2},\dots ,n,\mathrm{ k}\in {Z}_{+}.$ It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 980974, 12 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450111

Digital Object Identifier
doi:10.1155/2013/980974

Mathematical Reviews number (MathSciNet)
MR3093763

Zentralblatt MATH identifier
1294.34078

#### Citation

Luo, Zhenguo; Luo, Liping. Global Positive Periodic Solutions of Generalized $n$ -Species Competition Systems with Multiple Delays and Impulses. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 980974, 12 pages. doi:10.1155/2013/980974. https://projecteuclid.org/euclid.aaa/1393450111

#### References

• D. D. Bainov and P. S. Simeonov, System With Impulsive Effect: Stability, Theory and Applications, John Wiley & Sons, New York, NY, USA, 1989.
• V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
• A. M. Samoikleno and N. A. Perestyuk, Impulsive Differential Equations, vol. 28, World Scientific, Singapore, 1995.
• D. Bainov and P. Simeonov, Oscillation Theory of Impulsive Differential Equations, International Publications, 1998.
• J. R. Yan and A. M. Zhao, “Oscillation and stability of linear impulsive delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 187–194, 1998.
• G. Ballinger and X. Liu, “Existence, uniqueness and boundedness results for impulsive delay differential equations,” Applicable Analysis, vol. 74, no. 1-2, pp. 71–93, 2000.
• M. S. Peng and W. G. Ge, “Oscillation criteria for second-order nonlinear differential equations with impulses,” Computers & Mathematics with Applications, vol. 39, no. 5-6, pp. 217–225, 2000.
• B. Liu and J. Yu, “Existence of solution of $m$-point boundary value problems of second-order differential systems with impulses,” Applied Mathematics and Computation, vol. 125, no. 2-3, pp. 155–175, 2002.
• X. N. Liu and L. S. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons and Fractals, vol. 16, no. 2,pp. 311–320, 2003.
• H.-F. Huo, W.-T. Li, and X. Liu, “Existence and global attractivity of positive periodic solution of an impulsive delay differential equation,” Applicable Analysis, vol. 83, no. 12, pp. 1279–1290, 2004.
• J. Yan, “Existence of positive periodic solutions of impulsive functional differential equations with two parameters,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 854–868, 2007.
• M. Z. He, Z. Li, and F. D. Chen, “Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 11, no. 3, pp. 1537–1551, 2010.
• M. E. Gilpin and F. J. Ayala, “Global model of growth and competition,” Proceedings of the National Academy of Sciences United States of America, vol. 70, no. 2, pp. 3590–3593, 1973.
• M. Fan and K. Wang, “Periodic solutions of single population model with hereditary effect,” Mathematica Applicata, vol. 13, no. 2, pp. 58–61, 2000.
• H. I. Freedman and J. H. Wu, “Periodic solutions of single-species models with periodic delay,” SIAM Journal on Mathematical Analysis, vol. 23, no. 3, pp. 689–701, 1992.
• C. Alvarez and A. C. Lazer, “An application of topological degree to the periodic competing species problem,” Australian Mathematical Society, vol. 28, no. 2, pp. 202–219, 1986.
• S. Ahmad, “Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations,” Journal of Mathematical Analysis and Applications, vol. 127, no. 2, pp. 377–387, 1987.
• Z. Liu, M. Fan, and L. Chen, “Globally asymptotic stability in two periodic delayed competitive systems,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 271–287, 2008.
• D. J. Guo, Nonlinear Functional Analysis, ShanDong Science and Technology Press, 2001.
• M. A. Krasnoselskii, Positive Solution of Operator Equation, Noordhoff, Grőningen, The Netherlands, 1964.
• K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
• D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.