## 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

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

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