## Abstract and Applied Analysis

### Global Attractivity of an Integrodifferential Model of Mutualism

#### Abstract

Sufficient conditions are obtained for the global attractivity of the following integrodifferential model of mutualism: $d{N}_{\mathrm{1}}\mathrm{}(t)/dt={r}_{\mathrm{1}}{N}_{\mathrm{1}}(t)[(({K}_{\mathrm{1}}+{\alpha }_{\mathrm{1}}\mathrm{}{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{2}}\mathrm{}(s){N}_{\mathrm{2}}\mathrm{}(t-s)ds)\mathrm{‍}/(\mathrm{1}+{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{2}}\mathrm{}(s){N}_{\mathrm{2}}\mathrm{}(t-s)ds))\mathrm{‍}-{N}_{\mathrm{1}}(t)]$, $d{N}_{\mathrm{2}}\mathrm{}(t)/dt={r}_{\mathrm{2}}{N}_{\mathrm{2}}(t)[(({K}_{\mathrm{2}}+{\alpha }_{\mathrm{2}}\mathrm{}{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{}{J}_{\mathrm{1}}(s){N}_{\mathrm{1}}\mathrm{}(t-s)ds)\mathrm{‍}/(\mathrm{1}+{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{1}}\mathrm{}(s){N}_{\mathrm{1}}\mathrm{}(t-s)ds\mathrm{‍}))-{N}_{\mathrm{2}}(t)]$, where ${r}_{i},{K}_{i},$ and ${\alpha }_{i}$, $i=\mathrm{1,2}$, are all positive constants. Consider ${\alpha }_{i}>{K}_{i}$, $i=\mathrm{1,2}.$ Consider ${\mathrm{ J}}_{i}\in C([\mathrm{0},+\mathrm{\infty }),[\mathrm{0},+\mathrm{\infty }))$ and ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}{J}_{i}(s)ds=\mathrm{1}$, $i=\mathrm{1,2}.$ Our result shows that conditions which ensure the permanence of the system are enough to ensure the global stability of the system. The result not only improves but also complements some existing ones.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 928726, 6 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425048257

Digital Object Identifier
doi:10.1155/2014/928726

Mathematical Reviews number (MathSciNet)
MR3193560

Zentralblatt MATH identifier
07023330

#### Citation

Xie, Xiangdong; Chen, Fengde; Yang, Kun; Xue, Yalong. Global Attractivity of an Integrodifferential Model of Mutualism. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 928726, 6 pages. doi:10.1155/2014/928726. https://projecteuclid.org/euclid.aaa/1425048257

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