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Existence and Uniqueness of Positive (Almost) Periodic Solutions for a Neutral Multi-Species Logarithmic Population Model with Multiple Delays and Impulses

**Author(s)**Leave a comment

In this paper, by using the contraction mapping principle and
constructing a suitable Lyapunov functional, we established a set of easily
applicable criteria for the existence, uniqueness and global attractivity of
positive periodic solution and positive almost periodic solution of a neutral
multi-species Logarithmic population model with multiple delays and impulses.
The results improve and generalize the known ones in [1], as an
application, we also give an example to illustrate the feasibility of our main
results.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Open Journal of Applied Sciences*, Vol. 3 No. 2, 2013, pp. 247-262. doi: 10.4236/ojapps.2013.32032.

[1] | Q. Wang, Y. Wang and B. X. Dai, “Existence and Uniqueness of Positive Periodic Solutions for a Neutral Logarithmic Population Model,” Applied Mathematics and Computation, Vol. 213, No. 1, 2009, pp. 137-147. doi:10.1016/j.amc.2009.03.028 |

[2] | L. Berezansky and E. Braverman, “Explicit Conditions of Exponential Stability for a Linear Impulsive Delay Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 214, No. 2, 1997, pp. 439-458. doi:10.1006/jmaa.1997.5578 |

[3] | H. Fang, “Positive Periodic Solutions of N-Species Neutral Delay Systems,” Czechoslovak Mathematical Journal, Vol. 53, No. 3, 2003, pp. 561-570. doi:10.1023/B:CMAJ.0000024503.03321.b1 |

[4] | F. D. Chen, “On a Nonlinear Nonautonomous Predator— Prey Model with Diffusion and Distributed Delay,” Journal of Computational and Applied Mathematics, Vol. 180, No. 1, 2005, pp. 33-49. doi:10.1016/j.cam.2004.10.001 |

[5] | H. Fang and J. B. Li, “On the Existence of Periodic Solutions of a Neutral Delay Model of Single-Species Popula tion Growth,” Journal of Mathematical Analysis and Applications, Vol. 259, No. 1, 2001, pp. 8-17. doi:10.1006/jmaa.2000.7340 |

[6] | F. D. Chen, “Periodic Solutions and Almost Periodic Solutions for a Delay Multispecies Logarithmic Population Model,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 760-770. doi:10.1016/j.amc.2005.01.085 |

[7] | Z. H. Yang and J. D. Cao, “Positive Periodic Solutions of Neutral Lotka-Volterra System with Periodic Delays,” Applied Mathematics and Computation, Vol. 149, No. 3, 2004, pp. 661-687. doi:10.1016/S0096-3003(03)00170-X |

[8] | Y. K. Li, “On a Periodic Neutral Delay Logarithmic Population Model,” Journal of Systems Science and Complexity, Vol. 19, No. 1, 1999, pp. 34-38. (in Chinese) |

[9] | S. P. Lu and W. G. Ge, “Existence of Positive Periodic Solutions for Neutral Logarithmic Population Model with Multiple Delays,” Journal of Computational and Applied Mathematics, Vol. 166, No. 2, 2004, pp. 371-383. doi:10.1016/j.cam.2003.08.033 |

[10] | F. D. Chen, “Periodic Solutions and Almost Periodic Solutions of a Neutral Multispecies Logarithmic Population Model,” Applied Mathematics and Computation, Vol. 176, No. 2, 2006, pp. 431-441. doi:10.1016/j.amc.2005.09.032 |

[11] | R. E. Gaines and J. L. Mawkin, “Coincidence Degree and Nonlinear Differential Equations,” Springer-Verlag, Berlin, 1977. |

[12] | X. Z. Liu and G. Ballinger, “Existence and Continuability of Solutions for Differential Equations with Delay and State-Dependent Impulses,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 51, No. 4, 2002, pp. 633-647. doi:10.1016/S0362-546X(01)00847-1 |

[13] | B. G. Zhang and Y. J. Liu, “Global Attractivity for Certain Impulsive Delay Differential Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 52, No. 3, 2003, pp. 725-736. doi:10.1016/S0362-546X(02)00129-3 |

[14] | Y. J. Zhang, B. Liu and L. S. Chen, “Extinction and Permanence of a Two-Prey One-Predator System with Impulsive Effect,” Mathematical Medicine and Biology, Vol. 20, No. 4, 2003, pp. 309-325. doi:10.1093/imammb/20.4.309 |

[15] | J. R. Yan, “Existence of Positive Periodic Solutions of Impulsive Functional Differential Equations with Two Parameters,” Journal of Mathematical Analysis and Applications, Vol. 327, No. 2, 2007, pp. 854-868. doi:10.1016/j.jmaa.2006.04.018 |

[16] | W. M. Wang, H. L. Wang and Z. Q. Li, “Chaotic Behavior of a Three-Species Beddington-Type System with Impulsive Peturbations,” Chaos, Solitons & Fractals, Vol. 37, No. 2, 2008, pp. 438-443. doi:10.1016/j.chaos.2006.09.013 |

[17] | J. S. Yu, “Explicit Conditions for Stability of Nonlinear Scalar Delay Differential Equations with Impulses,” Non linear Analysis: Theory, Methods & Applications, Vol. 46, No. 1, 2001, pp. 53-67. doi:10.1016/S0362-546X(99)00445-9 |

[18] | 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 & Fractals, Vol. 16, No. 2, 2003, pp. 311-320. doi:10.1016/S0960-0779(02)00408-3 |

[19] | Y. J. Liu and W. G. Ge, “Stability Theorems and Existence Results for Periodic Solutions of Nonlinear Impulsive Delay Differential Equations with Variable Coefficients,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 57, No. 3, 2004, pp. 363-399. doi:10.1016/j.na.2004.02.020 |

[20] | 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, 1998, pp. 187-194. doi:10.1006/jmaa.1998.6093 |

[21] | V. Lakshmikantham, D. Bainov and P. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific Publisher, Singapore City, 1989. |

[22] | A. M. Samoikleno and N. A. Perestyuk, “Impulsive Differential Equations,” World Scientific, Singapore City, 1995. |

[23] | J. J. Nieto and R. Rodriguez-Lopez, “Periodic Boundary Value Problem for Non-Lipschitzian Impulsive Functional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 318, No. 2, 2006, pp. 593-610. doi:10.1016/j.jmaa.2005.06.014 |

[24] | H. F. Huo, “Existence of Positive Periodic Solutions of a Neutral Delay Lotka-Volterra Systems with Impulses,” Computers & Mathematics with Applications, Vol. 48, No. 12, 2004, pp. 1833-1846. doi:10.1016/j.camwa.2004.07.009 |

[25] | A. Ouahab, “Existence and Uniqueness Results for Impulsive Functional Differential Equations with Scalar Multiple Delay and Infinite Delay,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 67, No. 4, 2007, pp. 1027-1041. doi:10.1016/j.na.2006.06.033 |

[26] | F. D. Chen, “On the Existence and Uniqueness of Periodic Solutions of a Kind of Integro-Differential Equations,” Acta Mathematica Sinica, Vol. 47, 2004, pp. 973-985. (in Chinese) |

[27] | H. Q. Xie and Q. Y. Wang, “Exponential Stability and Periodic Solution for Cellular Neural Networks with Time Delay,” Journal of Huaqiao University, Vol. 25, 2004, pp. 22-26. (in Chinese) |

[28] | S. P. Lu and W. G. Ge, “Existence of Positive Periodic Solutions for Neutral Population Model with Multiple Delays,” Applied Mathematics and Computation, Vol. 153, No. 3, 2004, pp. 885-892. doi:10.1016/S0096-3003(03)00685-4 |

[29] | Y. H. Xia, “Positive Periodic Solutions for a Neutral Impulsive Delayed Lotka-Volterra Competition Systems with the Effect of Toxic Substance,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 1, 2007, pp. 204-221. doi:10.1016/j.nonrwa.2005.07.002 |

[30] | L. Barbalat, “Systemes d’Equations Differentielles d’Oscil lations Nonlineaires,” Revue Roumaine de Mathématique Pures et Appliquées, Vol. 4, 1959, pp. 267-270. |

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