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**Dynamics of mutualism-competition-predator system with Beddington-DeAngelis functional responses and impulsive perturbations.**
*(English)*
Zbl 1253.92046

Summary: We explore the dynamics of a class of mutualism-competition-predator interaction models with Beddington-DeAngelis functional responses and impulsive perturbations. Sufficient conditions for existence of positive periodic solution are established by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are given for the global stability and the globally exponential stability of system by employing comparison principle and Lyapunov method.

### MSC:

92D40 | Ecology |

34A37 | Ordinary differential equations with impulses |

37N25 | Dynamical systems in biology |

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\textit{X. Fan} et al., Abstr. Appl. Anal. 2012, Article ID 963486, 33 p. (2012; Zbl 1253.92046)

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### References:

[1] | G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling type II model,” Ecology, vol. 82, no. 11, pp. 3083-3092, 2001. |

[2] | Z. Zeng, L. Bi, and M. Fan, “Existence of multiple positive periodic solutions for functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1378-1389, 2007. · Zbl 1110.34043 |

[3] | A. Hastings, “Multiple limit cycles in predator-prey models,” Journal of Mathematical Biology, vol. 11, no. 1, pp. 51-63, 1981. · Zbl 0471.92022 |

[4] | F. Brauer and A. C. Soudack, “Coexistence properties of some predator-prey systems under constant rate harvesting and stocking,” Journal of Mathematical Biology, vol. 12, no. 1, pp. 101-114, 1982. · Zbl 0482.92015 |

[5] | M. Bohner, M. Fan, and J. Zhang, “Existence of periodic solutions in predator-prey and competition dynamic systems,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1193-1204, 2006. · Zbl 1104.92057 |

[6] | R. S. Cantrell and C. Cosner, “On the dynamics of predator-prey models with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 206-222, 2001. · Zbl 0991.34046 |

[7] | S. Froda and S. Nkurunziza, “Prediction of predator-prey populations modelled by perturbed ODEs,” Journal of Mathematical Biology, vol. 54, no. 3, pp. 407-451, 2007. · Zbl 1119.92064 |

[8] | S. Froda and A. Zahedi, “Simple testing procedures for the Holling type II model,” Theoretical Ecology, vol. 2, no. 3, pp. 149-160, 2009. |

[9] | M. R. Garvie and C. Trenchea, “Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response,” Numerische Mathematik, vol. 107, no. 4, pp. 641-667, 2007. · Zbl 1132.65092 |

[10] | J. R. Beddington, “Mutual interference between parasites or predator and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331-340, 1975. |

[11] | D. L. DeAngelis, R. A. Goldstein, and R. V. Neill, “A model for trophic interaction,” Ecology, vol. 56, pp. 881-892, 1975. |

[12] | S. Sivasundaram and S. Vassilyev, “Stability and attractivity of solutions of differential equations with impulses at fixed times,” Journal of Applied Mathematics and Stochastic Analysis, vol. 13, no. 1, pp. 77-84, 2000. · Zbl 0995.34041 |

[13] | V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific Publishing, Singapore, 1989. · Zbl 0719.34002 |

[14] | A. Anokhin, L. Berezansky, and E. Braverman, “Exponential stability of linear delay impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 193, no. 3, pp. 923-941, 1995. · Zbl 0837.34076 |

[15] | A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing, Singapore, 1995. |

[16] | A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, Visca Skola, Kiev, Ukraine, 1987. |

[17] | S. G. Hristova, “Integral stability in terms of two measures for impulsive functional differential equations,” Mathematical and Computer Modelling, vol. 51, no. 1-2, pp. 100-108, 2010. · Zbl 1190.34091 |

[18] | M. U. Akhmetov and N. A. Perestyuk, “Almost periodic solutions of nonlinear sampled-data systems,” Ukrainian Mathematical Journal, vol. 41, no. 3, pp. 259-263, 1989. · Zbl 0693.34054 |

[19] | M. De la Sen, “Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 621-650, 2006. · Zbl 1111.93072 |

[20] | M. De La Sen and N. Luo, “A note on the stability of linear time-delay systems with impulsive inputs,” IEEE Transactions on Circuits and Systems I, vol. 50, no. 1, pp. 149-152, 2003. · Zbl 1368.93272 |

[21] | J. Xu and J. Sun, “Finite-time stability of linear time-varying singular impulsive systems,” IET Control Theory & Applications, vol. 4, no. 10, pp. 2239-2244, 2010. · Zbl 1359.93397 |

[22] | Y. Zhang and J. Sun, “Stability of impulsive linear differential equations with time delay,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 10, pp. 701-705, 2005. |

[23] | X. Fan, F. Jiang, and H. Zhang, “Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 10, pp. 3363-3378, 2011. · Zbl 1213.92059 |

[24] | S. Ahmad and I. M. Stamova, “Asymptotic stability of an N-dimensional impulsive competitive system,” Nonlinear Analysis: Real World Applications, vol. 8, no. 2, pp. 654-663, 2007. · Zbl 1152.34342 |

[25] | X.-z. He and K. Gopalsamy, “Persistence, attractivity, and delay in facultative mutualism,” Journal of Mathematical Analysis and Applications, vol. 215, no. 1, pp. 154-173, 1997. · Zbl 0893.34036 |

[26] | B. S. Goh, “Stability in models of mutualism,” The American Naturalist, vol. 113, no. 2, pp. 261-275, 1979. |

[27] | R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977. · Zbl 0339.47031 |

[28] | S. Ahmad and I. M. Stamova, “Asymptotic stability of competitive systems with delays and impulsive perturbations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 686-700, 2007. · Zbl 1153.34044 |

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