##
**Weak average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input.**
*(English)*
Zbl 1133.92032

Summary: We investigated a predator-prey system in a polluted environment with impulsive toxicant input at fixed moments. We obtained two thresholds on the impulsive period by assuming the toxicant amount input is fixed to the environment at each pulse moment. If the impulsive period is greater than the big threshold, then both populations are weakly average persistent. If the period lies between of the two thresholds, then the prey population will be weakly average persistent while the predator population goes extinct. If the period is less than the small threshold, both populations tend to extinction. Finally, our theoretical results are confirmed by own numerical simulations.

### MSC:

92D40 | Ecology |

34A37 | Ordinary differential equations with impulses |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

PDF
BibTeX
XML
Cite

\textit{X. Yang} et al., Chaos Solitons Fractals 31, No. 3, 726--735 (2007; Zbl 1133.92032)

Full Text:
DOI

### References:

[1] | Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, Pitman monographs surveys pure appl math, 66, (1993) · Zbl 0793.34011 |

[2] | Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 |

[3] | Hodges, L., Environmental pollution, (1977), Holt Rinehart and Winston |

[4] | Freedman, H.I.; Shukla, T.B., Models for the effect of toxicant in single-species and predator-prey systems, J math biol, 30, 15-30, (1991) · Zbl 0825.92125 |

[5] | Hallam, T.G.; Clark, C.E.; Lassider, R.R., Effects of toxicants on populations: a qualitative approach I. equilibrium environment exposure, Ecol model, 8, 291-304, (1984) |

[6] | Hallam, T.G.; Clark, C.E.; Jordan, G.S., Effects of toxicants on populations: a qualitative approach II. first order kinetics, J math biol, 18, 25-37, (1983) · Zbl 0548.92019 |

[7] | Hallam, T.G.; Deluna, J.T., Effects of toxicants on populations: A qualitative approach III. environment and food chain pathways, J theor biol, 109, 411-429, (1984) |

[8] | Mukherjee, D., Persistence and global stability of a population in a polluted environment with delay, J boil syst, 10, 225-232, (2002) · Zbl 1099.92074 |

[9] | Dubey, B., Modelling the interaction of two biological species in a polluted environment, J math anal, 246, 58-79, (2000) · Zbl 0952.92030 |

[10] | Liu, H.P.; Ma, Z., The threshold of survival for system of two species in a polluted environment, J math biol, 30, 49-61, (1991) · Zbl 0745.92028 |

[11] | Ma, Z.; Cui, G.; Wang, W., Persistence and extinction of a population in a polluted environment, Math biosci, 101, 75-97, (1990) · Zbl 0714.92027 |

[12] | Shukla, J.B.; Dubey, B., Simultaneous effect of two toxicants on biological species: a mathematical model, J biol syst, 4, 109-130, (1996) |

[13] | Thomas, P.M.; Snell, T.W.; Joffers, M., A control problem in a polluted environment, Math biosci, 133, 139-163, (1996) · Zbl 0844.92026 |

[14] | Lu, Z.; Chi, X.; Chen, L., Impulsive control strategies in biological control of pesticide, Theor population biol, 64, 39-47, (2003) · Zbl 1100.92071 |

[15] | Liu, X.; Chen, L., Global dynamics of the periodic logistic system with periodic impulsive perturbations, Math anal appl, 289, 279-291, (2004) · Zbl 1054.34015 |

[16] | Jin, Z.; Han, M.; Ma, Z., The existence of periodic solutions of the n-species Lotka-Volterra competition systems with impulsive, Chaos, solitons & fractals, 22, 181-188, (2004) · Zbl 1058.92046 |

[17] | Yang, X.; Jin, Z.; Xue, Y., Changes of toxin in the organism in a polluted environment with impulsive toxicant input, Istm2005, 4, (2005) |

[18] | Liu, B.; Chen, L.; Zhang, Y., The effects of impulsive toxicant input on a population in a polluted environment, J biol syst, 11, 3, 265-274, (2003) · Zbl 1041.92044 |

[19] | Lakmeche, A.; Arino, O., Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynam continuous discrete impulsive syst, 7, 265-287, (2000) · Zbl 1011.34031 |

[20] | Panetta, J.C., A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull math biol, 58, 425-447, (1996) · Zbl 0859.92014 |

[21] | Lin, W., Description of complex dynamics in a class of impulsive differential equations, Chaos, solitons & fractals, 25, 1007-1017, (2005) · Zbl 1198.34014 |

[22] | Gao, S.; Chen, L.; Sun, L., Optimal pulse fishing policy in stage-structured models with birth pulses, Chaos, solitons & fractals, 25, 1209-1219, (2005) · Zbl 1065.92056 |

[23] | Zhang, S.; Chen, L., A Holling II functional response food chain model with impulsive perturbations, Chaos, solitons & fractals, 24, 1269-1278, (2005) · Zbl 1086.34043 |

[24] | Gao, S.; Chen, L., The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses, Chaos, solitons & fractals, 24, 1013-1023, (2005) · Zbl 1061.92059 |

[25] | Jin, Z.; Han, M.; Li, G., The persistence in a Lotka-Volterra competition systems with impulsive, Chaos, solitons and fractals, 24, 1105-1117, (2005) · Zbl 1081.34045 |

[26] | Gakkhar, S.; Singh, B., Complex dynamic behavior in a food web consisting of two preys and a predator, Chaos, solitons & fractals, 24, 789-801, (2005) · Zbl 1081.37060 |

[27] | Zhang, S.; Chen, L., Chaos in three species food chain system with impulsive perturbations, Chaos, solitons & fractals, 24, 73-83, (2005) · Zbl 1066.92060 |

[28] | Gao, S.; Chen, L., Dynamic complexities in a single-species discrete population model with stage structure and birth pulses, Chaos, solitons & fractals, 23, 519-527, (2005) · Zbl 1066.92041 |

[29] | Zhang, S.; Dong, L.; Chen, L., The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons & fractals, 23, 631-643, (2005) · Zbl 1081.34041 |

[30] | Moghadas, S.M.; Alexander, M.E., Dynamics of a generalized gause-type predator-prey model with a seasonal functional response, Chaos, solitons & fractals, 23, 55-65, (2005) · Zbl 1058.92049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.