Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances.

*(English)*Zbl 1257.92040Summary: On the basis of the theories and methods of ecology and ordinary differential equations, a three-species ecological model with allochthonous nutrient input, impulse perturbations and seasonal disturbances is studied analytically and numerically. Using mathematical theoretical analysis, we obtain the threshold expression of the release amount, allochthonous nutrient input and seasonal disturbances parameter under the condition of some species extinction and all species persistence, which in turn provide a theoretical basis for the numerical simulation.

Numerical analysis indicates that the key factors for long-term dynamical behavior are impulse perturbation and allochthonous nutrient input with seasonal disturbances. Nonetheless, it should be stressed that the allochthonous nutrient input with seasonal disturbances can aggravate periodic oscillations and promote the emergence of chaos.These results show that impulse perturbations cannot prevent the indirect effect on complex population dynamics caused by allochthonous nutrient input with seasonal disturbances, which further confirm that allochthonous nutrient input with seasonal disturbances can play an important role in population persistence and evolutionary. All these results are expected to be useful in the study of complex dynamics of ecosystems.

Numerical analysis indicates that the key factors for long-term dynamical behavior are impulse perturbation and allochthonous nutrient input with seasonal disturbances. Nonetheless, it should be stressed that the allochthonous nutrient input with seasonal disturbances can aggravate periodic oscillations and promote the emergence of chaos.These results show that impulse perturbations cannot prevent the indirect effect on complex population dynamics caused by allochthonous nutrient input with seasonal disturbances, which further confirm that allochthonous nutrient input with seasonal disturbances can play an important role in population persistence and evolutionary. All these results are expected to be useful in the study of complex dynamics of ecosystems.

##### MSC:

92D40 | Ecology |

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

37N25 | Dynamical systems in biology |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

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##### References:

[1] | Baek, H.; Do, Y., Seasonal effects on a beddington – deangelis type predator – prey system with impulsive perturbations, Abstract and applied analysis, 19, (2009), Article ID 695121 · Zbl 1187.34058 |

[2] | Baek, H.; Do, Y.; Saito, Y., Analysis of an impulsive predator-prey system with monod – haldane functional response and seasonal effects, Mathematical problems in engineering, 16, (2009), Article ID 543187 · Zbl 1180.92085 |

[3] | Baek, H.; Sang, P., Permanence and stability of an ivlev-type predator – prey system with impulsive control strategies, Mathematical and computer modelling, 50, 1385-1393, (2009) · Zbl 1185.34067 |

[4] | Bainov, D.D.; Covachev, V.C., Impulsive differential equations with a small parameter, (1994), World Scientific Singapore · Zbl 0828.34001 |

[5] | Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: asymptotic properties of the solutions, (1993), World Scientific Singapore · Zbl 0793.34011 |

[6] | Faria, L.; Costa, M., Omnivorous food web, prey preference and allochthonous nutrient input, Ecological complexity, 7, 107-114, (2010) |

[7] | Hou, J.; Teng, Z., Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates, Mathematics and computers in simulation, 79, 3038-3054, (2009) · Zbl 1166.92034 |

[8] | Hsu, B.; Hwang, T.W.; Kuang, Y., Global analysis of the michaelis – menten-type ratio-dependent predator – prey system, Journal of mathematical biology, 42, 489-506, (2001) · Zbl 0984.92035 |

[9] | Huxel, G.R.; McCann, K., Food web stability: the influence of trophic flows across habitats, American naturalist, 152, 460-469, (1998) |

[10] | Huxel, G.R.; McCann, K.; Polis, G.A., Effects of partitioning allochthonous resources on food web stability, Ecological research, 17, 419-432, (2002) |

[11] | Jiao, J.; Yang, X.; Cai, S.; Chen, L., Dynamical analysis of a delayed predator – prey model with impulsive diffusion between two patches, Mathematics and computers in simulation, 80, 522-532, (2009) · Zbl 1190.34107 |

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

[13] | Lv, S.; Zhao, M., The dynamic complexity of a three species food chain model, Chaos, solitons & fractals, 37, 1469-1480, (2008) · Zbl 1142.92342 |

[14] | McCann, K.S.; Hastings, A.; Huxel, G.R., Weak trophic interactions and balance of nature, Nature, 395, 794-798, (1998) |

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

[16] | Namba, T.; Tanabe, K.; Maeda, N., Omnivory and stability of food webs, Ecological complications, 5, 73-85, (2008) |

[17] | Negi, K.; Gakkhar, S., Dynamics in a beddington – deangelis prey-predator system with impulsive harvesting, Ecological modelling, 206, 421-430, (2007) |

[18] | Pei, Y.; Li, C.; Chen, L., Continuous and impulsive harvesting strategies in a stage-structured predator – prey model with time delay, Mathematics and computers in simulation, 79, 2994-3008, (2009) · Zbl 1172.92038 |

[19] | Polis, G.A.; Anderson, W.B.; Holt, R.D., Toward an integration of landscape and food web ecology: the dynamics of spatially subsidized food webs, Annual review of ecology and systematics, 28, 289-316, (1997) |

[20] | Rose, M.D.; Polis, G.A., The distribution and abundance of coyotes: the effects of allochthonous food subsidies from the sea, Ecology, 79, 998-1007, (1998) |

[21] | Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003 |

[22] | Sun, S.; Chen, L., Permanence and complexity of the eco-epidemiological model with impulsive perturbation, International journal of biomathematics, 1, 121-132, (2008) · Zbl 1166.92039 |

[23] | Takimoto, G.; Iwata, T.; Murakami, M., Seasonal subsidy stabilities food web dynamics: balance in a heterogeneous landscape, Ecological research, 17, 433-439, (2002) |

[24] | Tan, Y.; Tao, F.; Chen, L., Dynamics of a nonautonomous system with impulsive output, International journal of biomathematics, 1, 225-238, (2008) · Zbl 1155.92034 |

[25] | Yu, H.; Zhong, S.; Agarwal, R.P., Mathematics analysis and chaos in an ecological model with an impulsive control strategy, Communications in nonlinear science and numerical simulation, 2, 776-786, (2011) · Zbl 1221.37207 |

[26] | Yu, H.; Zhong, S.; Agarwal, R.P., Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Mathematical and computer modelling, 52, 25-36, (2010) · Zbl 1201.34018 |

[27] | Yu, H.; Zhong, S.; Ye, M., Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay, Mathematics and computers in simulation, 80, 619-632, (2009) · Zbl 1178.92058 |

[28] | Zavalishchin, S.T.; Sesekin, A.N., Dynamic impulse systems theory and applications, mathematics and its applications, (1997), Kluwer Dordrecht, p.394 · Zbl 0880.46031 |

[29] | Zhang, L.; Zhao, M., Dynamic complexities in a hyperparasitic system with prolonged dispause for host, Chaos, solitons & fractals, 42, 1136-1142, (2009) |

[30] | Zhao, M.; Lv, S., Chaos in a three-species food chain model with a beddington-deangelis functional response, Chaos, solitons & fractals, 40, 2305-2316, (2009) · Zbl 1198.37139 |

[31] | Zhao, M.; Wang, X.; Yu, H.; Zhu, J., Dynamics of an ecological model with impulsive control strategy and distributed time delay, Mathematics and computers in simulation, 82, 8, 1432-1444, (2012) · Zbl 1251.92049 |

[32] | Zhao, M.; Yu, H.; Zhu, J., Effects of a population floor on the persistence of chaos in a mutual interference host-parasitoid model, Chaos, solitons & fractals, 42, 1245-1250, (2009) |

[33] | Zhao, M.; Zhang, L., Permanence and chaos in a host-parasitoid model with prolonged diapause for the host, Communications in nonlinear science and numerical simulation, 14, 4197-4203, (2009) |

[34] | Zhao, M.; Zhang, L.; Zhu, J., Dynamics of a host-parasitoid model with prolonged diapause for parasitoid, Communications in nonlinear science and numerical simulation, 16, 455-462, (2011) · Zbl 1221.37208 |

[35] | Zhu, L.; Zhao, M., Dynamic complexity of a host-parasitoid ecological model with the hassell growth function for the host, Chaos, solitons & fractals, 39, 1259-1269, (2009) · Zbl 1197.37133 |

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