##
**Dynamic complexities in a single-species discrete population model with stage structure and birth pulses.**
*(English)*
Zbl 1066.92041

Summary: Natural populations, whose population numbers are small and generations non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. This paper investigates a recent study on the dynamic complexities in a single-species discrete population model with stage structure and birth pulses. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain a threshold conditions for its stability. Above this, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single-species discrete model with birth pulses are very complex, including (a) non-unique dynamics, meaning that several attractors and chaos coexist; (b) small-amplitude annual oscillations; (c) large-amplitude multi-annual cycles; (d) chaos. Some interesting results are obtained and they showed that pulsing provides a natural period or cyclicity that allows for a period-doubling route to chaos.

### MSC:

92D25 | Population dynamics (general) |

37N25 | Dynamical systems in biology |

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

PDF
BibTeX
XML
Cite

\textit{S. Gao} and \textit{L. Chen}, Chaos Solitons Fractals 23, No. 2, 519--527 (2005; Zbl 1066.92041)

Full Text:
DOI

### References:

[1] | EBenman, B.; persson, L., Size-structured populations: ecology and evolution, (1998), Spring Berlin |

[2] | Metz JAJ, Diekmann O. The dynamics of physiologically structured populations. Lecture Notes in Biomathematics, vol. 68, 1986 · Zbl 0614.92014 |

[3] | Aiello, W.G.; Freedman, H.I., A time-delay model of single-species growth with stage structure, Math. biosci., 101, 139-153, (1990) · Zbl 0719.92017 |

[4] | Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM. J. appl. math., 3, 855-869, (1992) · Zbl 0760.92018 |

[5] | Cai, Y.; Fan, J.; Gard, T.C., The effects of a state-structured population growth model, Nonlin. anal. th. mech. appl., 16, 20, 95-105, (1992) · Zbl 0777.92014 |

[6] | Freedman, H.I.; Wu, J.H., Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. appl. math., 2, 351-371, (1991) · Zbl 0732.92021 |

[7] | Hastings, A., Delay in recruitment at different trophic levels: effects on stability, J. math. biol., 21, 35-44, (1984) · Zbl 0547.92014 |

[8] | Gurney, W.S.C.; Nisbet, R.M.; Lawton, J.H., The systematic formulation of tractable single species population models incorporating age structure, J. animal ecol., 52, 479-485, (1983) |

[9] | Caswell, H., Matrix population models, () |

[10] | Tang, S.Y.; Chen, L.S., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 64, 169-184, (2002) |

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

[12] | Hastings, A., Delay in recruitment at different trophic levels: effects on stability, J. math. biol., 21, 35-44, (1984) · Zbl 0547.92014 |

[13] | Hastings, A.; Higgins, K.; Hinggins, K.; Hastings, A.; Botsford, L., Density dependence and age structure: nonlinear dynamics and population behavior, Science, Am. nat., 149, 247-269, (1997) |

[14] | Freedman, H., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023 |

[15] | Jury, E.I., Inners and stability of dynamic system, (1974), Wiley New York · Zbl 0307.93025 |

[16] | May, R.M., Biological population with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974) |

[17] | May, R.M.; Oster, G.F., Bifurcation and synamic complexity in simple ecological models, Am. nat., 110, 573-599, (1976) |

[18] | Eckmann, J.P., Routes to chaos with special emphasis on period doubling, () · Zbl 0616.58032 |

[19] | Collet, P.; Eckmann, J.P., Iterated maps of the interval as dynamical systems, Importance time delays, 70, 1434-1441, (1989) |

[20] | Hauser, M.J.B.; Olsen, L.F.; Bronnikova, T.V.; Schaffer, W.M., Routs to chaos in the peroxidase – oxidase reaction: period-doubling and period-adding, J. phys. chem. B, 101, 5075-5083, (1997) |

[21] | Hung, Y.F.; Yen, T.C.; Chern, J.L.; Kaneko, K., On the period-adding phenomena at the frequency locking in a one-dimensional mapping, Phys. lett. A, Prog. theor. phys., 69, 403-414, (1982) |

[22] | Guckenheimer, J.; Oster, G.; Ipaktchi, A., The dynamics of density dependent population models, J. math. biol., 4, 101-147, (1977) · Zbl 0379.92016 |

[23] | Gakkhar, S.; Naji, R.K., Chaos in seasonally perturbed ratio-dependent prey – predator system, Chaos, solitons and fractals, 15, 107-118, (2003) · Zbl 1033.92026 |

[24] | Upadhyay, R.K.; Rai, V.A.; lyengar, S.R., How do ecosystems respond to external perturbations?, Chaos, solitons and fractals, 11, 1963-1982, (2000) · Zbl 0985.37105 |

[25] | Tang, S.Y.; Chen, L.S., Quasiperiodic solutions and chaos in a periodically forced predator – prey model with age structure for predator, Int. J. bifurcat. chaos, 13, 973-980, (2003) · Zbl 1063.37586 |

[26] | Hasting, A., Comples interactions between dispersal and dynamics: lessons from coupled logistic equations, Ecology, 74, 1362-1372, (1993) |

[27] | Beddington, J.R.; Free, C.A.; Lawton, J.H., Dynamic complexity in predator – prey models framed in difference equations, Nature, 255, 58-60, (1975) |

[28] | Kaitala, V.; Ranta, E., Red/blue chaotic power spectra, Nature, 381, 198-199, (1996) |

[29] | Tang, S.Y.; Chen, L.S., Chaos in functional response host-parasitiod ecosystem model, Chaos, solitons and fractals, 13, 875-884, (2002) · Zbl 1022.92042 |

[30] | Rohani, P.; Miramontes, O., Immigration and the persistence of chaos in population models, J. theor. biol., 175, 203-206, (1995) |

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.