Chaos, noise, and extinction in models of population growth.

*(English)*Zbl 0699.92020Summary: A “hump-with-tail” map (i.e., with two extreme values) is derived to describe the growth of bobwhite quail populations. Although this map is shown to be capable of producing chaotic dynamics, analysis of field data suggests that the population is at a stable equilibrium. Thus the observed fluctuations in population density likely reflect random perturbations.

It is shown that populations governed by hump-with-tail maps are less prone to extinction in randomly fluctuating environments than hypothetical populations whose growth is governed by “one hump” maps, i.e., maps with a maximum as their only extremal value. Our observations emphasize the importance of considerations of noise in theoretical analysis of population growth and stability.

It is shown that populations governed by hump-with-tail maps are less prone to extinction in randomly fluctuating environments than hypothetical populations whose growth is governed by “one hump” maps, i.e., maps with a maximum as their only extremal value. Our observations emphasize the importance of considerations of noise in theoretical analysis of population growth and stability.

##### MSC:

92D25 | Population dynamics (general) |

##### Keywords:

difference equations; extreme values; growth of bobwhite quail populations; stable equilibrium; random perturbations; hump-with-tail maps; randomly fluctuating environments; noise
PDF
BibTeX
XML
Cite

\textit{J. G. Milton} and \textit{J. Bélair}, Theor. Popul. Biol. 37, No. 2, 273--290 (1990; Zbl 0699.92020)

Full Text:
DOI

##### References:

[1] | Beddington, J.R., Age distribution and the stability of simple discrete time population models, J. theoret. biol, 47, 65-74, (1974) |

[2] | Bélair, J.; Glass, L., Self-similarity in periodically forced oscillators, Phys. lett. A, 96, 113-116, (1983) |

[3] | Bélair, J.; Glass, L., Universality and self-similarity in the bifurcations of circle maps, Physica D, 16, 143-154, (1985) · Zbl 0593.58030 |

[4] | Bélair, J.; Milton, J.G., Itinerary of a discontinuous map from the continued fraction expansion, Appl. math. lett, 1, 339-342, (1988) · Zbl 0736.58025 |

[5] | Bellows, T.S., The descriptive properties of some models for density dependence, J. anim. ecol, 50, 139-156, (1981) |

[6] | Chitty, D., Mortality among voles (microtus agrestis) at lake vyrnwy, montgomeryshire in 1936-1939, Philos. trans. roy. soc. London ser. B, 236, 505-552, (1952) |

[7] | Clark, C., Mathematical bioeconomics: the optimal management of renewable resources, (1976), Wiley Toronto, Chap. 7 · Zbl 0364.90002 |

[8] | Collet, P.; Eckmann, J.-P., Iterated maps of the interval as dynamical systems, (1980), Birkhäuser Boston · Zbl 0456.58016 |

[9] | Crutchfield, J.P.; Farmer, J.D.; Huberman, B.A., Fluctuations and simple chaotic dynamics, Phys. rep, 92, 45-82, (1982) |

[10] | Devaney, R.L., An introduction to chaotic dynamical systems, (1986), Benjamin-Cummings Menlo Park, CA · Zbl 0632.58005 |

[11] | Edelstein-Keshet, L., Mathematical models in biology, (), 63 |

[12] | Errington, P.L., Some contributions of a fifteen year local study of the northern bobwhite to a knowledge of population phenomena, Ecol. monogr, 15, 1-34, (1945) |

[13] | Errington, P.L., Predation and vertebrate populations, Quart. rev. biol, 21, 144-177, (1946) |

[14] | Errington, P.L., Of population cycles and unknowns, (), 287-300 |

[15] | Feigenbaum, M.J., Universal behaviour in nonlinear systems, Physica D, 7, 16-39, (1983) · Zbl 0533.58025 |

[16] | Felsenstein, J., r- and K-selection in a completely chaotic population model, Amer. nat, 113, 499-510, (1979) |

[17] | Fretwell, S.D., Populations in a seasonal environment, (1972), Princeton, N.J. |

[18] | Glass, L.; Perez, R., Fine structure of phase locking, Phys. rev. lett, 48, 1772-1775, (1982) |

[19] | Green, R.G.; Evans, C.A.; Green, R.G.; Evans, C.A.; Green, R.G.; Evans, C.A., Studies on a population cycle of snowshoe hares on the lake Alexander area, J. wildlife manag, J. wildlife manag, J. wildlife manag, 4, 347-358, (1940) |

[20] | Hassell, M.P.; Lawton, J.H.; May, R.M., Patterns of dynamical behaviour in single-species populations, J. anim. ecol, 45, 471-486, (1976) |

[21] | Jenkins, D.; Watson, A.; Miller, G.R., Population studies on red grouse, lagopus lagopus scoticas, J. anim. ecol, 32, 317-376, (1963) |

[22] | Lasota, A.; Mackey, M.C., Noise and statistical periodicity, Physica D, 28, 143-154, (1986) · Zbl 0645.60068 |

[23] | May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 445-467, (1976) · Zbl 1369.37088 |

[24] | May, R.M., Chaos and the dynamics of biological populations, (), 27-44 · Zbl 0656.92012 |

[25] | May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Amer. nat, 110, 573-599, (1976) |

[26] | Mayer-Kress, G.; Haken, H., The influence of noise on the logistic map, J. statist. phys, 26, 149-171, (1981) · Zbl 0511.58025 |

[27] | Pandit, S.M.; Wu, S.M., Time series and systems analysis with applications, (1983), Wiley Toronto · Zbl 0372.62081 |

[28] | Pennycuick, C.J.; Compton, R.M.; Beckingham, L., A computer model for simulating the growth of a population, or of two interacting populations, J. theoret. biol, 18, 316-329, (1968) |

[29] | Pikovsky, A.S., A dynamical model for periodic and chaotic oscillations in the Belousov-Zhabotinsky reaction, Phys. lett. A, 85, 13-16, (1981) |

[30] | Pimm, S.L.; Redfearn, A., The variability of population densities, Nature, 334, 613-614, (1988) |

[31] | Usher, M.B., Developments in the Leslie matrix model, (), 29-60 |

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.