### Abstract

Language | English |
---|---|

Pages | 459-475 |

Number of pages | 17 |

Journal | Journal of Theoretical Biology |

Volume | 56 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1976 |

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### Keywords

- population dynamics
- growth rate
- average population

### Cite this

*Journal of Theoretical Biology*,

*56*(2), 459-475. https://doi.org/10.1016/S0022-5193(76)80086-0

}

*Journal of Theoretical Biology*, vol. 56, no. 2, pp. 459-475. https://doi.org/10.1016/S0022-5193(76)80086-0

**Population dynamics in a periodically varying environment.** / Nisbet, R.M.; Gurney, William.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Population dynamics in a periodically varying environment

AU - Nisbet, R.M.

AU - Gurney, William

PY - 1976/2

Y1 - 1976/2

N2 - We have investigated numerically the effect of periodic variation in the values of the intrinsic growth rate, r, and of the carrying capacity, K, for a species described by the logistic equation with a time delay, τ. If in the absence of driving oscillations the system would have a stable equilibrium, then oscillations in K normally cause population oscillations of the same frequency, accompanied by a drop in the time average population. Large amplitude oscillations in K may cause population cycles whose frequency is a subharmonic of the driver. The qualitative behaviour of the system is unchanged by oscillations in r alone, while the effect of simultaneous oscillations in r and K depends critically on the phase difference between them. If the population would, in the absence of driving oscillations, be in a limit cycle, then this limit cycle is normally robust against K-oscillations at frequencies very different from the limit cycle frequency. If the limit cycle frequency is close to the driving frequency or to one of its subharmonics, then the system may synchronize to the driver or to that subharmonic. Oscillations in r have little effect on a limit cycling system unless they are of large enough amplitude to decrease the effective value of the product rτ sufficiently to destroy the limit cycle. On the basis of the above results we comment on the analysis of cycles in field and laboratory situations, especially those where synchronization may occur. We finally use our results to discuss the direction of evolution of r in a regularly varying environment.

AB - We have investigated numerically the effect of periodic variation in the values of the intrinsic growth rate, r, and of the carrying capacity, K, for a species described by the logistic equation with a time delay, τ. If in the absence of driving oscillations the system would have a stable equilibrium, then oscillations in K normally cause population oscillations of the same frequency, accompanied by a drop in the time average population. Large amplitude oscillations in K may cause population cycles whose frequency is a subharmonic of the driver. The qualitative behaviour of the system is unchanged by oscillations in r alone, while the effect of simultaneous oscillations in r and K depends critically on the phase difference between them. If the population would, in the absence of driving oscillations, be in a limit cycle, then this limit cycle is normally robust against K-oscillations at frequencies very different from the limit cycle frequency. If the limit cycle frequency is close to the driving frequency or to one of its subharmonics, then the system may synchronize to the driver or to that subharmonic. Oscillations in r have little effect on a limit cycling system unless they are of large enough amplitude to decrease the effective value of the product rτ sufficiently to destroy the limit cycle. On the basis of the above results we comment on the analysis of cycles in field and laboratory situations, especially those where synchronization may occur. We finally use our results to discuss the direction of evolution of r in a regularly varying environment.

KW - population dynamics

KW - growth rate

KW - average population

U2 - 10.1016/S0022-5193(76)80086-0

DO - 10.1016/S0022-5193(76)80086-0

M3 - Article

VL - 56

SP - 459

EP - 475

JO - Journal of Theoretical Biology

T2 - Journal of Theoretical Biology

JF - Journal of Theoretical Biology

SN - 0022-5193

IS - 2

ER -