### Abstract

A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.

Original language | English (US) |
---|---|

Pages (from-to) | 369-380 |

Number of pages | 12 |

Journal | Journal of Mathematical Biology |

Volume | 3 |

Issue number | 3-4 |

DOIs | |

State | Published - 1976 |

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### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

**Predator prey interactions with time delays.** / Cushing, Jim M.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 3, no. 3-4, pp. 369-380. https://doi.org/10.1007/BF00275066

}

TY - JOUR

T1 - Predator prey interactions with time delays

AU - Cushing, Jim M

PY - 1976

Y1 - 1976

N2 - A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.

AB - A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.

UR - http://www.scopus.com/inward/record.url?scp=0017109785&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0017109785&partnerID=8YFLogxK

U2 - 10.1007/BF00275066

DO - 10.1007/BF00275066

M3 - Article

C2 - 1035612

AN - SCOPUS:0017109785

VL - 3

SP - 369

EP - 380

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 3-4

ER -