Predator prey interactions with time delays

Research output: Contribution to journalArticle

12 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)369-380
Number of pages12
JournalJournal of Mathematical Biology
Volume3
Issue number3-4
DOIs
StatePublished - 1976

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Predator-prey
Carrying Capacity
predator-prey relationships
Conservation of Natural Resources
Prey
carrying capacity
Time Delay
Time delay
Critical value
Predator
Interaction
predators
Lotka-Volterra Model
Lotka-Volterra
birth rate
Birth Rate
Globally Asymptotically Stable
Periodic Solution
Unstable
Tend

ASJC Scopus subject areas

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

Cite this

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

In: Journal of Mathematical Biology, Vol. 3, No. 3-4, 1976, p. 369-380.

Research output: Contribution to journalArticle

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