Journal Article

Predation thresholds for survival of endangered species


in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 14, issue 3, pages 241-250
Published in print September 1997 | ISSN: 1477-8599
Published online September 1997 | e-ISSN: 1477-8602 | DOI:
Predation thresholds for survival of endangered species

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  • Applied Mathematics
  • Biomathematics and Statistics


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In this paper we derive estimates of productivity and survival from a model of a population of an endangered species and focus especially on the significance of predation. This model is based on an age structure of three classes: dependent young, juveniles, and adults differentiated by gender. In the first two classes, age is an independent variable and we use a partial-differential-equation framework for these age compartments. Assuming predation and natural death rates are independent of population size, it is possible to calculate analytically the threshold of the predation rate—especially on dependent young and juveniles—below which survival is assured and above which the cohort is doomed to extinction. The model leads to a linear-delay differential equation of a standard type. Transform techniques reveal that the transient behaviour will vary like exp(λt), where λ is related to a balance between birth rates and predation rates in various age classes. The sign of λ governs the behaviour, and λ equal to zero determines the threshold between survival and extinction for this model. The motivation for the development of this model is the need to determine the thresholds for survival of the kiwi, a flightless bird found only in New Zealand. Although protected, it is subject to extreme predation by introduced mammals and is declining throughout the mainland. The model shows that predation rates of kiwi are significantly above the threshold level required for survival, and that the species now needs to be managed to avert extinction. This methodology can be applied to other endangered species.

Keywords: age-structured model; linear-differential delay equation; McKendrick's equation; New Zealand kiwi; Apteryx species

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

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