This material is published by Ecological Modelling (2003, 162: 233-245).
Consequences for predators of rescue and Allee effects on prey
Adam Kent, C. Patrick Doncaster and Tim Sluckin
The size of a population can be augmented by enriching the carrying capacity of its limiting resource, or by subsidising the renewal of the resource. Rosenzweig's (1971) well known 'paradox of enrichment' models the first case, in which enrichment can force consumers and their limiting resource into destabilising limit cycles, whereas impoverishment stabilises the dynamics. We model the case of resource subsidy, where the resource is a limiting prey to predators. In contrast to enrichment, the system is stabilised by an influx of prey in the form of a rescue effect, and destabilised by an outflux of prey in the form of an Allee effect. Limit cycles are not sustained by the Allee effect; instead both populations collapse to zero over a large region of the predator-prey phase plane. The catastrophic extinction of prey requires the presence of both an Allee effect on prey and a predator with a type II functional response, though neither needs to contribute a large impact to prey dynamics. The novel implication is that consumers exaggerate the impact of Allee effects on a renewing resource. Conversely, an Allee effect in the form of a cull of resource, even of small value, can trigger local extinction of resource-dependent consumers.
s: non-dimensionalised prey abundance.
n: non-dimensionalised predator abundance.
D: intrinsic prey flux into (+ve) or out of (-ve) a prey population.
C: ratio of intrinsic search to handling time of each prey by each predator.
sigma: predator's relative marginal subsistence demand for prey.
v: conversion ratio of consumed prey into new predator biomass.
- phase plane of s and n.
- stability of s and n with respect to D, C and sigma.
- continuous rates of change with time in s and n.
Scaling of the model and linear stability analysis.
Further model details at the Register of Ecological Models (REM).
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