# 1. General Model Information

### Name: Prey flux in predator-prey dynamics

### Acronym: PREY_FLUX

**Main medium:** all

**Main subject:** population dynamics, agriculture

**Organization level:** Population, Ecosystem

**Type of model:** ordinary differential equations

**Main application:** research, decision support/expert system

**Keywords:** extinction threshold, functional response, Lotka-Volterra, open system

### Contact:

** Dr C. Patrick Doncaster **

School of Biological Sciences,

University of Southampton,

Bassett Crescent East, Southampton SO16 7PX, UK

Phone: +44 (0)23 80594352

Fax: +44 (0)23 80594269

email: cpd@soton.ac.uk

Homepage: http://www.soton.ac.uk/~cpd/

### Author(s):

Kent, A., Doncaster, C. P., Sluckin, T.
### Abstract:

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.
The 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.

# II. Technical Information

### II.1 Executables:

**Operating System(s):** Microsoft Windows

### II.2 Source-code:

**Programming Language(s):** Mathcad 2000

cpd@soton.ac.uk

### II.3 Manuals:

cpd@soton.ac.uk

### II.4 Data:

This is a data-free model, but example values for constants are given figure legends of Kent, A., Doncaster, C. P., Sluckin, T. (In press).

# III. Mathematical Information

### III.1 Mathematics

Scaling of model and linear stability analysis are provided in appendices to Kent, A., Doncaster, C. P., Sluckin, T. (In press).

### III.2 Quantities

s: non-dimensionalised prey abundance. n: non-dimensionalised predator abundance.

#### III.2.1 Input

D: intrinsic prey flux into (+ve) or out of (-ve) a prey population. C: ratio of intrinsic search to handling tim 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.

#### III.2.2 Output

- continuous rates of change with time in s and n.
- phase plane of s and n.
- stability of s and n with respect to D, C and sigma.

# IV. References

Rosenzweig, M.L., 1971. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385-387.

Kent, A., Doncaster, C. P., Sluckin, T. (In press) Consequences for predators of rescue and Allee effects on prey. Ecological Modelling.

# V. Further information in the World-Wide-Web

# VI. Additional remarks

This develops Rosenzweig's (1971) 'paradox of enrichment'. It is intended as a conceptual aid to understanding how the presence of predators can exaggerate the effects of resource fluxes and Allee/rescue effects on the dynamic stability of prey.

Last review of this document by: Mon Sep 9 14:38:18 2002

Status of the document: Contributed by C. Patrick Doncastrer

* last modified by
Joachim Benz Wed Sep 18 15:03:02 CEST 2002 *