# Predator-prey fluctuations in patchy environments

Gurney, William and Nisbet, R.M.
(1978)
*Predator-prey fluctuations in patchy environments.*
Journal of Animal Ecology, 47 (1).
pp. 85-102.
ISSN 0021-8790

## Abstract

(1) We have constructed a model of a predator-prey system which persists in a patchy environment by balancing a high rate of local extinction with an equally high rate of recolonization of unoccupied patches. (2) Our model focuses attention only upon the presence or absence of each species on a given patch and not on population numbers. We assume: (a) that colonization and extinction are random processes which may be described by `transition probabilities'. (b) that the probability per unit time of the prey species colonizing a particular empty patch is simply proportional to the fraction of patches occupied by prey. (c) that the predator species can only colonize patches already occupied by prey, and that the probability per unit time that a particular suitable patch will be so colonized is determined by the proportion of patches occupied by predators. (d) that prey extinction occurs only through over-exploitation by the predator and that the prey population of a patch occupied by both species has a constant probability per unit time of suffering this fate. (e) that predator extinction only occurs through starvation and thus that the population of a patch occupied only by predators has a constant probability per unit time of becoming extinct. We thus construct a representation of our model couched in terms of three coupled stochastic differential equations. (3) We obtain estimates of the intensity of the fluctuations in the numbers of occupied patches which arise as a result of the `filtering' of noise (associated with randomness in the processes of migration and local extinction) by an apparently deterministic dynamical system. The intensity of these fluctuations is controlled by the same combinations of parameters as determine the average number of occupied patches and we can therefore derive experimentally testable relationships between the intensity of fluctuations and the average number of occupied patches. (4) We find that the fluctuations in the numbers of predator-occupied patches (E) and of prey-occupied patches (F) are small and non-cyclic if the average values of E and F are comparable with the total number of patches in the system (N), while if E and F are small compared to N the fluctuations are large and cyclic. (5) In systems containing hundreds (rather than thousands) of patches the fluctuations which accompany cyclic behaviour are sufficiently large to guarantee the global extinction of one or both species within a relatively short time. (6) Our model successfully predicts the relationship between the average number of occupied patches and the intensity of fluctuations in patch occupancy observed in Huffaker's laboratory studies of the predator-prey interaction of Typhlodromus occidentalis (Nesbitt) and Eotetranychus sexmaculatus (Riley) in an experimental `universe' composed of a regular array of oranges.

Item type: Article ID code: 41607 Dates: DateEventFebruary 1978PublishedKeywords: predator-prey system, colonization, patchy environments, Probabilities. Mathematical statistics, Ecology, Evolution, Behavior and Systematics, Animal Science and Zoology Subjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 19 Oct 2012 17:59 Last modified: 20 Mar 2021 01:25 URI: https://strathprints.strath.ac.uk/id/eprint/41607