Attributes of plausible predator-prey models

Royama (1992) and Berryman et al. (1995) proposed the following minimal set of attributes that all consumer-resource models should possess:

Attribute1: The death rate of an individual in the resource population should increase with the density of the consumer population; i.e., the more predators there are, the greater your probability of being eaten.

Attribute 2: The birth rate of individual consumers should increase with the density of the resource population; i.e., the more food one gets, the more offspring one produces.

Attribute 3: The reproductive rate of a consumer should decrease with its own density; i.e., the more competitors there are the less food one gets and, through attribute 2, the fewer offspring one produces (this is usually called intraspecific competition for resources).

Attribute 4: Consumers must have a finite demand for resources (appetites) and, consequently, a finite maximum reproductive rate; i.e., consumers can only eat so much before they become satiated or full.

Conditions on the prey equation

The above attributes place the following conditions on the prey R-function:

Condition 1: When the prey population is constant, its per-capita rate of change should decline with increasing consumer density (because of Attribute 1).

Condition 2: When the predator population is constant, the per-capita rate of change of the prey should rise at first with prey density (because of Attribute 4), and then decline as prey become very dense (because of Attribute 3).

Conditions on the predator equation

The corresponding conditions for the predator R-function under the assumption that they are not subjected to attack by higher trophic levels are:

Condition 3: When the prey population is constant, the per-capita rate of change of the predator should decline with increasing predator density (after Attribute 3).

Condition 4: When the predator population is constant, its per-capita rate of change should rise with prey density (because of Attribute 2) to a maximum positive value (because of Attribute 4).

These four conditions are required of all reasonable models for consumer-resource interactions. In addition, it seems sensible, though perhaps not essential, to require that:

Condition 5: The equations for resource and consumer populations should have the same basic form, or be structurally homogeneous. Homogenous equations are desirable because any member of a trophic chain or web can be both a consumer on those below it and a resource to those above and, for this reason, the structure of the equation for any species should not depend on its position in the food chain.

Testing models for adherence to plausibility criteria

Models can be checked for their plausibility by examining the predator-prey R-functions. For example, the R-function for a L-V prey population is

so we test this equation against conditions 1 and 2 (above):

1. First, set the prey population constant and check to see if R_h is inversely related to C. If so then condition 1 is satisfied. We can see the L-V prey R-function meets this condition.
2. Now set the predator population constant and check to see if R_h increases at first with H but then declines as H gets large. If so then condition 2 is satisfied. As the L-V prey R-function does not depend on prey density (H) this condition is not satisfied.

The R-function for a L-V predator is evaluated in a similar manner

1. Set the prey population H constant and check to see if R_c declines with C. If so then condition 3 is satisfied. We can see the predator R-function is constant when H is constant and so the L-V model violates criterion 3.
2. Set the consumer population C constant and check to see if R_c rises with H to some maximum value, corresponding to the maximum fecundity of the species. If so then condition 4 is satisfied. We can see that this criterion is not met in the L-V predator R-function because R_c rises indefinitely with prey density. In other words, L-V predators are insatiable.

Berryman, A. A., A. P. Gutierrez and R. Arditi. 1995. Credible, parsimonious and useful predator-prey models-a reply to Abrams, Gleeson, and Sarnelle. Ecology 76: 1980-1985.

Royama, T. 1992. Analytical population dynamics. Chapman and Hall, London.