Use of a halfsaturation constant in predatorprey interactions

Turchin and Ellner (2000) have asserted that the predator-prey equations used by Rosenzweig and MacArthur (1963) should be adopted as "the standard" for predator-prey interactions, since they eliminate the assumption of the linear functional response and are "perhaps the simplest model that can actually be applied to real life systems" (Turchin 2003, p. 95). The only differences between these equations and Equations 10.13a and 10.14a are that the handling term in the functional response has been replaced by d + N, where d is the half saturation (half maximum killing rate) parameter, and E has been replaced by c, the maximum killing rate when the search and capture components have been minimized. In other words, we are using the alternative functional response shown as Equation 10.8b:

An alternative form of the predator equation (10.16a) is to add what is known as the ZPG component consumption rate (pp). This parameter represents the minimum rate of prey consumption needed for a predator to survive and just replace itself. This is usually easier to estimate than the predator death rate in the absence of prey, mp. The result is Equation 10.16b, a version of which we will use in Chapter 11 on herbivore-plant interactions:

Now compare the predator equation (10.16a) to Equations 7.19 and 7.20, which describe the effects of resource depletion on the growth rate of a consumer in the context of competition.

In a revised version of Equation 7.20, instead of a competing species, N2, we are substituting P, the predator population. The death rate, m, is now mp. We have replaced Xpc, which measures the maximum rate of conversion of the resource (prey) into predators, by b, which was the maximum growth rate of the competing species.

The prey population (N in equation 10.16a) is considered a resource, so is replaced by R, the concentration of the resource. The half-saturation constant d is replaced by KR. The two equations are now identical. Equation 7.20 measures the growth rate of a consumer in terms of its maximum growth rate, the concentration of the resource, the halfsaturation constant for that resource, and the death rate of the consumer. The predator equation (10.16a) measures the growth rate of the predator in terms of maximum efficiency in turning prey into predator individuals (xpc), the concentration of the resource (N), the half-saturation constant, and the death rate of the predator. We now have a general mechanistic equation, useful in both competitive interactions and predator-prey interactions for the growth rate of a consuming population.

Because the half-saturation parameter is more easily estimated in the field than is a specific handling time value, this approach (Eqns. 10.15 and 10.16) was employed by Gilg et al. (2003) in their study of lemming-predator interactions in Greenland, as well as by Turchin and Ellner (2000).

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