Holling (1959a) defined three kinds of behavioral (functional) responses as shown in the figure but only the Type II (cyrtoid) and Type III (sigmoid) are commonly encountered. Note that the sigmoid response can cause the prey death rate to rise with prey population density (- feedback) while the other types only give rise to ⁺feedback. Sigmoid behavioral responses are often created by consumers which switch from less abundant to more abundant prey species or which aggregate in regions where prey are more dense. Switching and aggregation are commonly observed in intelligent generalist predators such as birds and mammals, but aggregation can also occur as a result of well-developed prey locating adaptations, as found in some parasitoids. For example, Gould et al. (1990) transplanted large numbers of gypsy moth egg masses from outbreak regions into areas with very sparse gypsy moth populations. Within a season, all these local gypsy moth infestations wiped out by the aggregation of insect parasitoids.

Holling (1959b) derived the cyrtoid response from experiments in which blindfolded people acting as "predators" searched a table top with their finger tips for sandpaper disks ("prey"), the so-called disk equation. We can derive the "disk equation" in the following way: Define the number of prey attacked by a predator which instantaneously assimilates its prey by

where N_a is the number of prey attacked per predator per unit of time, N is prey density (with t-1 omitted for simplicity), and a is the rate of attack. Now consider the time taken by predators that have to handle, kill and devour its prey before it can search for another. This is called the "handling time" of the predator. If the predator requires t_a time units to handle its prey then, of the total time T exposed to prey, it will spend T - t_aN_a time actually searching for prey. Thus, the proportion of the total time spent in searching for prey is (T - t_aN_a)/T. If we now insert this modifier into the above equation we obtain the disk equation as the behavioral response of a consumer

Real (1977) showed that the two major predator responses can be described by the generalized Michaelis-Menton equations of enzyme kinetics

where x is the encounter rate between predators and prey where the predator reaches maximum efficiency. When x = 1 we have a cyrtoid behavioral response and when x >1 the response is sigmoid (see figure). If the behavioral response gives the number of prey killed per predator, then the per-capita death rate of the prey attacked by P predators is

It is easy to see that, when x = 1 (cyrtoid response) then N^x-1 = 1, the prey death rate is

and, therefore, the prey's death rate decrease with prey density as shown in the figure. If, for the sake of argument, the prey's birth rate is assumed constant, then we may have the situation shown in the next figure. Here the system is in equilibrium when births equal deaths or

where E is the value of N at equilibrium. Notice that this is an unstable equilibrium or threshold (see bottom figure). Rearranging this equation yields

Notice that the unstable threshold E gets higher as predator density increases.

When x > 1 (sigmoid response), however, the death rate rises with prey density as long as N < F, and then declines when N > F (see figure). Sigmoid behavioral responses will be discussed in greater detail later in this course.

Another kind of behavioral response was developed by Ivlev (1955). In his own words: "the actual ration of food eaten by the predator over a certain period of time will, under favorable feeding conditions, tend to approach a certain definite size, above which it cannot under any circumstances increase and which also corresponds to the physiological condition of full satiation." In other words, Ivlev was thinking about how consumers met their energy demands and how the attack rate depends on how hungry the consumer is. From this he proposed that the amount of food obtained by an individual consumer (i.e., the behavioral response), changes in the following way with respect to prey density

where a is the availability or "apparency" of the resource, W is the demand for food, and measures the difference between the amount of food needed and that obtained, or the hunger of the consumer. Watt (1959) took a similar approach but, in addition, argued that the attack rate will also be related to the density of consumers. Simplifying Watt's argument we let the attack rate be inversely related to P so that

and, integrating over prey density, we obtain a functional response model derived from the consumer point of view

• Gould, J. R., Elkinton, J. S. and W. E. Wallner. 1990.Density-dependent suppression of experimentally created gypsy moth, Lymantria dispar (Lepidoptera:Lymantriidae) populations by natural enemies. Journal of Animal Ecology 59: 213-233.
• Holling, C. S. 1959a. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Canadian Entomologist 91: 293-320.
• Holling, C. S. 1959b. Some characteristics of simple types of predation and parasitism. Canadian Entomologist 91: 385-398.
• Ivlev, V. S. 1955. Experimental ecology of the feeding of fishes. Yale University Press, New Haven.(English translation published in 1961)
• Real, L. 1977. The kinetics of functional response. American Naturalist 111: 289-300.
• Watt, K. E. F. 1959. A mathematical model for the effect of densities of attacked and attacking species on the number attacked. Canadian Entomologist 91: 129-144.

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