Solomon (1949) separated the response of consumers to the density of their resources into functional and numerical responses, with the former describing how the consumption rate of individual consumers changes with respect to resource density and the latter how the per-capita reproductive rate changes with resource density. The term behavioral response may be more appropriate than functional response because the response describes the hunting and attack behavior of the consumer, in contrast to the reproductive (numerical) response of the population. Holling (1959a) identified three basic types of behavioral responses:
Type I (linear) response in which the attack rate of the individual consumer increases linearly with prey density but then suddenly reaches a constant value when the consumer is satiated.
Type II (cyrtoid) functional response in which the attack rate increases at a decreasing rate with prey density until it becomes constant at satiation. Cyrtoid behavioral responses are typical of predators that specialize on one or a few prey.
Type III (sigmoid) functional response in which the attack rate accelerates at first and then decelerates towards satiation . Sigmoid functional responses are typical of generalists natural enemies which readily switch from one food species to another and/or which concentrate their feeding in areas where certain resources are most abundant.
These are called prey-dependent responses because the feeding rate of consumers is dependent only on the density of prey.
Holling (1959b) derived a mathematical model for the cyrtoid prey-dependent response from experiments in which blindfolded people acted as "predators" by searching a table top with their finger tips for sandpaper disk "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 Ha is the number of prey attacked per predator per unit of time, H is prey density, which is assumed constant over the hunting period (= instantaneous hunting model), 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 ta. time units to handle its prey then, of the total time T exposed to prey, it will spend T - taHa time actually searching for prey. Thus, the proportion of the total time spent in searching for prey is (T - taHa)/T. If we now insert this modifier into the above equation, we obtain
where d = T/ta and F = T/ata. Real (1977) showed that the type II and III responses can be described by the generalized Michaelis-Menton equations of enzyme kinetics
where x is the encounter rate between predators and prey needed before the predator reaches maximum efficiency. When x = 1 we have a cyrtoid behavioral response and when x >1 the response is sigmoid. Holling (1959a) demonstrated, experimentally, that small mammals feeding on sawfly cocoons have sigmoid behavioral responses due to their switching from one prey to another as prey density increases.
For more on Michaelis-Menten-Holling functional responses check out Alexei Sharov's WWW course at Virginia Tech.
Some authors prefer to use behavioral responses in which the quantity of food per predator, or the prey/predator ratio, is substituted for prey density in the equation; e.g., H/C is used instead of H on the x-axis of the response graph or in the equations (Arditi and Ginzburg 1989). In this case the cyrtoid functional response becomes
This kind of functional response is called "ratio dependent" in contrast to the more conventional "prey dependent" form. A more general ratio-dependent functional response has been proposed by DeAngelis et al. (1975).
There has been considerable debate concerning the relative merits and demerits of ratio- versus prey-dependent behavioral responses with Arditi et al. (1991), Akcakaya et al. (1995) and Berryman et al. (1995) supporting the ratio-dependent view and Abrams (1994), Gleeson (1994) and Sarnelle (1994) the prey-dependent view.
Abrams, P. A. 1994. The fallacies of ratio-dependent predation. Ecology 75: 1842-1850.
Akcakaya, H. R., R. Arditi and L. R. Ginzburg. 1995. Ratio-dependent predation: an abstraction that works. Ecology 76: 995-1004.
Arditi, R. and L. R. Ginzburg. 1989. Coupling in predator-prey dynamics: ratio-dependence. Journal of Theoretical Biology 139:311-326.
Arditi, R., L. R. Ginzburg and H. R. Akcakaya. 1991. Variations in plankton densities among lakes: a case for ratio-dependent models. American Naturalist 138: 1287-1296.
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.
DeAngelis, D. L., R. A. Goldstein, and R. V. O'Neill. 1975. A model for trophic interactions. Ecology 56: 881-892.
Gleeson, S. K. 1994. Density dependence is better than ratio dependence. Ecology 75: 1834-1835.
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.
Real, L. 1977. The kinetics of functional response. American Naturalist 111: 289-300.
Sarnelle, O. 1994. Inferring process from pattern: trophic level abundances and imbedded interactions. Ecology 75: 1835-1841.
Solomon, M. E. 1949. The natural control of animal populations. Journal of Animal Ecology 18: 1-35.
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.
©1998 Alan A. Berryman