Functional response

A functional response in ecology is the intake rate of a consumer as a function of food density (the amount of food available in a given ecotope). It is associated with the numerical response, which is the reproduction rate of a consumer as a function of food density. Following C. S. Holling, functional responses are generally classified into three types, which are called Holling's type I, II, and III. These were formulated using laboratory experiments where participants collected disks from a board of increasing disk density. Thus, the resulting formulae are often referred to as Holling's Disk Equations.

Type I The type I functional response assumes a linear increase in intake rate with food density, either for all food densities, or only for food densities up to a maximum, beyond which the intake rate is constant. The linear increase assumes that the time needed by the consumer to process a food item is negligible, or that consuming food does not interfere with searching for food. A functional response of type I is used in the Lotka–Volterra predator–prey model. It was the first kind of functional response described and is also the simplest of the three functional responses currently detailed.

Type II The type II functional response is characterized by a decelerating intake rate, which follows from the assumption that the consumer is limited by its capacity to process food. Type II functional response is often modelled by a rectangular hyperbola, for instance as by Holling's disc equation, which assumes that processing of food and searching for food are mutually exclusive behaviours. The equation is

f ( R ) = a R 1 + a h R {\displaystyle {\begin{aligned}f(R)&={\frac {aR}{1+ahR}}\end{aligned}}}

where f denotes intake rate and R denotes food (or resource) density. The rate at which the consumer encounters food items per unit of food density is called the attack rate, a. The average time spent on processing a food item is called the handling time, h. Similar equations are the Monod equation for the growth of microorganisms and the Michaelis–Menten equation for the rate of enzymatic reactions. In an example with wolves and caribou, as the number of caribou increases while holding wolves constant, the number of caribou kills increases and then levels off. This is because the proportion of caribou killed per wolf decreases as caribou density increases. The higher the density of caribou, the smaller the proportion of caribou killed per wolf. Explained slightly differently, at very high caribou densities, wolves need very little time to find prey and spend almost all their time handling prey and very little time searching. Wolves are then satiated and the total number of caribou kills reaches a plateau.

Type III The type III functional response is similar to type II in that at high levels of prey density, saturation occurs. At low prey density levels, the graphical relationship of number of prey consumed and the density of the prey population is a super-linearly increasing function of prey consumed by predators:

f ( R ) = a R k 1 + a h R k , k > 1 {\displaystyle {\begin{aligned}f(R)&={\frac {aR^{k}}{1+ahR^{k}}},\;\;\;\;\;\;\;k>1\end{aligned}}}

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