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Functional response

The ecologist Crawford S. Holling (1959) argued that animals may not just respond numerically to an abundance of food by increased reproduction, but also functionally by altering their consumption rate. He distinguished three types of functional responses.

Type I

Type I functional response is characterized by a linear increase of intake rate with the amount of food available. Linear increase assumes that food processing or food searching time and other limitations are negligible. Animals just eat what they can get. This type I of functional response is used in the Lotka–Volterra predator–prey model.

Type II

More realistically is type II functional response, which is characterized by a decelerating intake rate, assuming that the per capita consumption rate \(cons\) asymptotically approaches a maximum \(g\) as food density \(A\) increases. Simply put, not all there is can be eaten. This type II response is often modeled with a Monod equation with a half-saturation constant \(H\).

\[cons = g * \frac{A}{A+H}\]
Monod

Type III

If consumption rises more steeply around some threshold value, type III functional response could be appropriate. Animals may need time to learn how to find food and therefore initially have lower intake rates. From a certain time onwards however, the type II dynamics take over and the per capita consumption rate \(cons\) again approaches a maximum \(g\) as food density \(A\) increases. Simply put, at first there is food in abundance, but the problem is how to obtain it. Later there is still enough food, but then gradual saturation sets in. This type III response is usually modeled with a Hill equation, which again has a half-saturation constant \(H\). Additionally now, there is an exponent \(p\) making the curve sigmoidal. The higher \(p\), the steeper the increase. Note, that the Monod function is a special case of the Hill function, obtained by setting \(p\) to 1.  

\[cons = g * \frac{A^p}{A^p+H^p}\]
Hill

Holling, C. S. (1959). Some characteristics of simple types of predation and parasitism. The Canadian Entomologist 91 (7). p. 385–398.