Ivlev (1955) derived the functional response of a consumer by arguing that the rate of consumption should depend on how hungry it is. Let d be the amount of food needed to satiate a consumer (its demand) and f the amount it actually obtains, then the hunger of the consumer is given by the fraction of its demand that is met, (d - f) / d . If the consumption rate of the consumer depends on its hunger, and if v is the efficiency with which consumers extract resources from their environment, sometimes called the "apparency of the resources", then the realized rate of consumption will be v(d - f) / d .

Now suppose we have a population of H consumers (herbivores) feeding on P resources (plants), then the change in individual consumption, df_h, with change in resource density, dP, will be

where the subscript h defines the parameters for the consumer population, H. Rearranging terms and integrating both sides gives Ivlev's version of the behavioral response model

This equation describes a cyrtoid or Type II prey-dependent functional response because the feeding rate declines with increasing resource abundance until it reaches a constant rate d_h. It is also "prey-dependent" because the consumption rate depends only on the density of resources, P, and does not involve the number of consumers utilizing those resources, H.

Ivlev's equation describes the effect of consumer satiation on the rate of resource consumption in a similar way to Holling's disk equation; i.e., it describes a cyrtoid or Type II functional response. However, although the model has an identical form, it is derived from different assumptions and has a different explicit mathematical structure. Notice also that Ivlev's model is derived from a differential equation that has not been integrated over time and, as such, must be considered an instantaneous consumption equation. In other words, it must be integrated over time before it can be applied to real data.

Reference

Ivlev, V. S. 1955. Experimental ecology of the feeding of fishes. Yale University Press, New