A derivation of Holling's type I, II and III functional responses in predator-prey systems

Predator-prey dynamics is most simply and commonly described by Lotka-Volterra-type ordinary differential equations (ODEs) for continuous population density variables in the limit of large population sizes. One popular extension of these ODEs is the so-called Rosenzweig-MacArthur model in which various interaction rates between the populations have a nonlinear dependence on the prey concentration. Nonlinear 'functional responses' of this type were originally proposed by Holling on the basis of a general argument concerning the allocation of a predator's time between two activities: 'prey searching' and 'prey handling'.Although these functional responses are constructed in terms of the behaviour of an individual predator, they are routinely incorporated at the population level in models that include reproduction and death. In this paper we derive a novel three variable model for the simplest possible mathematical formulation of predator-prey dynamics that allows the interplay between these various processes to take place, on their different characteristic timescales. We study its properties in detail and show how it reduces to Holling's functional responses in special limits. As a result we are able to establish direct links between individual-level and population-level behaviour in the context of these well-known functional responses.