Functional Response

The Nicholson-Bailey model shows unstable host-parasite (prey-predator) dynamics. Later authors began to examine aspects of host-parasite interactions that could provide stability to the model -- and presumably reflect the assumed stability of host-parasite systems in Nature. One area that has been examined extensively is the relationship between the number of hosts (prey) attacked per parasite (predator) and host (prey) density. In the Nicholson-Bailey model, the relationship between the number of host attacked per parasite and host density was assumed to be linear. Later workers showed that this assumption was not true, which then stimulated significant research into the nature of the attack-host density relationship. Later termed the "functional response", the relationship between the number of hosts attacked per parasite and host density has become one of the most-commonly measured aspects of host-parasite interactions.

In this window we will review the original derivation of the functional response, including its mathematical foundation and behavioral basis. In the references section are more readings that expand upon our presentation and offer criticism to our current understanding of the functional response.

Derivation of the Functional Response Models

In the late 1950's, a Canadian researcher, C. S. Holling provided a mechanism and mathematical description of the observed relationship between the number of hosts attacked per parasite and host density. Holling defined the instantaneous search rate, a' as:

         a'= N_e / N_tT                                                   [1.]

where;

         T = total time available to find prey

         N_e = number of host encountered and,

         N_t = number of hosts.

Defining time (T) to be comprised of two separate activities involving search for host (T_s) and handling encountered hosts (T_h) such that:

         T = T_s + T_h . N_e                                             [2.]

Holling concluded that as parasites encounter and handle more hosts as host density increases, they would have less time available to search for new hosts to attack. Thus the "time" important to the functional response would be the search time, Ts, defined as:

         T_s = T - T_h . N_e                                             [3.]

Re-arranging equation 1, and assuming that an encounter (N_e) leads to an attack (N_a) gives:

         N_a = a' T N_t                                                 [4.]

Replacing time, T, with search time T_s (equation 3) gives:

          N_a = a' T_s N_t                                               [5.]

Inserting equation 3 for T_s, we arrive at the definition of the functional response as:

          N_a = a' (T- T_h . N_a) N_t                                [6.]

which can be simplified to:

          N_a = N_t a' T / (1 + a' N_t T_h)                        [7.]

Equation 7 has been referred to as Holling's "disk equation", because he used an elegant laboratory experiment involving a subject "attacking" sand-paper disks to validate his model of the functional response. Plotting the "disk equations" shows (Fig. 1) that as host density increases, the number of hosts attacked levels at a plateau, where the number of attacks remains relatively constant despite increases in host density.

Figure 1

Subsequent measurement of the functional response of many arthropod predators (Luck 1984), showed that most possess a Type II response (but see O'Neil 1989). Analysis of the Nicholson-Bailey model including the Holling functional response has shown that the model is not stabilized by inclusion of the Types I and II models. Although the Type III response does have elements that could provide stability (it is "density-dependent" over a range of host densities) the inherent time-lag between attacks on hosts and the production of offspring precludes stability for specific host-parasite systems.