The Functional and Numerical Response

The Lotka-Volterra model assumes that the prey consumption rate by a predator is directly proportional to the prey abundance. This means that predator feeding is limited only by the amount of prey in the environment. While this may be realistic at low prey densities, it is certainly an unrealistic assumption at high prey densities where predators are limited, for example, by time and digestive constraints. The need for a more realistic description of predator feeding came from the experimental work of G. F. Gause on protist prey-predator interactions. He observed that to explain his experimental observations, the linear functional dependencies of the Lotka-Volterra model must be replaced by nonlinear functions.

To understand the nature of prey-predator interactions, M. E. Solomon introduced concept of functional and numerical responses. The functional response describes prey consumption rate by a single predator as a function of prey abundance, while the numerical response describes the effect of prey consumption on the predator recruitment. Most simple prey-predator models such as the Lotka-Volterra model assume that production of new predators is directly proportional to the food consumption. In this case, the numerical response is directly proportional to the functional response. The constant of proportionality, e in model [1], is the efficiency with which prey are converted to newborn predators.

C. S. Holling introduced three types of functional responses (Figure 2). The type I functional response is the most similar to the Lotka-Volterra linear functional response, but it assumes a ceiling on prey consumption rate fI (R) = minfAR, const}

where const is the prey consumption rate when prey abundance is high (Figure 2a). This functional response is found in passive predators that do not hunt actively (e.g., web-building spiders and filter feeders).

The type II functional response assumes that predators are limited by total available time T. During this time predators are assumed either to search for prey (for Ts time units), or to handle prey (for Th units). If the predator search rate is A and R is the current prey density then the encounter rate of a searching predator with prey is AR. If handling of a single prey item takes h time units then Th = h A R Ts. Thus, T = Ts + Th = Ts(1 + h AR) and the number of consumed prey by a predator during time T is ARTs. The average consumption rate over time interval T is then

which is the Holling type II functional response (Figure 2b). This functional response is concave and for large prey abundances it converges to 1/h, which is the upper limit on consumption. The form of the Holling type II functional response is equivalent with the

Holling type I

Holling type I

5 10 15

Prey abundance

5 10 15

Prey abundance

10 15

Prey abundance

10 15

Prey abundance

Holling type II

Holling type II

40 60 Prey abundance

40 60 Prey abundance

20 40 60 80 Prey abundance

20 40 60 80 Prey abundance

Figure 2 The three Holling type functional responses (left panel (a), type I; (b) type II; (c), type III). Rcrit in panel (c) is the critical prey density below which the functional response is stabilizing. The right panel shows the effect of the functional response on the equilibrium stability. Stability condition [8] requires that the ratio of consumed prey to total prey abundance is an increasing function of prey abundance. Parameters: A = 1, h = 0.1.

Figure 2 The three Holling type functional responses (left panel (a), type I; (b) type II; (c), type III). Rcrit in panel (c) is the critical prey density below which the functional response is stabilizing. The right panel shows the effect of the functional response on the equilibrium stability. Stability condition [8] requires that the ratio of consumed prey to total prey abundance is an increasing function of prey abundance. Parameters: A = 1, h = 0.1.

Michaelis-Menten rate of substrate uptake as a function of the substrate concentration.

The Holling type II functional response assumes that the predator search rate A is independent of the prey density. However, there are several ecological processes that can make this parameter itself a function of prey abundance, that is, A(R). These processes include, for example, predator inability to effectively capture prey when at low densities, predator learning, searching images, predator switching between several prey types, optimal predator foraging, etc. When substituted to the Holling type II functional response, this added complexity can change the concave shape of the functional response to a sigmoid shape (Figure 2c). Sigmoid fUnctional responses are called the Holling type III fUnctional responses. A prototype of such a functional response is obtained when A is replaced by ARm ~1 in the Holling type II functional response, which then leads to

Figure 3 The Beddington-DeAngelis functional response [3]. Parameters: X = 1, h = 0.1, z = 0.2.

Figure 3 The Beddington-DeAngelis functional response [3]. Parameters: X = 1, h = 0.1, z = 0.2.

a particular form of the Holling type III functional response

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