Functional responses
A functional response describes how the number of prey consumed by an individual predator per unit time changes as prey abundance changes.
What exactly is the function describing?
Let \(N\) denote prey abundance or density. Write
\[f(N)=\text{prey consumed per predator per unit time}.\]If there are \(P\) predators, the total prey removal term is commonly
\[f(N)P.\]Thus the functional response is a per-predator feeding rate. Multiplying by predator abundance converts it into total predation pressure.
Why this matters in predator–prey models
The classical Lotka–Volterra prey equation uses the predation term
\[aNP.\]This is equivalent to assuming
\[f(N)=aN.\]So the classical model already contains a functional response: the simplest linear one.
Type I functional response
The Type I response is
\[\boxed{f(N)=aN}.\]Here \(a\) measures how rapidly feeding increases with prey density.
If prey abundance doubles, the feeding rate doubles. If prey abundance triples, the feeding rate triples.
The limitation of Type I
If \(N\) becomes arbitrarily large, \(aN\) also becomes arbitrarily large. A real predator cannot consume infinitely many prey per unit time.
After capturing prey, a predator may need time to subdue, eat and digest it. This motivates the Type II response.
Type II functional response
A standard Type II response is
\[\boxed{f(N)=\frac{aN}{1+ahN}}.\]Here \(a\) is an attack/search parameter and \(h\) is the average handling time per prey item.
At low prey density, encounters are uncommon and the response is approximately linear. At high prey density, prey are easy to find but handling time limits how quickly they can be consumed.
Where does the Type II formula come from?
Suppose a predator has total available time \(T\). Let \(T_s\) be time available for searching. If encounters occur at rate \(aN\) while searching, the expected number of prey eaten is
\[C=aNT_s.\]If each captured prey requires handling time \(h\), total handling time is \(hC\), so
\[T_s=T-hC.\]Substituting gives
\[C=aN(T-hC).\]Rearranging,
\[C(1+ahN)=aNT,\]and therefore the consumption rate \(C/T\) is
\[\boxed{\frac{C}{T}=\frac{aN}{1+ahN}}.\]The maximum Type II feeding rate
When prey are extremely abundant, searching time becomes relatively unimportant. The predator is almost continuously handling prey. Mathematically,
\[\lim_{N\to\infty}\frac{aN}{1+ahN}=\frac{1}{h}.\]So \(1/h\) is the maximum feeding rate in this model.
Half-saturation prey density
The Type II response reaches half of its maximum when
\[N=\frac{1}{ah}.\]This quantity helps describe how quickly the predator approaches saturation. A predator with a high attack rate or long handling time reaches half-saturation at a lower prey density.
Type III functional response
A Type III response is sigmoidal: feeding is weak at very low prey abundance, rises rapidly at intermediate abundance, and then saturates.
A common form is
\[\boxed{f(N)=\frac{aN^2}{1+ahN^2}}.\]The squared prey term is one simple mathematical way to produce the low-density suppression and S-shaped response.
Why might predation be weak at low prey density?
Several biological mechanisms can produce a Type III-like response. Predators may have difficulty locating rare prey, prey may have effective refuges, predators may learn to capture a prey type only after encountering it frequently, or predators may switch toward a more abundant alternative prey species.
Compare Types I, II and III
| Response | Low prey density | High prey density | Main idea |
|---|---|---|---|
| Type I | approximately linear | continues linearly in the basic form | consumption proportional to prey density |
| Type II | approximately linear | saturates | handling time limits feeding |
| Type III | suppressed | saturates | low-density refuge, learning or switching plus saturation |
Do not confuse functional and numerical responses
A functional response describes how much prey an individual predator consumes as prey density changes.
A numerical response describes how predator abundance changes in response to prey availability, through reproduction, survival, immigration or movement.
Putting a functional response into the prey equation
If prey grow logistically and predators consume prey at rate \(f(N)\), a general prey equation is
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-f(N)P.\]For Type II this becomes
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-\frac{aNP}{1+ahN}.\]The shape of \(f(N)\) can therefore change the dynamics of the entire predator–prey system.
Predator growth from consumption
If a fraction \(e\) of consumed prey is converted into predator population growth, a corresponding predator equation can be written
\[\frac{dP}{dt}=e f(N)P-mP.\]For a Type II response,
\[\frac{dP}{dt}=e\frac{aN}{1+ahN}P-mP.\]Functional responses can change stability
Replacing the linear Lotka–Volterra interaction with a saturating response changes the feedback between predator and prey. Depending on parameters, the resulting system may approach an equilibrium, show damped oscillations, or sustain cycles.
This is why choosing a functional response is not merely a cosmetic change to the equation.
Predator saturation can reduce control of abundant prey
Under a Type II response, once predators are close to their maximum feeding rate, further increases in prey density produce little increase in consumption per predator.
This can allow prey to escape strong predator control at high abundance unless predator numbers also increase.
Type III can protect rare prey
Because consumption is weak at low prey density, a Type III response can provide a form of low-density protection. Predation pressure becomes stronger only after prey become more common.
This can have stabilising effects in some predator–prey systems, although the full outcome depends on the rest of the model.
Parameters and units
Units depend on whether \(N\) represents abundance or density and on the precise formulation. Parameter units must be chosen so that \(f(N)\) has units of prey consumed per predator per unit time.
Dimensional consistency is an important check when constructing or fitting a functional-response model.
Estimating a functional response from data
Experiments can expose predators to different prey densities over a fixed time and record the number consumed. Candidate functional-response curves can then be fitted to the observations.
Care is needed because prey may be depleted during the experiment, predators may interfere with one another, and attack behaviour may change with hunger or experience.
Predator interference
The simple functions above depend only on prey abundance. At high predator density, predators may interfere with one another, reducing individual feeding rates.
More advanced functional responses can therefore depend on both \(N\) and \(P\), rather than prey density alone.
Functional responses beyond predator–prey ecology
Similar saturating functions appear throughout mathematical biology whenever a biological rate cannot increase indefinitely. Examples include resource uptake, enzyme kinetics and some host–parasite interactions.
The biological interpretation changes, but the general modelling idea of a rate that increases and then saturates is widely useful.
Stochastic interpretation
In a stochastic individual-based or continuous-time Markov model, the functional response can determine the rate at which predation events occur. Actual predation events are then random even when the underlying rate is specified deterministically.
This distinction is useful: the functional response specifies the event rate, while the stochastic model determines the random sequence of events.