From: Alan McGowen <alanm@igc.org>
Newsgroups: sci.environment
Date: 05 Apr 92 17:16 PDT
Subject: ECO CENTRAL 4
Nf-ID: #N:cdp:1466601309:000:6881
Nf-From: cdp.UUCP!alanm    Apr  5 17:16:00 1992


ECO CENTRAL 4

               Predator Functional Response
               and Prey Refugia

Functional Response Curves

In the predator-prey models I presented in ECO CENTRAL 1 and 2, 
the predator functional response was assumed to be a capture 
probability A times the prey density V. This is not too 
unrealistic when the prey are scarce, but if there are more prey 
than the predators can consume -- i.e. if AV is large enough that 
the predators become *satiated*, the functional response curve 
will show a drop below AV as V increases. 

We can make a simple model of the predator handling function.
Let m be the number of prey eaten per predator, h be the time 
required to capture and eat the prey (called the predator's 
*handling time* for the prey, and T the total time available for 
feeding and foraging. Thus the time during which the predator is
actually eating (or handling the prey) is hM, where M is the 
number of prey eaten, and the time during 
which the predator is searching is T - hM. The total number of
prey encountered and caught by each predator will be 
proportional to the time spent searching  and the prey density V:
M = A(T - hM)V, or

1) M = ATV/(1 + AhV)  (Type II)

This functional response is known as type II, and is the basis of 
a great deal of modeling.  Type I is a linear functional response 
ATV  up  to some maximum value AVmax,  and then a  constant  value 
thereafter:

2) M  =  ATV    for V < Vmax    (Type I)
      =  ATVmax for V >= Vmax  

A type I functional response occurs when the handing time is 0:
for example, a filter feeder such as a baleen whale or a sessile 
marine animal exhibits a type I functional response. 

Predators that are capable of learning exhibit yet another kind 
of functional response. Below a certain encounter rate, the prey 
is taken opportunistically and the functional response is very 
low. As V increases, the predator begins incorporating the prey 
into its  *search image* and begins to seek it actively -- at 
this point the number of prey taken rises rapidly with V until 
satiation sets in. This sort of curve is sigmoid. Optimal 
foraging theory, which attempts to calculate the amount of effort 
a predator should devote to each of n different prey types with 
differing nutritional values and capture probabilities, also 
predicts sigmoid, or type III functional responses for each prey 
type. There are also circumstances in which individuals exhibit 
type II curves but the population as a whole exhibits a type III 
curve when the varying individual curves are summed. A purely 
empirical form of type III is given by:

3) M = ATV^q/(1 + AcV^q)   (Type III)

where generally q is set to 2.0. In general there is no reason to 
expect A, c, or q to be independent of V.

Spatial Inhomogeneity 

It is also possible to incorporate spatial inhomogeneity of the 
prey distribution into the functional response. Suppose the 
environment is divided into V equal-size plots so that on average 
each plot has one prey individual. The predator searches plots, 
and if it encounters a prey, attempts to take it. Since the prey 
are discrete, each plot contains exactly 0, or 1 or 2 etc prey. 
Let N be the number of plots which contain non-zero prey 
individuals. The probability that a randomly encountered plot 
contains prey is N/V, and the probability that the plot contains 
exactly x prey is 

Prob(x) = (N/V)^x exp(-N/V) / x!

and the probability that the plot contains no prey is

Prob(0) = exp(-N/V)

and that the plot contains prey is

Prob (x != 0) = 1 - exp(-N/V).

The number of prey taken per plot is the capture probability A 
times the number of plots containing prey 1 - exp(-N/V), and the 
total number taken is this times the number of plots V: 

4) M = AV(1 - exp(-N/V))   (spatial inhomogeneity)

The shape of this curve is very similar to that of 1) or 3). Note 
that the existence of prey refugia is handled by 4). If there are
n refugia, the number of patches containing no accessible prey is 
exp(-(N-n)/V), and 4) becomes

5) M = AV(1 - exp(-(N-n)/V)) (prey refugia) 

Thus the dynamics of a predator-prey system with refugia will
be similar to that of a system with predator satiation, since 5) 
is similar in shape to 2). Functional response curves 1) - 4)
are known to introduce limit cycles to the solution set of
predator-prey models. 5) is formally the same as 4),
and it will also exhibit cycles for some parameter values.

  
Predator-prey Cycles

These functional response curve can be plugged back into the 
predator-prey model with logistic prey growth to get a model with 
more realistic predator behavior, for example, using the type II 
curve:

6) dV/dt = r(1 - V/K)V - ATVP/(1 + AhV)

7) dP/dt = BATP/(1 + AhV) -dP.

When functional response curves such as 1) - 5) are incorporated 
into the model, in addition to stable equilibria, limit cycles 
appear for some combinations of parameters. These are different 
from the cycles exhibited by the first model I presented in ECO 
CENTRAL 1. The cycles there had amplitudes that were set by the 
initial conditions of the system: limit cycles have amplitudes 
that are set by the model parameters. In a sense cycles represent 
a stable oscillation, since when the system is perturbed away 
from the limit cycle, it returns to it.

Do limit cycles actually occur in nature? It is thought that they 
do, but there is surprisingly little hard evidence that true limit 
cycles, as opposed to random fluctuations, do occur. Analysis of 
the famous lynx - hare cycles. records of which were kept by the 
Hudson Bay Company, suggest that the cycling was not due to 
noise; however, it has been suggested that the cycling may have 
been a result of the trapping activities of humans rather than of 
the endogenous dynamics of the system. However, other systems 
have been found  which seem to exhibit cycles, e.g. lemmings, 
preyed upon by owls and jaegers in Alaska, and consuming grass, 
appear to exhibit a cycle involving all three animals and the 
grass. Putative cycles always seem to occur in boreal 
environments. To the best of my knowledge, no evidence for cycles 
(as opposed to fluctuations) has been found in tropical moist 
forests or coral reefs. 

The effect of a limit cycle is to bring the population to a 
periodic minimum. It thus destabilizes the population against 
extinction due to stochastic effects, if the amplitude is large. 
This will also be true of limit cycles in the system with prey 
refugia. Since predator numbers are usually smaller than prey 
numbers, they are more often the population that becomes extinct.
Thus, the incorporation of refugia into a model like that 
presented in ECO CENTRAL 2 could, by introducing cycling, 
increase the chances of extinction at least of the predators: it 
could *de*stabilize the system against stochastic extinction.

-------
Alan McGowen