Interactions Between Searching Predators and Hidden Prey - Search Theory: A Game Theoretic Perspective - page 224

15.2.2 The Model of Cooper and Frederick

In the [ 14 ] model prey flee when the cost of fleeing precisely matches the benefit.
The concept of benefits and costs here are rather abstract. The cost and benefit of
fleeing immediately can only be evaluated if we know what happens if the prey
does not flee immediately, which in turn depends upon when it does eventually flee.
Similarly between two potential fleeing distances, the actions of the prey may affect
subsequent costs and benefits. A more pertinent consideration is, what is the optimal
time to flee for a prey individual to maximise its fitness?
Cooper and Frederick [ 3 ] modelled this by developing a model of the foraging
scenario in [ 14 ] using an explicit fitness function. In their model again both prey and
predator can see the other, and the fitness of an individual, if it survives, depends
upon its resource level. It is assumed that the predator approaches the prey at a
constant speed, so that there is a simple relationship between the time since the start
of the encounter and the distance between prey and predator, and the resource level
is given by

F 0 + B ( d ) − E ( d ) , (15.1)

where d is the distance of the predator from the prey, F 0 is the fitness at the start
of the encounter (when d = d d ), E ( d ) is the energetic cost of escaping at distance

d ,and B is the benefit of waiting from the start of the encounter until the prey

has reached distance d , which is increasing with d (so B

( d )

0). We note that
this benefit is achieved through extra foraging opportunity, and so more properly
depends upon the time since the start of the encounter rather than the distance. In
this instance, since there is a deterministic relationship between time and distance
this is not problematic, but a more realistic model (e.g. with variable predator speed)
would need to contain time as a separate factor.
When the prey flees is has probability of survival P s

( d d )=

( d ) which increases with d .

The total fitness at distance d is thus

F ( d )= { F 0 + B ( d ) − E ( d ) } P s ( d ) . (15.2)

We note that in reality P s may also increase with the level of resources, which in-
creases with time (and so decreases with d ), and so it is possible that in some situ-
ations P s might not increase monotonically with d (although this would likely have
to be associated with a very slow predator approach).
Cooper and Frederick [ 3 ] used the following example functions

B ∗ 1

n

d
d d

f
d m ,

e − cd B ( d )= − , E ( d )= P s ( d )= 1 − , (15.3)

giving the fitness as

F 0 +

B ∗ 1

n

d
d d

f
d m

e − cd F ( d )= − − ( 1 − ) . (15.4)