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The predator pays an opportunity cost
per unit time it spends looking for the prey
i.e. if it leaves, the expected reward it would get elsewhere per unit time is
λ
λ
.The
model parameters are summarised in Table
15.3
.
Parameter
Meaning
α
The probability that a prey individual is present
β
The probability of survival if the prey breaks cover and runs
γ
(
<
β
)
The probability of prey survival if the predator spots the prey
G
(
t
)
The probability that the predator finds the prey (if present) by time
t
g
(
t
)
The rate of search success of the predator, the derivative of
G
(
t
)
λ
The cost paid by the predator per unit time during the search
ν
The predator's discovery rate at
t
=
0
Table 15.3
The parameters for the stationary predator and hidden prey model
The prey's strategy is the time
s
after the predator's arrival when it will run. The
predator's strategy is the time
t
after its arrival to give up if no prey has been found.
For given values of
s
and
t
, the prey reward (simply its survival probability) is
R
(
s
,
t
)
where
G
(
t
)
γ
+(
1
−
G
(
t
))
s
>
t
R
(
s
,
t
)=
(15.17)
G
(
s
)
γ
+(
1
−
G
(
s
))
β
s
≤
t
The predator reward
P
(
s
,
t
)
is its probability of catching the prey minus
λ
times
the expected time spent in the search. This is
⎧
⎨
0
g
α
{
G
(
t
)(
1
−
γ
)
}−
λ
{
(
1
−
α
G
(
t
))
t
+
α
(
x
)
xdx
}
s
>
t
P
(
s
,
t
)=
α
{
G
(
s
)(
1
−
γ
)+(
1
−
G
(
s
))(
1
−
β
)
}−
λ
{
(
1
−
α
)
t
(15.18)
⎩
+
α
(
0
g
(
x
)
xdx
+(
1
−
G
(
s
))
s
)
}
s
≤
t
Our game is a type of asymmetric generalized war of attrition. The longer the
predator waits, the greater the conditional probability that in fact there is no prey
in cover, so the predator must eventually stop searching. Thus under most circum-
stances the expected incremental payoff to the predator of waiting extra time will
decrease and there is likely to be a unique point at which it is best for the predator
to leave. We suppose that the predator chooses pure strategy
t
.
For the prey the risk is high early on, but it knows that it can outwait the predator.
A natural strategy for the prey is to either run immediately, or not at all. In fact,
similarly to the game of [
2
], there is no point it waiting a short time, and exposing
itself to the risk of being attacked, only then to flee before the predator gives up. The
only possible choices are thus to flee immediately or to choose a leaving time which
is longer than the predator is prepared to wait (and any such time is equivalent to
never to flee). We suppose that the prey chooses
s
=
0 with probability
p
,and
s
=
∞
with probability 1
−
p
. We denote this strategy by
p
.
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