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15.3.3 The Predator Can Delay Its Attack
In this case, the predator can still see behind it, but need not attack as soon as it
has seen the prey. Now the prey can play the same strategies as considered above,
and the predator plays a strategy
U
, which involves delaying an attack until position
U
∈
[
−
,
]
if the prey is detected before this position, or attacking immediately on
detecting the prey, if detection occurs after it reaches this position.
If
V
1
1
≤
U
, then the prey will always flee before a predator attacks, and will receive
payoff
R
(
V
,
U
)=(
1
−
c
)
{
1
−
f
(
d
(
V
)+
Δ
)
}.
(15.10)
The predator receives payoff
P
(
V
,
U
)=
f
(
d
(
V
)+
Δ
)
.
(15.11)
U
then either the predator will detect the prey before position
U
and will
initiate an attack at position
U
, or the predator will detect the prey and thus initiate
an attack at some position between
U
and
V
, or the predator will not detect the
prey before position
V
and the prey will flee at position
V
. These three possibilities
(respectively) lead to three terms in the payoffs to prey and predator:
If
V
>
V
U
(
dA
(
v
)
(
,
)=
(
)(
−
)
{
−
(
(
))
}
+
−
)
{
−
(
(
))
}
R
V
U
A
U
1
c
1
f
d
U
1
c
1
f
d
v
dv
dv
+(
1
−
A
(
V
))(
1
−
c
)
{
1
−
f
(
d
(
V
)+
Δ
)
},
(15.12)
V
d
(
A
(
v
))
P
(
V
,
U
)=
A
(
U
)
f
(
d
(
U
))+
f
(
d
(
V
))
dv
+(
1
−
A
(
V
))
f
(
d
(
V
)+
Δ
)
.
dv
U
(15.13)
Similarly if the prey plays strategy N, the rewards are
1
U
(
dA
(
v
)
R
(
N
,
U
)=
A
(
U
)(
1
−
c
)
{
1
−
f
(
d
(
U
))
}
+
1
−
c
)
{
1
−
f
(
d
(
v
))
}
dv
dv
+
{
1
−
A
(
1
)
},
(15.14)
1
d
(
A
(
v
))
P
(
N
,
U
)=
A
(
U
)
f
(
d
(
U
))+
f
(
d
(
V
))
dv
.
(15.15)
dv
U
By a similar argument to that used before,
R
(
−
1
,
U
)
is greater than the weighted
average of similar terms that constitute
R
1. This is true for
any
U
, so whatever the predator's strategy, the prey's best strategy will be to either
never flee unless attacked (Strategy
N
) or flee immediately on detecting the predator
(strategy -1).
If strategy -1 is adopted, then the payoffs to both parties are independent of the
predator's strategy. For the case where the prey picks strategy
N
,Itwasshownin[
2
]
(
V
,
U
)
for all
V
> −
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