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We thus have a game where the predator chooses a time
t
, and the prey chooses
a probability
p
. We note that we have not ruled out the possibility of other strategy
combinations associated with a mixed predator strategy. If the predator used a mixed
strategy involving at least two different strategies
t
1
and
t
2
t
1
, then the payoffs to
playing either of these as a pure strategy would have to be equal in any population
in equilibrium. This in turn means that there must be some probability of the prey
fleeing between times
t
1
and
t
2
. Why would a prey individual flee at such interme-
diate times, having exposed itself to initial risk of being attacked (similarly to the
argument above)? One reason could be that the predator gets better at searching as
it aclimatises to the location. This cannot be ruled out, so it may be that such mixed
strategies occur (although it is not obvious that they can). As we shall see, under
reasonable assumptions (including that the predator does not get better at searching
the longer it waits) we find a unique strategy of this type for all parameters. We also
assume that if the predator and prey choose to leave at the same time, then an attack
initiated by the prey is the result (i.e. as it prepares to go the predator sees the prey
flee and launches an attack).
The payoffs for prey and predator become
>
R
(
p
,
t
)=
p
β
+(
1
−
p
)
G
(
t
)
γ
+(
1
−
p
)(
1
−
G
(
t
))
,
(15.19)
P
(
p
,
t
)=
α
{
p
(
1
−
β
)+(
1
−
p
)
G
(
t
)(
1
−
γ
)
}−
λ
{
(
1
−
α
)
t
t
+(
1
−
p
)
α
(
g
(
x
)
xdx
+(
1
−
G
(
t
))
t
)
}.
(15.20)
0
Against a given
t
the optimal prey strategy is the value of
p
which maximises the
expression for
R
(
,
)
=
(
=
)
p
t
in (
15.19
), which is
p
0
p
1
when
1
−
β
G
(
t
)
<
(
>
)
−
γ
,
(15.21)
1
and all values are equivalent if (
15.21
) becomes an equality.
The optimal predator strategy occurs where
P
achieves its maximum value
in (
15.20
). A local maximum of this expression occurs when
(
p
,
t
)
dP
(
p
,
t
)
=
α
(
1
−
p
)(
1
−
γ
)
g
(
t
)
−
λ
(
1
−
α
)
−
λ
(
1
−
p
)
α
(
1
−
G
(
t
)) =
0
.
(15.22)
dt
(
)
For most reasonable functions
G
this decreases with
t
, so there will be at most one
such value. If there is such a value this is the optimal predator choice of
t
,otherwise
(if the right hand side of (
15.22
)isnegativeat
t
t
0 is optimal.
We thus look for combinations of
t
and
p
which satisfy (
15.21
)and(
15.22
). It
is easy to see that there is no stable solution with
p
=
0) then
t
=
1, since this means that it is
impossible for prey to be in hiding after time 0, so that
t
=
=
0 maximises (
15.20
), but
for
t
0 maximises (
15.19
)). Similarly no
stable solution can exist involving a mixed strategy 0
=
0 it is best for the prey to wait (i.e.
p
=
<
p
<
1and
t
=
0, again since
p
=
0 maximises (
15.19
)when
t
=
0. This leaves three possibilities,
p
=
0
,
t
=
0;
p
=
0
,
t
>
0and0
<
p
<
1
,
t
>
0.
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