Interactions Between Searching Predators and Hidden Prey - Search Theory: A Game Theoretic Perspective - page 233

15.4.3 An Example Payoff Function

We shall consider the function g ( t )= ν exp ( − ν t ) . This represents a predator who

spots a prey individual at constant rate , conditional on the fact that there was

initially a prey individual to be spotted, and that it has not already fled or been
spotted by the predator. We obtain the following results, which are illustrated for a
particular choice of parameters in Fig. 15.2 .

ν

= , = = =

1. p 0 t 0 defines an equilibrium if t 0 maximises ( 15.20 )when p 0, (note

that p = 0 automatically maximises ( 15.19 )for t = 0). Using ( 15.22 ), this gives

α ( 1 − γ ) ν − λ ( 1 − α ) − λα < 0 ⇒ (15.23)

λ

ν <

− γ ) .

(15.24)

α ( 1

2. p = 0 , t > 0 defines an equilibrium if ( 15.22 ) is satisfied for a value of t > 0

when p = 0, and for this t , p = 0 is optimal. For p = 0( 15.22 ) becomes

e − ν t e − ν t α ( 1 − γ ) ν − λ ( 1 − α ) − λα = 0 ⇒ (15.25)

1
ν

ln α (

1 − γ ) ν − λα

λ (

= .

t (15.26)

1 − α )

=

The value of t in ( 15.26 ) is only positive if ( 15.24 ) does not hold. p 0is

optimal if ( 15.21 ) holds i.e.

1 − β

e − ν t 1 − <

− γ .

(15.27)

1

These two conditions thus yield

1
1

λ 1 − α

α

1
β − γ

− γ ) < ν < λ − γ +

. (15.28)

α ( 1

3. p 0 defines an equilibrium if ( 15.21 )and( 15.22 ) are satisfied with

equality. These yield

> 0 , t >

1
ν

1 − γ

β − γ

t = ln (15.29)

and

λ ( 1 − α )( 1 − γ )

p = 1 −

− γ ) ν − λ ) ,

α ( β − γ )(( 1

the second of which requires

1
1

1 − α

α

1
β − γ

λ

− γ +

< ν . (15.30)