Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 166

or patch as described in the model. In the rabbit and grass model, one could assign
a variable x i for each patch. The values of the finite field k }

then represent how many rabbits and how much grass is currently on the patch. The
polynomials f i determine the number of rabbits and amount of grass on patch i at
the next iteration, given the current number of rabbits and grass. We will explain in
more detail how to derive the polynomials for an agent-based model in Section 5.9 ,
however their existence is assured by the following result.
Theorem 5.5 ([ 26 ]).

={ 0 , 1 , 2 ,..., p − 1

k r

Let f : → k be any function on a finite field k. Then there

k r k r

exists a unique polynomial g .

Any such mapping over a finite field can be described by a unique polynomial.
Using Lagrange interpolation , we can easily determine the polynomial. Let f

: → k, such that ∀ x ∈ , f ( x ) = g ( x )

k r : →

k be any function on k . Then

r

1

1 p −

g ( x ) = f ( c i 1 ,..., c ir ) − ( x j − c ij ) (5.1)

( c i 1 ,..., c ir ) ∈ k r j = 1

is the unique polynomial that defines the same mapping as f .
Example 5.6.

2
3

Suppose k = F 3 , r = 2, and the mapping f is defined on F =

{ 0 , 1 , 2 }×{ 0 , 1 , 2 } as follows:

f ( 0 , 0 ) = 0 ,

f ( 0 , 1 ) = 1 ,

f ( 0 , 2 ) = 2 ,

f ( 1 , 0 ) = 1 ,

f ( 1 , 1 ) = 2 ,

f ( 1 , 2 ) = 0 ,

f ( 2 , 0 ) = 2 ,

( , ) = ,

f 2 1 0

f ( 2 , 2 ) = 1 .

Then the polynomial g that defines the same mapping as f is constructed as follows:

g ( x , y ) = 0 + 1

x 2

2

( 1 − )( 1 − ( y − 1 ) ) +

2

x 2

2

( 1 − )( 1 − ( y − 2 ) ) +

1

2

y 2 ( 1 − ( x − 1 ) )( 1 − ) +

2

2 2

( 1 − ( x − 1 ) )( 1 − ( y − 1 ) ) +

0 +