Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 171

Table 5.2 Relative position of
neighbors to agent x .
l 8 l 1 l 2

l 7 x = l 0 l 3

l 6 l 5 l 4

￿ Downhill:

⎛ ⎞

8 i − 1

I p − 1
i

I p − 1
j

⎝ ( ⎠

f ( x ) = 1 − ) l i

i = 1 j = 0

⎛

⎛ ⎞

p − 1

C m − 1 8

⎝

I p − 1
0

⎝ ⎠

+ ( 1 − ) · l 0 +

i (

k − I i )

m = 1 k = 0 j =

⎛

8 i − 1

⎝

p − 1 p − 1

·

1 (

1 − ( m − I i ) ) l i

0 (

m − I j )

i = j =

⎞ ⎞

p − 1

⎠ ⎠ ,

+ ( 1 − ( m − I 0 ) ) l 0 (5.8)

where the concentration ranges from 0 to C C represents the highest possible con-

centration, I i is the concentration level at neighboring patch i , and l i is the relative
location of patch i from the current patch, see Table 5.2 . I 0 is the concentration of
the current patch, I 1 ,...,

,

I 8 are the concentrations of its eight neighbors.
Next, we show that Eq. ( 5.7 ) describes the uphill movement.

1 ⎛

⎛ ⎞

1

8 8 i − 1

p −

⎝ ⎝ (

p − 1 p − 1

⎠

f ( x ) = −

0 (

C − I i ) 1 − ( C − I i ) ) l i

0 (

C − I j )

i = i = 1 j =

1

1

1

1 l 0

C − 1 8

p − p −

+ − ( C − I 0 ) + −

0 (

m − I i )

m = 1 i =

C 8 8

p − 1 p − 1

×

0 (

k − I i )

1 (

1 − ( m − I i ) ) l i

k = m + 1 i = i =

⎞
⎠ .

i − 1

p − 1 p − 1

×

0 (

m − I j ) + ( 1 − ( m − I 0 ) ) l 0 (5.9)

j =

It is straightforward to see that Eq. ( 5.9 ) describes the movement to the neighboring
patch with the highest concentration level: 1

− i = 0 (

p − 1 is 0 unless at least

C − I i )