Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 169

Table 5.1 Movement of an agent
on a grid with n patches.
At Time t At Time ( t + 1 )

Patch 0 Patch 1

Patch 1 Patch 2

Patch 2 Patch 3

. .

Patch i Patch i + 1

. .

Patch n − 1 Patch 0

￿ Agent moves forward one patch per time step:

p − 1

f ( x ) = x + 1 + n

1 (

i − x ). (5.2)

i = 0 , i = n −

￿ Agent moves backward one patch per time step:

p − 1

( ) = − −

1 (

− ).

f x x 1 n i x (5.3)

i =

￿ We can use these equations to construct a more general movement function
f ( x , u ) , where u ∈{− 1 , 0 , 1 } specifies the direction of the agent by taking u =− 1

when the agent moves back one step, u = 0 when the agent stays as is, and u = 1

when it moves one step forward. Note that a p − 1

= 1 for all a = 0 ∈ F p .

p − 1 p − 1

p − 1 p − 1

f ( x , u ) = x + u + un ( u + 1 )

1 (

i − x ) + un ( u − 1 )

1 (

i − x ).

i = 0 , i = n − i =

(5.4)

￿ Agent moves forward m patches per time step:

⎛ ⎞

m p − 1

⎝ ⎠ .

f ( x , m ) = x + m + n

j (

i − x ) (5.5)

j = 1 i = 0 , i = n −

￿ Agent moves backward m patches per time step:

⎛ ⎞

p − 1

m − 1

⎝ ⎠ .

f ( x , m ) = x − m − n

j (

i − x ) (5.6)

j = 0 i = 0 , i =