Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 157

model sizes. Scaling a and b accordingly gives the following revised costs for the
scaled model:

c ( u ) = 200 ( 75 + 15 + 20 + 27 + 36 ) + 2000 ( 0 + 1 + 1 + 0 + 1 ) = 40600 ,

c (v) = 200 ( 75 + 100 + 20 + 4 + 5 ) + 2000 ( 1 + 0 + 0 + 0 + 0 ) = 42800 .

Here we obtain essentially the same cost as the original model, and thus eliminate the
discrepancy in cost.
The “Rabbits and Grass” model lends itself nicely to scaling because the initial
distribution and location of rabbits and grass is random. Thus, when reducing the
dimensions, we may still choose which patches have grass at random, and likewise
may randomly choose where to place our rabbits at the start of the simulation. But
what about models that are not so spatially homogeneous? Suppose the landscape
included a river, or a rocky area in the upper right-hand corner of the map? In such
cases, we cannot simply reduce dimensions because the spatial layout of the model
may be critical to the dynamics. Thus in general, a more sophisticated approach is
required for spatially heterogeneous models.
Suppose that in the “Rabbits and Grass” model, the field consists of a hill whose
peak is at the center of the field. Going out from the peak of the hill, the altitude
decreases; thus at the periphery of the map, the land is flat. Suppose further that we
nowdistinguish between various levels of grass: each patchmay have little or no grass,
some grass, or a lot of grass. Finally, suppose the grass grows more abundantly at
higher altitudes: thus at the peak of the hill there is a lot of grass, and at the periphery
there is less. If we wish to model rabbits on such a landscape, it is important to
maintain these characteristics as we scale the field.
The first step of our approach is to create a matrix that represents the physical
landscape. We may do this by using the values 0, 1, 2, 3 (for example) to represent how
much grass a given patch contains (with 3 being the most abundant and 0 representing
little or no grass). Suppose our original model is 10

× 10 and has a layout as shown

in Figure 5.1 .
In order to scale the model, we reduce the landscape using the nearest neighbor
algorithm. First, we decide what dimensions we would like our reduced model to
have; suppose it is n

n points over
the original landscape (see Figure 5.2 ). Finally, we select values for each point by
choosing the value of the neighbor nearest to that point (see Figure 5.3 ).
Note that this is only one algorithm that can be used for scaling a spatially heteroge-
neous model. Other methods include bilinear interpolation and bicubic interpolation;
the reader is urged to explore the details of these algorithms on their own as this
chapter provides only an introductory look at scaling.
Aggregation is another method of reducing computation and run time by simpli-
fication of the agent-based model. Rather than physically scaling the entire model,
we may aggregate certain agents into groups and view each group as an agent. Thus
there are fewer entities to keep track of, and fewer decisions that need to be made.
This strategy is particularly helpful in models consisting of many agents of the same

× n . We then overlay an evenly spaced grid of n ×