Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 154

the top of the model interface will increase run time significantly). What do you notice
about the cost? Is it stabilizing? What is the least number of runs required in order to
achieve a reliable cost? How would you justify your answer mathematically?

Exercise 5.12. What effect does altering poison-cost have on the cost of a fixed
schedule? What effect does rabbit-cost have? What do you think are reasonable
choices for these values if this were an actual field containing rabbits and grass?
Justify your answer.

5.6 SCALING AND AGGREGATION
Our investigation of agent-based models and how to formulate control problems for
them motivates this section and the next. Searching for an optimal solution in such
models often requires running many thousands of simulations, thus performing such
simulations as quickly as possible is a primary concern. In this section, we discuss
means of reducing the run time and complexity of agent-based models via scaling
and aggregation.
Scaling is a method of shrinking the size of an agent-based model in order to
improve run time. In Exercise 5.1 , we explored the “Rabbits Grass Weeds” model.
The dimensions of this model are 43

45 patches for a total of 1935 patches. The
default number of rabbits is 150. In an attempt to scale this model, we may reduce
the dimensions (and correspondingly, the initial number of rabbits). If we change the
dimensions to 25

×

× 25 for a total of 625 patches, we may choose the initial number of

625
1935

rabbits to be 150 48. Our hope, then, is that the pertinent model dynamics

remain the same at this reduced size, since in that case we can run all subsequent
trials at this reduced size for a substantial decrease in run time.
The first question we need to answer is exactly what it means for the model dynam-
ics to remain the same, and howwe can verify that this is indeed the case? Since we are
using the model with a specific control objective in mind, and the value of a solution
relies on the associated cost function, this cost function will help us determine how to
quantify whether our scaled model retains the pertinent dynamics. In Section 5.5 ,we
used the “Rabbits and Grass” model to determine control schedules for poisoning the
rabbits. In that case, our cost function relied on two parameters: the rabbit population,
and the number of days in which poison was used. In particular, the day-by-day grass
levels are not relevant to the cost function, nor are the energy levels of the rabbits;
thus these parameters need not be preserved in the scaled model. In the scaled model,
the number of days in which poison is used is unchanged, so this variable is also
unaffected by scaling. Then the only parameter we need to preserve is the rabbit
population. To be more specific, we need to answer the following question: what are
the smallest dimensions we can use in the model (by scaling down the number of
rabbits accordingly) so that the average day-by-day rabbit population dynamics in
the scaled model follow the same pattern as those dynamics in the original model?
We answer this question by simulating many runs at each size and keeping track of

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