Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 144

analysis of agent-based models. In essence, the only approach to model analysis is
model simulation. That is, from given initializations of agent states and environmen-
tal parameters, one observes the time evolution of the system and hopes to observe
patterns that might help one draw conclusions about, e.g., steady states of the model.
Through choosing many initializations and doing many simulation runs from a given
initialization in the case of a stochastic model, one aims to obtain global dynamic
information about the model. Little else can be done because, in essence, the model
consists of a computer program rather than a set of mathematical equations, and there
are few things one can do with a computer program other than run it.
The lack of mathematical analysis tools extends in particular to the arena of opti-
mal control. Existing approaches are heuristic in nature, based on domain knowledge.
The goal of this chapter is to describe some of these approaches and to outline steps
one can take to expand the repertoire of available tools to include techniques based
on mathematics. The way to do this is to translate an agent-based model into a math-
ematical object, such as a system of equations of some sort, that makes it amenable
to mathematical analysis tools. There are several possible ways in which to do this,
and research is only in its early stages. Thus, the reader should see this chapter as a
snapshot of an exciting research area that is evolving rapidly and providing rich oppor-
tunities for contributions at all levels. This chapter showcases one possible approach
and the steps that have been accomplished on the road to developing mathematical
tools for optimal control of agent-based models.
5.2 A FIRST EXAMPLE
Go to http://ccl.northwestern.edu/netlogo/models/RabbitsGrassWeeds . There you
will find an agent-based model of rabbits in a field of grass and weeds. At each time
step (or “tick”) the rabbits move in a random direction (they lose energy by moving).
If there is grass at their location, they eat it and gain energy. If their energy level climbs
above a certain threshold, they give birth (in this model, a new rabbit is spontaneously
created at the location of the parent). Upon birth, the parents' energy is halved, and
the offspring is created with this halved energy level. Upon each tick, empty squares
(or “patches”) have a certain fixed probability of grass re-generating. Weeds can also
be introduced in order to further complicate the dynamics; their behavior is similar
to that of the grass.
Near the top of the page you will find an option that allows you to run this model in
your web browser. Spend some time reading through the description in order to get a
feel for how this model works. Click setup and then go to run the simulation (press
go again to stop the simulation). Note that you can speed up or slow down the model
by using the slider at the top of the interface. Each time you wish to re-initialize the
model and start over, you must click setup again.
In thismodel, each of the grid squares (henceforth referred to as patches ) are agents,
and each rabbit is an agent as well. Notice that the rabbits move around randomly,
eating grass as they encounter it. Note too that the patches are colored greenwhen grass