Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 142

interactions of discrete molecules. Modeling such systems using differential equa-
tions is based on two assumptions: first, there are many individuals involved, so that
we can view them collectively as a continuous quantity; second, that we are able to
describe the individual interactions in a “global” manner, as a term in a set of differ-
ential equations, usually involving one or more global parameters. Sometimes, both
of these assumptions are justified, such as for large populations of bacteria or large
quantities of chemicals in a fermenter. But at other times, one or the other, or both,
of these assumptions fail. For example, in a cell, there might only be two or three
molecules of a particular protein present at any given time, so that its role in a reg-
ulatory network becomes discrete and stochastic, and cannot be accurately modeled
with continuous functions. And, as another example, in an ecosystem with several
species, continuous models sometimes cannot accurately capture extinction events
when one species reaches a very low count.
For these reasons, and others, several other frameworks have been used to model
biological systems. One of these consists of so-called “agent-based,” also called
“individual-based” models. These types of model have a long history, going back
to the 1940s and the work of John von Neumann [ 4 ].
Von Neumann was interested in the process of self-replication and aimed to con-
struct a machine that could faithfully replicate itself. A theoretical instantiation of
such a machine turned into the concept of a cellular automaton as a computational
model. A very well-known example of a cellular automaton is John Conway's Game
of Life [ 5 ]. Since it includes many of the basic concepts of agent-based models, we
describe it briefly.
The Game of Life takes place on a chess-board-like 2-dimensional grid. This grid
can either be finite or extend infinitely in all directions, thereby yielding two different
versions of the game. Each square, or cell , on the grid can be either black or white.
Since the Game of Life is intended to simulate life, a cell is instead referred to as
either alive (black) or dead (white). Each cell away from the boundary of the grid has
eight neighbors on the grid that physically touch it, four with which it shares an edge,
and four that touch only on the corners. Cells on the boundary have fewer neighbors.
To make all cells uniform, one can make the assumption that a cell at an edge of the
grid, but away from the corner, has as additional neighbors the corresponding cells
on the opposite edge of the grid. Thus, we effectively “glue” opposite edges together,
so that the grid is situated on a torus rather than in the plane. Now that all cells have
eight neighbors, we establish a rule that determines the evolution of the cells on the
grid by determining what the status of a given cell is, depending on the status of its
eight neighbors. The rule is quite simple.
￿ Any live cell with fewer than two live neighbors dies;
￿ Any live cell with two or three live neighbors stays alive;
￿ Any live cell with more than three live neighbors dies;
￿ Any dead cell with three live neighbors comes alive.
Thus, the rules are reminiscent of a population whose survival is affected by under-
and overpopulation. If we now initialize this “Game” by assigning a “live” or “dead”