Barnes D.J., Chu D. Introduction to Modeling for Biosciences

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1.4 Validity and Purpose of Models 11

Apart from merely varying the basic assumptions of the model it is also good

modeling practice to vary the modeling technique itself. Usually one and the same

problem can be approached using more than one method. Biochemical systems,

for example, can be modeled in agent-based systems, differential equations, using

stochastic differential equations, or simulated using the Gillespie [21] and related

algorithms. Each of these approaches has its own advantages. If the modeler uses

more than just a single approach, then this will quite naturally lead to varying as-

sumptions across the models. For example, differential equation models usually rest

on the assumption that stochastic ﬂuctuations are not important, whereas stochastic

simulations using Gillespie’s algorithm are designed to show ﬂuctuations. Specif-

ically during early stages of a modeling project, it is often enlightening to play

with more than one technique. Maintaining various models of the same system is,

of course, also a good way to cross-validate the models and overall can lead to a

higher conﬁdence into the results generated by them. Of course, building several

models requires considerably more effort than building just one!

1.4 Validity and Purpose of Models

A saying attributed to George E.P. Box [12] is: “Essentially, all models are wrong,

but some are useful.” It has been discussed already that a model that is correct, in

the sense that it represents every part of the real systems, would be mostly useless.

Being “wrong” is not a ﬂaw of a model but an essential attribute. Then again, there

are many models that are, indeed, both wrong and useless. The question then is, how

can one choose the useful ones, or better, the most useful one from among all the

possible wrong ones?

A comprehensive theory of modeling would go beyond the scope of this intro-

ductory chapter, and perhaps also over stretch the reader’s patience. However, it is

worth brieﬂy considering some types of models classiﬁed according to their useful-

ness, though the following list is certainly not complete. The reader who is interested

in this topic is also encouraged to see the article by Groß and Strand on the topic of

modeling [25].

A simple, but helpful way to classify models is to distinguish between, (i) pre-

dictive, (ii) explanatory and (iii) toy models. The latter class is mostly a subset of

explanatory models, but a very important class in itself and, hence, worth the extra

attention. As the name suggests, predictive models are primarily used for the pur-

pose of predicting the future behavior of a system. Intuitively, one would expect that

predictive models must adhere to the most exacting standards of rigour because they

have to pass the acid test of correctness. There is no arguing with data. Therefore,

the predictive model, one might think, must be the most valid one; in some sense

the most correct one. In reality, of course, the fact that the model does make correct

predictions is useful, but it is only one criterion, not always the most important one,

and never sufﬁcient to make a model useful.

A well known class of models that are predictive, but otherwise quite uninforma-

tive, are the so-called empirical formulas. These are quite common in physics and

12 1 Foundations of Modeling

are models that have been found empirically to correctly predict the results of exper-

iments. Sometimes, these empirical models even make assumptions that are known

to be completely incorrect, but they are still used in some circumstances, simply for

predicting certain values. An example of such a model is the “liquid drop model” in

nuclear physics which crudely treats the nucleus as an incompressible drop of ﬂuid.

Despite its crude nature, it does still have some useful predictive properties. There

are many other such models. Purely predictive models do not tell us anything about

the nature of reality, even though they can be quite good at generating numerical

values that correspond well to the real world. This is unsatisfactory in most circum-

stances. Models should do more than that—they should also explain reality in some

way.

Explanatory models do explain reality in some way, and are very important in

science. Unlike predictive models, their primary purpose is to show how certain as-

pects of nature that have been observed experimentally can be made sense of. Often

explanatory models are also predictive models; in these cases, the predictive accu-

racy can be a criterion for the usefulness of a model. However, one could easily

imagine that there are explanatory models that do not predict reality correctly. Par-

ticularly in biology, this case is quite common because of the chronic difﬁculty of

measuring the correct parameters of systems. A quantitative prediction is then often

impossible. In those cases one could decide to be satisﬁed with a weaker kind of

prediction, namely qualitative prediction. This means that one merely demonstrates

that the model can reproduce the same kinds of behavior without insisting that the

model reproduces some measured data accurately. Again, such qualitative predic-

tion is not necessarily inferior to a quantitative prediction; this is particularly true

when the quantitative prediction relies on ﬁtting the model to empirical data, which

risks being the equivalent of an empirical formula.

The main evaluation criterion for explanatory models is how well they illuminate

a particular phenomenon in nature. Prediction is one way to assess this, but another

is the structural congruence between model and reality. Explanatory models are

often used to ask whether a certain subset of ingredients is sufﬁcient to explain

a speciﬁc observed behavior. A good example in this respect is models in game

theory. A typical objective of research in game theory is to understand under which

conditions co-operation can evolve from essentially selﬁsh agents. Game-theoretical

models practically never intend to predict how evolution will continue, but simply

attempt to understand whether or not a speciﬁc kind of behavior has the potential to

evolve given a set of speciﬁc conditions. Nearly all models in this ﬁeld are “wrong”

but they are productive, or, as Box said, “useful.” They tell us something about

the basic properties of evolution and, as such, add to human understanding and the

progress of science, without, however, allowing us to predict the future, or (in many

cases) retrodict the past. Within their domain, these and many other explanatory

models are of equal usefulness as predictive models. Explanatory models are not

the poor relation of their more glorious predictive cousins, and their value can be

independently justiﬁed.

A sub-class of explanatory models are “toy models.” These are models that are

primarily used to demonstrate some general principle without making speciﬁc ref-

erence to any particular natural system. The main advantage of toy models is that

1.4 Validity and Purpose of Models 13

they are very general in scope, while at the same time abstracting away from all

the complications of reality that normally make the life of a modeler difﬁcult. Toy

models take parsimony to the extreme. Due to their simplicity, such toy models can

often be formulated mathematically and, as such, provide general insights that can-

not be obtained from more speciﬁc models. Nonetheless, insights extracted from

such models can often be transferable to real cases.

A particularly famous example of such a toy model is Per Bak’s sand pile model

described in his very well written book, “How nature works” [1]. The basic idea of

this model is to look at a pile of sand on which further grains of sand are dropped

from time to time. At ﬁrst, the pile grows, but after some time it stops growing

and newly dropped grains may fall down the sides of the pile, potentially causing

“avalanches”. There are events, therefore, where adding a single grain causes a large

number of existing grains to become unstable and glide down the slope of the pile.

Bak’s original model was a computer model and does not actually describe the be-

havior of real sand. However, an experimental observation with rice [17] has been

found to behave in a similar way. So, Bak’s computer model is wrong—dramatically

so—yet, it is useful.

The beauty of Bak’s model is that it provided an explanatory framework that

allowed the uniﬁcation of a very large number of phenomena observed in nature,

ranging from extinction in evolution, to earthquakes and stock market crashes. None

of these events were accurately modeled by Bak’s sand pile, but ideas from the

model could be applied to them. The avalanche events had their counterparts in real

systems, be it waves of extinction events in evolution or earthquakes that suddenly

relax tensions that have become built up in the Earth’s tectonic plates. As such,

Bak’s toy model opened up a new way to look at phenomena, and allowed novel

scenarios to be considered.

Not all toy models are of the same generality as Bak’s. Even in more restricted

scope, it is good idea to start modeling enterprises using toy models which can then

be reﬁned. The criterion of the usefulness of such toy models is clearly the extent to

which they help generate new understanding. They also harbor a danger, however;

if taken too far then toy models can actually obscure rather than illuminate.

In summary: There is no single criterion to assess the quality or usefulness of

a model. In every modeling enterprise, the speciﬁc choices made to produce the

model must be justiﬁed each time. Predictive power in a model is a good thing, but

it is not the only criterion, neither should it be used as the acid test of the quality of

a model—at least not in all cases. Sometimes, the most powerful models are highly

simpliﬁed and do not predict anything, at least not quantitatively.

Having now established what we believe to be the most fundamental guiding

principles for developing good modeling skills, it is time to get going with the prac-

tice. The rest of the book primarily concerns itself with technique. What we mean is

that it will introduce the reader both to the underlying ideas and the scope of mod-

eling techniques, but it will also present details of practical tools and environments

that can be used in modeling. In addition to the focus on technique, walk-through

examples will be provided to show the reader how the various guidelines of good

modeling established in this chapter translate into actual modeling practice.

Chapter 2

Agent-Based Modeling

Traditionally, modeling in science has been mathematical modeling. Even today, the

gold standard for respectability in science is still the use of formulas, symbols and

integrals to communicate concepts, ideas and arguments. There is a good reason for

this. The language of mathematics is concise and precise and allows the initiated

to say with a few Greek letters what would otherwise require many pages of text.

Mathematical analysis is important in science, but it would be wrong to make its use

an absolute criterion for good science.

Many of the phenomena modern science studies are complicated, indeed so com-

plicated that the brightest mathematicians have no hope of successfully applying

their craft to describe these phenomena. Most of reality is so complex that even

formulating the mathematical model is close to impossible. Yet still, many of these

phenomena are worth our attention and meaningful knowledge can be obtained from

studying them.

This is where computer models become useful tools in a scientist’s hands. Simu-

lation models can be used to describe and study phenomena where traditional math-

ematical approaches fail. In this chapter we will introduce the reader to a speciﬁc

class of computer models that is very useful for exploring a multitude of phenomena

in a wide range of sciences.

Computers are still a relatively recent addition to the methodological armory of

science. Initially their main use was to extend the range of tractability of mathemat-

ical models. In the pre-computer era, any mathematical model had to be solved

by laborious manual manipulations of equations. This is, of course, time inten-

sive and error prone. Computers made it possible to out source the tedious hand-

manipulations and, more importantly, to generate numerical solutions to mathemat-

ical models. What takes hours by hand can be done within an instant by a computer.

In that way, computers have pushed the boundaries of what could be calculated.

2.1 Mathematical and Computational Modeling

In what follows, we are not going to be interested in computer-aided mathematical

modeling, i.e., methods to generate numerical solutions. Instead, we will be look-

D.J. Barnes, D. Chu, Introduction to Modeling for Biosciences,

DOI 10.1007/978-1-84996-326-8_2, © Springer-Verlag London Limited 2010

16 2 Agent-Based Modeling

ing at a class of computer models that is formal by virtue of being speciﬁed in a

programming language with precise and unambiguous semantics. Yet the models

we are interested in are also non-mathematical, in the sense that they represent and

model phenomena that go well beyond what can even be formulated mathematically,

let alone be solved.

Computer models are formal models even when they are not mathematical mod-

els. They are written in a precise language that is designed not to leave any room

for ambiguity in its implementation. Every detail of the model must be expressed in

this language and no aspect can be left out. All the computer does, once the model is

formulated, is to mercilessly draw conclusions from the model’s speciﬁcation. For-

mal analysis of this kind is a useful tool for generating a deep understanding of the

systems that are studied, and is far superior to mere verbal reasoning. It is therefore

worth spending time and effort to develop computational representations of systems

even when (or precisely when) a mathematical analysis seems hopeless.

How do we know that a system is not amenable to mathematical analysis but can

be modeled computationally? Maybe, it is easier ﬁrst to understand what it is that

makes a system suitable for mathematical analysis. Possibly the most successful

and inﬂuential mathematical model in science is that formulated by Newton’s laws

of motion. These laws can be applied to a wide variety of phenomena, ranging from

the trajectory of a stone thrown into a lake, to the motion of planets around their

stars. One feature that Newton’s laws share with many models in physics is that

they are deterministic. The deﬁning feature of a deterministic system is that, once

its initial conditions are ﬁxed, its entire future can be calculated and predicted. The

word “initial” implies somehow that we are seeking to identify the conditions that

apply at the “beginning” of a system. In fact, initial conditions are often associated

with the condition at the time t = 0; in truth, this is only a convenient label for

the time at which we have a complete speciﬁcation of the system, and does not,

of course, denote the origin of time. All that matters is that we have, for a single

time point, a complete speciﬁcation of the system in terms of all the positions and

velocities of its components. Given those then we can compute the positions and

velocities of the system for all future (and, indeed, past) times, if the system is

deterministic.

A good example of such deterministic behavior is that of the planets within our

solar system. Since we know the laws of motion of the planets, we can predict

their positions for all time, if we only know their positions at one speciﬁc time

(that we would arbitrarily label as time t = 0). The positions of the planets are

relatively easy to measure, so determining the initial conditions is not a problem.

Equipped with this knowledge, astronomers can make very accurate descriptions

of phenomena such as eclipses or the reappearance of comets and meteors (that

might or might not collide with the Earth). Applying the same laws further, we can

predict when a certain beach will reach high tide or how long it will take before a

tsunami hits the nearest coast. Determinism is also the essential requirement for our

ability to engineer electrical and electronic circuits. Their behaviors do not follow

from Newton’s laws, but they are also deterministic, as are many of the phenomena

described by classical physics.

2.1 Mathematical and Computational Modeling 17

2.1.1 Limits to Modeling

2.1.1.1 Randomness

In the intellectual history of science, determinism was dominant for a long time but

was eventually replaced by a statistical view of the world—at least in physics. Firstly

thermodynamics and then quantum mechanics led to the realization that there is in-

herent randomness in the world. The course of the world is, after all, not determined

once and forever by its initial conditions. This insight was conceptually absorbed

into the scientiﬁc weltanschauung and, to some extent, into the body of physical

theory. Despite the conceptual shift from a deterministic world-view to a statistical

one, mathematical modeling, even in the most statistical branches of science, contin-

ues not to represent the randomness in nature, at least not directly. Take as evidence

that the most basic equation in quantum mechanics (the so-called Schrödinger equa-

tion) is a deterministic equation, even though it represents fundamentally stochastic,

indeterministic, physical phenomena. Randomness in quantum mechanics only en-

ters the picture through the interpretation of the deterministic equation, but the very

random behavior itself is not modeled.

To be fair, in physics and physical chemistry there are models that attempt to

capture randomness, for example in the theory of diffusion. However, the models

themselves only describe some deterministic features of the random system, but not

the randomness itself. Mathematical models tell us things such as the expected (or

mean) behavior of a system, the probability of a speciﬁc event taking place at a

speciﬁc time, or the average deviation of the actual behavior from the mean behav-

ior. All these quantities are interesting, but they are also deterministic. They can be

formulated in equations and, once their initial conditions are ﬁxed, we can calcu-

late them for all times. Mathematics allows us to extract deterministic features from

random events in nature. True randomness is rarely seen in mathematical models.

An example might illustrate how inherently stochastic systems can be described

in a deterministic way: Think of a small but macroscopic particle (for example,

a pollen grain) suspended in a liquid. Observing this particle through a microscope

will show that it receives random hits from time to time resulting in its moving about

in the liquid in a seemingly random fashion. This so-called Brownian motion cannot

be described in detail by mathematical modeling precisely because it is random.

Nevertheless, very sophisticated models can give an idea of how far the particle

will travel on average in a given time (mean), by how much this average distance

will be different from the typical distance (standard deviation), how the behavior

changes when force ﬁelds are introduced, and so on. Note that the results of this

mathematical modeling, as important as they are to characterize the motion of the

particle, are themselves deterministic. This does not make them irrelevant; quite the

opposite. The point to take from this discussion is that the description of the random

particle is indeed a description of the deterministic aspects of the random motion,

and not of the randomness itself.

One could argue that there is not much more we could possibly want to know

about the pollen grain beside the deterministic regularities of the system. What good

18 2 Agent-Based Modeling

is it to know about the idiosynchracies of the random drift of a particular particle?

And yes, perhaps, in this case we really only want to know about the statistical

regularities of the system, which are deterministic. In that respect, the randomness

of Brownian motion is quite reducible and we are well served with our deterministic

models of the random phenomenon.

Another example of systems where stochasticity is reducible are gases in statis-

tical mechanics. A gas (even an ideal one) consists of an extremely large number of

particles, each of which is characterized at any particular time by its position and

momentum. Keeping track of all these individual particles and their time evolution

is hopeless. Collisions between the particles lead to constant re-assignments of ve-

locities and directions. In principle, one could calculate the entire time-evolution of

the system if given the initial conditions—ideal gases are deterministic systems. In

reality, of course, this would be an intractable problem. Moreover, the initial condi-

tions of the system are unknown and unmeasurable.

As it turns out, however, this impossibility of describing the underlying motion

of particles in gases is not a real limitation. All we really need to care about are

some stochastic regularities emerging from the aggregate behavior of the colliding

molecules. Macroscopically, the behaviors of gases are quite insensitive to the de-

tails of the underlying properties of the individual particles and can be described

in sufﬁcient detail by a few variables. It is well known that an ideal gas in thermal

equilibrium can be described simply in terms of its pressure, volume and tempera-

ture:

PV ∝T

There is no need to worry about all the billions of individual molecules and their

interactions. We do not need to know where every molecule is at any given time.

All we care about is how fast they are on average, the probability distribution of

energies over all molecules, and the expected local densities of the gas. Once we

have those details we have reduced its random features to deterministic equations.

This approach of reducing randomness really only works if the individual ran-

dom behavior of a speciﬁc particle in our system is unimportant. In physics, this

will often be the case, but there are systems that are irreducibly random;systems

where the random path of one of its components does matter and a deterministic de-

scription of the system is not enough. One example is natural evolution in biological

systems. Random and unpredictable mutations at the level of genes cause changes

at the level of the phenotype. Over evolutionary time scales, there will be many

such mutations, striking randomly, and often leading to the demise of their bearer.

Sometimes a mutation will have no noticeable effect at all but, on rare occasions, a

mutation will be beneﬁcial and lead to an increase in ﬁtness.

Imagine now that we wish to attempt to model this process. If we tried to come

up with a deterministic model of mutations and their effects, after much effort and

calculation we would perhaps know how beneﬁcial mutations are distributed, for

how long we need to wait before we observe one, and so on. This might be what

we want to know, but maybe we want to know more. Imagine that we would like to

model the actual evolution of a species, that is, we are interested in creating a model

2.1 Mathematical and Computational Modeling 19

that allows us to study how and when traits evolve and what these traits are. Imagine

that we want to model the evolutionary arms race between a predator and its prey.

Imagine that we want to have a model that allows us to actually see evolution in

action. In this case then, we do not care about all the mutations that led nowhere.

All we are interested in is those few, statistically insigniﬁcant, events that resulted

in a qualitative change in our system: a new trait, or a new defense mechanism, for

instance. What we would be interested in are the particulars of the system, not its

general features.

Mathematical models are normally not very good at representing particulars of

random systems; different approaches are required if we are interested in those.

We can say, therefore, that if our system is irreducibly random then mathematical

approaches will be limited in their usefulness as models.

2.1.1.2 Heterogeneity

Another aspect of natural systems that limits mathematical tractability is system

heterogeneity. A system is heterogeneous if it: (i) consists of different parts, and

(ii) these parts do not necessarily behave according to the same rules/laws when they

are in different states. As a simple example, one can think of structured populations

of animals, say lions. Normally lions live in packs, and each pack has an intrinsic

order that determines how individuals act in the context of the entire group. This

group structure has evolved over time and is arguably signiﬁcant for the survival of

lions. It is also difﬁcult to model mathematically.

One could try to circumvent this and ignore the detail. Often such an approach

will be successful. If one wanted to model the population dynamics of lions in the

Serengeti, it may be sufﬁcient to look at the number of prey, the efﬁciency with

which prey is converted into offspring by lions, and the competition (in the form of

leopards, cheetahs, etc.). With this information, we might then formulate a reason-

able model of how the lion population will develop over time in response to various

environmental changes. Yet, this model would have ignored much of the structure

of lion populations.

The interactions between the individual animals would be difﬁcult to model in

a mathematical model. Lions behave differently depending on their age, their rank

within the group and their gender. Keeping track of this in a set of equations would

quickly test the patience and skill of the modeler. Moreover, lions reproduce and

die.

Equations seem the wrong approach in this case. At the same time, it is not

unthinkable that one may want to model the behavior of packs of lions formally.

One might, for example, be interested in the life strategies of lions and complement

empirical observations with computer models. The heterogeneity of the pack is irre-

ducible in such a model. It would make no sense to assume an “average” lion with

some “mean” behavior. The dynamics of the packs and how the behavior patterns

contribute to the evolutionary adaptability of lions rest crucially on the lions being a

structured population. We are not aware of attempts to model life histories of lions,

20 2 Agent-Based Modeling

but in the context of social insects and ﬁsh there have been many attempts to use

computer models to understand group dynamics (see [3, 26]); but never have these

studies used purely equation-based approaches.

While we can clearly see that most systems are heterogeneous, often they are

reducibly so. The difference between the component parts can often be ignored and

be reduced to a mean behavior while still generating good and consistent results.

Whether or not a system is reducibly or irreducibly heterogeneous depends on the

particular goals and interests of the modeler and is not a property of the system per

se.

In physics, the heterogeneity of most problems is reducible which has to do with

the type of questions physicists tend to ask. In biology things are different. Many

of the phenomena bioscientists are interested in are essentially about heterogeneity.

Reducing this heterogeneity is often meaningless. This is one of the reasons why

it has been so difﬁcult to base biology on a mathematical and formal theoretical

basis, whereas it has been so successful in physics. Irreducible heterogeneity makes

mathematical modeling very difﬁcult and life consequently harder.

2.1.1.3 Interactions

Finally, a third complicating property of systems is component interaction. Unlike

heterogeneity and randomness, interactions have been acknowledged as a problem

in physics for a long time. In its most famous incarnation this is known as “the n-

body problem”. Theoretical physics has the tools to ﬁnd general solutions for the

trajectories of two gravitating bodies. This could be two stars that are close enough

for their respective gravitational ﬁelds to inﬂuence each others’ motions. What about

three bodies? As it turns out, there is no nice (or even ugly) formula to describe this

problem; and the same is true for more than three bodies.

Interaction between bodies poses a problem for mathematical modeling. How do

physicists deal with it? The answer is that they do not! In the case of complex multi-

body interactions it is necessary to solve such problems numerically. Furthermore,

there are cases of systems that are so large that one can approximate the interac-

tion between parts with a “mean ﬁeld”, which essentially removes any individual

interactions and makes the system solvable. There are many cases of such reducible

interactions in physics, where the mean-ﬁeld approximation still yields reasonable

results. Before the advent of computers, irreducible interactions were simply ig-

nored because they are not tractable using clean mathematical approaches.

In biology, there are few systems of interest where interactions are reducible.

Nearly everything in the biological world, at all scales of magniﬁcation, is inter-

action: be it the interactions between proteins at a molecular level, inter-cell com-

munications at a cellular level, the web of interactions between organisms at the

scale of ecology and, of course, the interaction of the biosphere with the inanimate

part of our world. If we want to understand how the motions of swarms of ﬁsh are

generated through the behavior of individual ﬁsh, or how cells interact to form an

embryo from an unstructured mass of cell, then this is irreducibly about interactions

between parts.

2.2 Agent-Based Models 21

The beauty of mathematical modeling is that it enables the modeler to derive

very general relationships between variables. Often, these relationships elucidate

the behavior of the system over the entire parameter space. This beauty comes at a

price, however; the price of simplicity. If we want to use mathematics then we need

to limit our inquiry to the simplest systems, or at least to those systems that can

be reduced to the very simple. If this fails then we need to resort to other methods,

for example computer models. While computational models tend not to satisfy our

craving for the pure and general truths that mathematical formulas offer, they do

give us access to representations of reality that we can manipulate to our pleasure.

In a sense, computational models are a half-way house between the pure mathe-

matical models as they are predominantly used in theoretical physics, and the world

of laboratory experimentation. Computer models are formal systems, and thus rig-

orous, with all the assumptions and conditions going into the models being perfectly

controllable. Experimentation with real systems often does not provide this luxury.

On the other hand, in a strict sense, computer models can only give outcomes for

a particular set of parameters, and make no statement about the parameter space

as a whole. In this sense, they are inferior to mathematical models that can pro-

vide very general insights into the behavior of the system across the full parameter

space. In practice, the lack of generality is often a problem, and it makes com-

puter models a second-best choice—for when a mathematical analysis would be

intractable.

2.2 Agent-Based Models

The remainder of this chapter focuses on a particular computer modeling technique

—agent-based modeling (ABM)—which can be very useful in modeling systems

that are irreducibly heterogeneous, irreducibly random and contain irreducible in-

teractions. The principle of an ABM is to represent explicitly the heterogeneous

parts of a system in the computer model, rather than attempting to “coarse grain.” In

essence this is achieved by building a virtual copy [7] of the real system; the model

explicitly represents components of the real system and keeps track of individual

behaviors over time. So, in a sense, each individual lion of the pack would have a

virtual counterpart in the computer model. As such, ABMs are quite unlike mathe-

matical models, which represent components by the values of variables rather than

by behaviors. In an ABM, the different components (the “agents”) represent entities

in the real world system to be modeled. As well as the individual entities, an ABM

also represents the environment in which these entities “live.” Each of these mod-

eled entities has a state and exhibits an explicit behavior. An agent can interact with

its environment and with other entities.

The behavior of agents is often “rule-based.” This means that the instructions

are formulated as if-then statements, rather than as mathematical formulas. So, for

example, in a hypothetical agent-based model of a lion, one rule might be:

If in hunting mode and there is a prey animal closer than 10 meters, then attack.