Individual-based ecological models:  Spatial models for undergraduate
investigation

Louis J. Gross 
Departments of Mathematics and Ecology and Evolutionary
Biology University of Tennessee, Knoxville, TN 37996-1300
gross@math.utk.edu

Handout for Mathematics for Life Sciences Students Conference, Iowa
State University, May 17-18, 1996

An ongoing discussion amongst those who work on theory in biology deals
with the relative benefits of reductionist versus holistic viewpoints.
Reductionists tend to argue for breaking systems down to their smallest
feasible objects of interest, trying to build up from mechanisms at
this small scale to larger scale phenomena. Those with a more holistic
view prefer to deal with aggregated properties of systems first, and
then add to it complications imposed by smaller scale structure within
the system. These contrasting views of the process of theory
construction are mirrored in the general process of model construction
in any field. There are those who much prefer to build highly
simplified models first, and then add new variables and constructs in
order to increase the realism, while others prefer to start by
including every aspect of the system which may be important, perhaps
eventually discarding much of it as the process proceeds. The
alternative views may thus converge towards similar models (as measured
by the number of state variables say), but not necessarily. My
experience is that, at least in theoretical biology, those trained as
mathematicians tend towards the holistic end of this spectrum (and it
is indeed more of a continuum than just two distinct contrasting
views), while those with more of an engineering background tend towards
the reductionist view.

A prime example of the contrast between these approaches occurs in
population biology. The vast majority of theory in population biology
has started from very simple differential equations in which a single
variable represents population density, solutions of these are analyzed
mathematically, and potentially compared to abundance estimates from
field or lab observations. Although these models have had a great
influence on theory in ecology, their aggregated form is particularly
difficult to relate to observational biology. As this aggregated view
of a population is highly simplified, a wide variety of extensions have
been made to incorporate (1) size, age or physiological structure
(leading to coupled systems of differential equations or partial
differential equations); (2) space (leading to metapopulation models in
which a population is broken down into distinct patches, or in the
continuous space case to partial differential equations); (3) discrete
generations (leading to difference equation models and matrix models);
(4) stochastic effects (leading to birth and death processes and
stochastic differential equations). In all these cases, as extensions
are added, the models become less analytically tractable, and become
considerably more difficult for students to follow, particularly life
science students with a typically minimal exposure to college level
mathematics.

A consequence of the above is that undergraduate life science students,
if exposed to any mathematical models in ecology at all, will generally
see only a few cases of very general models, which even they realize
are caricatures only minimally related to real biology. Alternatively,
they may be exposed to computer programs (such as Populus) which
calculate the mathematical behavior of more complicated models, though
they typically have little ability to follow the mathematics which
underlies the models. Happily, there are now some options which allow
undergraduates to deal personally with much more realistic models based
on more of a reductionist world view. Exposing students to these models
not only allows them to individually investigate some quite fascinating
ecological phenomena, but also shows them first hand that modern theory
construction is not bound by the limits of analytical mathematics.

Computational ecology will continue to grow in importance, and it is
now quite feasible to expose undergraduates to its promise (as well as
its limitations). This further illustrates for students that there is
not one single "correct" approach to science, and that there are a
diverse array of mathematical and computational tools available -  they
are not necessarily limited by the smattering of calculus and basic
discrete math that is all the quantitative training most students
obtain. Other advantages of exposing students to these computational
approaches is that it gives them control over "an experiment" - due to
it's stochastic nature no two students (or lab groups) will obtain
exactly the same results, and each can be very free to investigate the
effects of a wide variety of different effects. This can also of course
be a disadvantage, as it gives the potential for students to control
experiments with many degrees of freedom. Some constraints and guidance
in the labs is therefore required.

One recent reductionist approach in ecology analyzes systems based upon
the actions of individuals. These individual-based models track the
behavior, growth, reproduction and death of individuals, from which
they build up the dynamics of aggregated units such as populations and
communities. These individual-based models are beginning to have
significant impact on a variety of theoretical and practical questions
in ecology, and there are now a few such models which are available for
teaching purposes. The program we will use in this Workshop is
EcoBeaker, written by Eli Meir and published by Sinauer Press, which
has associated with it a variety of ecological scenarios preprogrammed
to illustrate key ecological concepts (see the WWW site
http://www.zoology.washington.edu/ecobeaker/ecobeaker.html). Other 
individual-based modeling resources include those at the National 
Micropolution Simulation Resource (with a biomedical emphasis - see
http://www.nmsr.labmed.umn.edu/nmsr/NMSR.html), the Swarm program based
at the Sante Fe Institute (see http://www.santafe.edu/projects/swarm/),
David Griffeath's Ecomachine project (see his Primordial Soup Kitchen
Page at http://math.wisc.edu/~griffeat/kitchen.html), and Rick
Durrett's stochastic spatial models (see
http://www.tc.cornell.edu/er94/ff02spring/ff04models.html). A basic,
but now outdated, reference is DeAngelis, D. L. and L. J. Gross (1992)
(editors) (1992) Individual-Based Models and Approaches in Ecology,
Routledge, Chapman and Hall, New York.

There are several ways to construct individual-based models. My purpose
here is to just mention the basics for models with explicit spatial
structure. In this case, some type of spatial grid is set up, in which
each location (or pixel in a computer map) corresponds to a spatial
location with a certain spatial extent for each grid cell. Each grid
cell then is characterized by a state, with each state corresponding to
(a) presence or absence of a given species, (b) number of a given
species present in the cell, (c) numbers of each of several species
present in the cell, etc. There are then two basic approaches: (i)
model how each grid cell changes based upon some set of rules and the
states of surrounding grid cells, or (ii) model how each individual
organism being considered moves among the grid cells and how the
organisms state changes through time. Type (i) is the cellular automata
approach, in which each cell is in one of a relatively small number of
states and the rules for changes in state depend on the current state
and that of nearest neighbors. The program SimLife from Maxis is of
this type, in which each grid location consists of a single individual
of a given species. Type (ii) is a truly individual-based approach, in
which individual organisms can move anywhere across the spatial set of
grid cells, change their respective states (e.g. size, fat content,
number of offspring, etc.) and have their location attached to them as
just another state variable. In this approach, each grid cell just
combines the individuals located in it at any particular time.

The program EcoBeaker allows one to combine both of the above
approaches. It allows allows one to set up changes in states of a grid
cell as a simple transition matrix (possibly dependent on neighboring
cells) as for example used in the Situation File to illustrate the
Intermediate Disturbance Hypothesis. In this File, each grid cell is
assigned to a state based upon the species present (e.g. Grasses,
Blackberry Bushes, Oak Trees, Fire, etc.), with a certain probability
of transition to a new state the following year.  Alternatively, the
program allows species to be set up as Individualistic, in which they
have certain movement rules, and in which there can be several
individuals of different species extant in a given grid cell at any
particular time. The Situation File to Illustrate Competitive Exclusion
is one example of this, in which the various Rabbit species have
differing daily energy requirements, and movement rules. In addition to
allowing students to change the parameters associated with any
particular species, to add new species, and to code entirely new
scenarios (admittedly this is something very few students would
attempt), EcoBeaker also allows students to investigate the effects of
different sampling methods.

While EcoBeaker is structured to be a very easily applied tool, the
complexity of model output is also instructive for students. They have
to decide what are the appropriate output variables to plot, and to
come up with their own methods to summarize results. Though reminiscent
of real field experiments, the program output is obviously more
constrained. It does serve to illustrate however that, though such
models are not that difficult to construct, the interpretation of
results is non-trivial, particularly when trying to pick apart the
effects of different model parameters. This is of course just as true
for more realistic individual-based models, designed not for education
but for application to ecosystem management. On this point, see L. J.
Gross (1994) Limitations of reductionist approaches in ecological
modeling: model evaluation, model complexity and environmental policy.
In Wildlife Toxicology and Population Modeling: Integrated Studies of
Agroecosystems (Eds. R. J. Kendall and T. E. Lacher), pp. 509-518.
Lewis Publishers and CRC Press, Boca Raton.