A PRIMER ON QUALITATIVE MODELING OF COMPLEX ECOSYSTEMS
2001 Workshop
Philippe Rossignol, Dept. Entomology, OSU (rossignp@bcc.orst.edu)
Hiram Li
Jeff Dambacher
A mathematical model in community ecology, or population dynamics model,
typically consists of a set of equations and numerical solutions. From the solutions,
biologists can assess for the presence or absence of equilibrium, the set of solutions to the
variables when change is zero. Simply put, can all populations in the system coexist at some
constant positive levels? The presence of such a solution then requires establishing a quality,
that is, stability, a measure of system recovery following a disturbance. This analysis would
be relatively simple if the exact quantitative relationships between each population could be
measured reliably. In a complex system, even with only three variables, this level of
quantification is often not possible in practice.
Fortunately, it is not necessary to measure exact quantities in most cases because the
mathematical criteria of determining stability of a system require only the sign of the
relationships, either positive, negative, or absent. The reason is that it is only the quality of
feedback or the direction of a returning signal that defines stability. The mathematics that
will be presented at the workshop is from graph theory and matrix algebra, and is the basis of
qualitative modeling in ecology.
DATA AND MODELS
Qualitative analysis is now becoming accepted as a standard approach in biology.
The most popular text in the mathematical ecology (Edelstein-Keshet, 1988, Mathematical
Models in Biology) places an "emphasis throughout on qualitative methods...". Here is Dr.
Jorgensen's (IET, Inc.) presentation of qualitative and quantitative analyses. Qualitative
analysis addresses issues of efficacy (does it work or not?). We will present novel advances
that provide a greater than ever flexibility to the method. Quantitative analysis addresses
efficiency (how well does it work?). The second assumes that the first has been assessed,
which is often not the case. Cost, difficulty and loss of generality and realism will increase
from nominal to ratio type of data. Many questions, such as effects of input and of new
introductions or behavior of functional responses gain little from a parametric approach.
Parametric studies are best undertaken to resolve an ambiguity identified through qualitative
analysis. Data collected a priori of a mathematical or statistical hypothesis are usually
meaningless.
Type of Data
Nominal
Ordinal
Interval
Ratio
Statistical Analysis
Categorical
Non-parametric
Non-parametric
Parametric
Model
Natural history
Qualitative
‘Weighted’ qualitative
Quantitative
PHILOSOPHY OF MODELING
The importance of models, and the strategies used in constructing them has been
analyzed by Richard Levins. His philosophy of modeling, which is widely accepted, is that
human requirements are for understanding, predicting and modifying the world around us.
The scientific method and modeling in particular provides us with generality, realism and
precision that generally, but not exactly, correspond with our requirements. However, it
seems that only two of those objectives are achievable through modeling. In fact, anymore
would probably be counterproductive, since it may require a model as complex as the system
itself.
Imagine an ecologist-observer in the center of the circle below, which represents the
overlap of her intellectual and physical universes. Since focus on a specific problem is
required, she must restrict herself to no more than parts of two sections and sacrifice some
field.
Due to the nature of the ecology, models of complex systems are best at providing
general and realistic solutions, and weakest at precision. Unfortunately, precision is often
what is most required by farmers, fishermen and managers who deal with resources from
natural systems. Precise models are available but are typically expensive and severely lack
in generality, both spatial and temporal. It can be a situation fraught with potential conflict
between what scientists can best provide and what society most urgently requires. This
dilemma has been discussed on a philosophical level (Levins 1966, Orzack and Sober 1993,
Levins 1993).
BASIC LEXICON OF QUALITATIVE ANALYSIS
Variables: represented by large circles; this is a variable named ‘1’. A variable is typically a
single species population, but can represent other entities as well, such as
guilds, chemicals, physiological states or regulatory commissions, etc. Thus,
it can be intensity of predation instead of predator population size, or it can be
convenient shorthand for a stable complex sub-system. Define it well and
make it measurable and variable.
Links are rates of transfer between variables. A positive effect between variables is
represented by a traditional arrow or a plus sign while a negative effect is
represented by a 'bubble' arrow or negative sign between two variables. Links
can usually be interpreted in terms of consumer uptake or predatory rates, thus
ultimately of birth or death rates. An absence of arrow means no direct
interaction between two variables. A path is a series of links that never
crosses a variable twice. A loop or cycle is a path that returns to the starting
variable without crossing any variable twice.
Self-effects are represented by self-feeding arrows.
A negative self-effect refers to density-dependence or self-regulation. This
phenomenon is common in many populations, usually at the lower trophic
levels. It is usually due to logistic growth, in which the population grows
exponentially at very low levels, but where the number of deaths increases to
the square of density. The equilibrium reached, where births equal deaths,
defines the carrying capacity. Use it if a predator has other prey than ones in
the system, or if a resource or prey is tied to another stable system.
A positive self-effect, or self-excitation, is typically not an inherent property
of most natural systems. 'Pioneering' species can display this trait under
certain conditions. Anthropogenic input, such as some types of harvest, is the
most common justification for its use in modeling.
Press or permanent change in birth or death rate refers to input from outside
(as through an experiment, natural conditions or selection). It represents a
change in mortality or fecundity. It is represented simply as an unlinked
positive or negative arrow on a variable.
Feedback
Negative feedback returns a signal in the opposite sign of input, hence ‘negative’.
Note however that the signal needs not return as a negative, but rather the negative of input.
A thermostat is an instrument of negative feedback. Populations that can exist in the absence
of other community members exhibit negative feedback.
Positive feedback returns in the same sign and thus amplifies input. 'Feedback' in a
public address system is positive.
Pairwise relationships
Type of pair-wise relationship
Name
Predator-prey
Feedback
Negative
Interference
Positive
Mutualism
Positive
Commensalism
0
Amensalism
0
SOFTWARE REQUIREMENTS
This primer presents elements for constructing qualitative models of ecological
communities. Their analysis will be elaborated during the workshop. We have converted
these techniques to software containing a symbolic processor, MATHCAD, which removes
tedious hand calculations and minimizes errors (any version from 3.0 on is adequate).
MAPLE is recommended for advanced analysis. We will also provide a graphical program in
POWERPLAY that automatically carries out relevant analyses in MAPLE. You will find that
modern computer programs, available only in the last decade, allow researchers to reach
rapid and clear conclusions and to quickly analyze a large number of complex systems.
MODEL CONSTRUCTION AND PRELIMINARY ANALYSIS
To analyze a model in MATHCAD, you must first enter it into a matrix, the so-called
community (or Jacobian or cross-impact) matrix. An element of the matrix corresponds to a
particular link and is labeled ai,j, the i refering to the row and the j to the column. The
element should be read as 'the effect to i from j ', thus identifying the arrow. Be careful!