Almost all errors in analysis occur at this stage of converting the model into a matrix. The
following matrix formally identifies each element.
 a1 1 a1 2 a1 3 


 a2 1 a2 2 a2 3 
a

 3 1 a3 2 a3 3 
MATHCAD procedures
If you have access to MATHCAD, a model is entered and analyzed in the following
way. Using a simple tiered food-chain model, say of a self-regulated plant 1, its herbivore 2
and a specialized predator 3, hand draw the conceptual model.
This type of representation is called a
signed digraph (for di-rected graph).
From this graph, the community matrix can be represented symbolically. Locate the Matrix
icon (View Toolbars Matrix or Ctrl-M) and call up a 3x3 matrix. Into each element
(appearing as black squares), enter the following, corresponding to the diagram above (do not
use subscripts for this analysis). Here -a11 (i.e. negative link to 1 from one) represents the
 a11 a12 0 
 a21 0 a23 
 0 a32 0 


self-effect on variable 1, -a12 is the negative link to 1 from 2, a21 is the positive link to 2
from 1, and so on. This matrix represents the set of Lotka-Voltera simultaneous equations
for this system.
Practice the following operations. On each diagonal element, add ‘-’ by entering –L (in
lower case), selecting L and pressing Ctrl-G (changes font to Greek alphabet). The
solutions to the symbol  are called eigenvalues. Eigenvalues are analogous to the
growth rate and represent the recovery characteristic of each population following a
disturbance.
 a11   a12 0 


0   a23 
 a21
 0

a32 0   

Then ‘grab’ the whole matrix, through either dragging the pointer or pressing the space
bar if your cursor is already in the matrix. Calculate the determinant from Symbolic
Matrix Determinant. You will get a complicated looking expression.
2
3
a11   a11 a23 a32     a23 a32  a21 a12 
Place your cursor on the  key in Symbolic Collect and the polynomial form will be
obvious; this result is the characteristic polynomial, (analogous to the Euler equation)
and solutions to  are the eigenvalues.
3
2
  a11   (a21 a12  a23 a32)   a11 a23 a32
Keeping your pointer on the , key in Symbolic Polynomial Coefficients. The resultant
column vector lists the coefficients of the characteristic polynomial, which are the values
 a11 a23 a32 


 a21 a12  a23 a32 


a11


1


1
of feedback at various levels, from F0 = -1 upwards to F4. For even numbered systems, of
4 variables for example, multiply the vector by –1 for the correct result (select the vector,
then key in *-1; select the whole expression again and Shift-F9). Feedback will be the
basis for assessing the presence of a stable equilibrium. As a preliminary evaluation, all
feedback must be negative for stability to occur, which is the case here.
The limit of matrices in MATHCAD is 10x10,although it may not be able to solve
symbolically systems much smaller. The workshop will introduce MAPLE techniques to
handle very large system.
Practical model building advice
Mathematical modeling is considered essential in modern ecology and conservation
biology. A mathematical model should be conceived as a hypothesis, not perceived a result.
The difference from a more general conception of a hypothesis is that a mathematical
construct gives a narrow and therefore very rigorous hypothesis. Statistics is one form of
mathematical modeling that all scientists have come to accept, but relies heavily on data and
does not deal with large systems very well. Qualitative modeling is a formal technique that
draws inferences from community structure and is well suited for the types of problems that
face community ecologists.
Some rules of thumb, the so-called Quirk-Ruppert rules, to keep in mind when
analyzing or building models are:
-at least one diagonal entry ( a11…) must be negative, or self-regulated, for stability to
occur
-a system with a row or column of zeros is unstable
-interference or mutualism between two species reduces stability due to their positive
feedback. The most stable pairwise relationship is predator-prey because of its negative
feedback
-loops or cycles of three or more parameters introduce instability. In other words, links
that skip a level, such as omnivory, introduce instability
-the presence of unstable relationships does not mean that the system is unstable overall,
but rather that conditions exist that must be satisfied for stability to occur
Also keep in mind that:
-a subsystem with overall negative feedback that is linked to a variable may be
represented as a self-effect. One need not represent the whole subsystem. Thus, a
predator that relies on prey outside the specific system under study would have a selfeffect attached to it.
-a straight chain of predator-prey relationships, since it meets stability criteria, may be
condensed into a single parameter within a larger system
-a system with all interactions as negative or zero, or all positive or zero is unstable
-a system with a variable linked to two or more satellite systems is unstable. A satellite is
a subsytem of zero feedback linked to the system through a single variable, its principal.
This is actually easy to tell because the row and column corresponding to the satellite will
be all zeros except at its principal. Typically, a satellite is a specialized predator;
parameter 3 in the model above is a satellite of parameter 2.
-a satellite will buffer the principal so that no change in other variables will affect the
principal except through the satellite and no change to the principal will affect the system
except the satellite.
-no parameter can be devoid of an input link
-there are two major types of competition. Interference or contest competition is direct
and represented as double negative links. Resource or scramble competition is indirect
and arises when two or more populations consume the same resource; the resultant
instability gave rise to Gause’s Law, also called Competitive Exclusion Principle.
2
THE PARADOX
When ‘introduced’ three decades ago, mathematical modeling of communities
challenged ecology with a serious and difficult paradox, the stability-complexity paradox,
wherein complexity appears incompatible with stability. The mathematical considerations
that coerce this conclusion are often referred to as the Routh-Hurwitz criteria. The intuitive
and so-called Eltonian perspective that had predominated until then had asserted that stability
of ecosystems is maintained because of complexity. Partly as a result of numerous attempts
to circumvent this paradox, stability has become a vague term in ecology and 100+
definitions have been suggested (Grimm and Wiesel 1997).
The community matrix approach is powerful because it reduces the approach to
Malthusian parameters and therefore is interpretable from an evolutionary point of view. It
also sets formal definitions as a starting point. We shall discuss over the workshop how this
approach can be extended to deal with the paradox.
3
EXAMPLES
R
1
Ex. 1: Plankton model (from Stone 1990). Plankton
(P) and bacteria (B) consumes a common resource
(N: nitrogen), while plankton leach carbon to
bacteria.
Two predators, rotifers (R) and
zooplankton (Z), prey on them and leach nitrogen
back into the system when they die.
Z
3
B
2
P
4
N
5
M
1
Ex. 2: Old field model (Schmitz 1990). Four plants
(F, P, S, H) compete for nitrogen (N) and interfere
with each other as well. A grasshopper (M)
consumes each plant. Schmitz completely specified
quantities of the community matrix and then
pressed the system.
F
2
P
3
S
4
H
5
N
6
Ex. 3. Danish shallow lakes (Jeppesen, 1998). A
complex food web of five trophic levels, ranging
from nutrient, macroplankton, phytoplankton,
invertebrates, zooplankton, cyprinids, predatory
fishes and various aquatic birds. Eutrophication is
considered a bottom-up press on nutrients
Ex. 4. Snowshoe hare field press (Krebs et al. 1995,
model from Dambacher et al. 1998). Plants (1) are
consumed by hares (2) who are in turn preyed upon
by various predators (3). The predators have other
important sources of prey and are therefore ‘selfregulated’. Krebs pressed the system in various
ways. Dambacher suggested a bottom-up
commensal link between plants and predators,
representing the likely benefit that plant cover
provides to predators.
Ex. 5. Scallop harvest regulation model (Puccia and
Levins 1985). In order to preserve Nantucket
scallops, two models are proposed. In one, the
commission responds to declining population by
enhancing juvenile breeding. In the other, it
reduces harvest pressure by increasing the cost of
permits. Which will be effective?
4
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Goodman, D. 1975. The theory of diversity-stability relationships in ecology. Quart. Rev.
Biol. 50: 237-266
Grimm, V. & C. Wiesel. 1997. Babel, or the ecological stability discussions: an inventory
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Jeppesen, E. 1998. The ecology of shallow lakes - trophic interactions in the pelagial.
National Environmental Research Institute, Technical Report No. 247. Silkeborg,
Denmark.
Krebs, C. J., S. Boutin, R. Boonstra, A. R. E. Sinclair ad J.N.M. Smith. 1995. Impact of food
and predation on the snowshoe hare cycle. Science 269: 1112-1115
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pp.
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542-44
Puccia, C. J. & R. Levins. 1985. Qualitative Modeling of Complex Systems. An Introduction
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Puccia, C. J. & R. Levins. 1991. Qualitative Modeling in Ecology: Loop Analysis, Signed
Digraphs, and Time Averaging. Chap. 6, pp. 119-143 In, Qualitative Simulation Modeling
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Vandermeer, J. 1981. Elementary Mathematical Ecology. John Wiley and Sons. 294 pp.
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