A Real World Example
The models we have dealt with so far, the Malthus, logistics and
LotkaVolterra models are very useful in learning the art and science
of modeling, but have some weaknesses that limit their usefulness in
modeling real biological systems. In the October 6, 1977 issue of
Nature Magazine (Vol. 269, pg. 471) a review article appears which
discusses several interesting models which have had important
applications in the real world. The models presented are one and two
species models using differential equations. The extensive
bibliography at the end of the article provides an excellent reading
list for those interested in pursuing other realistic examples of
mathematical modeling.
The Spruce Budworm Model
One of the models discussed in this paper is of interest to us because
it involves an insect pest found in northern Minnesota pineries, the
spruce budworm (Choristoneura fumiferana). The model was
developed in Canada to describe infestations observed there, but is
certainly relevant to northern Minnesota. Our interest lies in seeing
how models can be used to evaluate management decisions in the natural
resource realm.
The spruce budworm crawls upon and consumes the leaves of coniferous
trees. Excessive consumption can damage and kill the host. The
budworms themselves are eaten primarily by birds, who eat many other
insects as well. The budworms prefer larger trees. A key factor in
determining the spruce budworm population is the leaf surface area per
tree. Larger trees have larger leaf surface areas, resulting in
larger spruce budworm populations.
The Canadians had observed that the spruce budworm population
underwent irruptions approximately every 40 years. For unknown
reasons the budworm population would explode, devastating the
pineries, and then return to their previous managable levels. The
loss of timber represented a significant cost to the Canadian wood
products industry and various management techniques, pesticide
application, for example, were tried without success.
In an effort to understand the cycles of spruce budworm populations,
and with an eye toward developing inexpensive and effective management
of the problem, several scientists at the University of British
Columbia (R. Morris, D. Ludwig, D. Jones and C.S. Holling) studied the
problem and produced a series of mathematical models.
As is often the case in real world modeling, the models became simpler
as the researchers learned which processes were critical to the
dynamics of the system and which could be removed from the model
without seriously affecting it usefulness.
The simplest of these models is a single species model, measuring only
spruce budworm populations. The actual differential equation is
dN/dt = r*N*(1  N/K)  (beta*P*N*N)/(No*No + N*N)
where

N, the spruce budworm population, is the only variable in the
problem.
The parameters in the problem are

r, the natural growth rate, as in the logistic model;

K, the carrying capacity, as in the logistic model;

beta is a measure of predation efficiency. If birds are good
at catching spruce budworms, this number will be larger than if birds
often miss the budworm they are attacking.

P is the bird population, considered a constant in this model;

No is called the switching value, more on this soon.
This real world model is simply the logistic model with one additional
term which is designed to incorporate the effects of predation. The
predation term deserves some attention.
The parameter beta, represents the predation efficiency of
birds. P represents the number of birds, and the remaining
term, N*N/(No*No + N*N) is called a Holling Type III
predation function. It measures how intensively the birds will select
spruce budworms for predation. The idea is that birds are lazy, they
will go where food density is high allowing them to consume much
while expending minimal energy. If the spruce budworm density is low,
birds will opt for some other prey which most likely lives in other
parts of the trees. For example, if birds decide that beetles are
abundant, they will congregate around tree trunks and branches where
beetles can be found, leaving the spruce budworms unmolested.
On the other hand, once the budworm population increases the birds
will leave the beetle habitat behind and begin focusing on the easier
prey, budworms in this scenario. Thus predation on budworms exhibits
this switching phenomenon, and it is this behavior which is
represented by the Hollings Type III function. The importance of the
parameter No, known as the switching value, is that
when N = No the value of the Type III predation function is
exactly onehalf. This indicates the population at which predators
begin showing increased interest in harvesting budworms.
Through observation it had been observed that two of the model
parameters, No and K, are directly dependent on the
average leaf surface area per tree. Letting S be the average
leaf surface area, then
No = 0.5 * S
K = 4 * S.
This relationship between leaf area and No and K
allows us to investigate the effect of growing trees on the budworm
population.
Exercises

Using the following parameter values, build a Stella version of the
spruce budworm model. Include a converter for S, the average
leaf surface area. Values for this parameter will be supplied later.

P = 10

r = 0.1

beta = 20

S = (see below)

Using an initial population of 30 spruce budworms, determine the
stable equilium for the budworm population for the following values of
S = 1700, 2150, 2500. You may want to use a Stella table as
well as a graph in your analysis.

Repeat the analysis done above for the same values of S but
with an initial population of 25000 budworms. When comparing your
results with those from the previous problem, you should notice
something interesting.

Try to determine the range of S values for which this interesting
phenomenon occurs.

Play with various values of S and initial budworm populations
until you feel you understand what is happening. Once you
understand what is happening, formulate a theory as to why it
is happening.
More Analysis
So far we have held leaf size constant during each run of the model.
Since forests, left unmanaged, typically grow with time it would be
interesting to see how tree growth affects our model. For starters we
will make some very simple assumptions concerning tree growth, namely
that S increased by a fixed amount each year. In particular,
we will assume that S increases from 100 to 6000 over the
course of a single model run.
To implement this in our Stella model we must change the formula
stored in the S converter. So far we have used converters to
store only constants. Converters can also store formulas, which is
what we need if S is going to change during a model run.
To have S change gradually from 100 to 6000 over the course
of a single model run, the following formula should be placed in the
S converter:
100 + 5900*TIME/STOPTIME
The quantities TIME and STOPTIME can be chosen from
the scrollable windown in the converter dialog box or they can be
typed in by hand. The quantity TIME is the model time
during the current time step of the model; it changes with each time
step of the model. STOPTIME is the ending time of the model
run. Its value is constant through a given run. The converter
formula given above causes S to have value 100 at the
beginning of a model run (TIME = 0) and a value of 6000 at
the end of the run (TIME = STOPTIME).
A similar formula could be used to model a declining forest, where
S might fall from 6000 to 100 over the course of a model run.
Exercises

Note how the spruce budworm population evolves when we start with a
low average leaf surface area (i.e. young trees) and gradually grow
the forest. Pay particular attention to the population behavior in
the middle of the 100 to 6000 range.

Repeat the same experiment gradually reducing the average leaf
surface area from 6000 to 100. This could correspond to the effects
of an insect infestation or overharvesting the forest. Again, pay
particular attention to the budworm population near the midpoint of
the range.

Does the budworm population take the "high road" or the "low road" as
the forest grows? As the forest declines?

Making the assumption that the forest grows (S gets bigger)
when budworm populations are "small" and diminishes when budworm
populations get "large," what kind
of growth patterns for budworms and spruce trees would be expected if
the forest were left unmanaged?
How does this compare with the cyclical pattern actually observed in
Canada's forests?
Two Species Budworm Model
The budworm model we have developed so far has focused almost entirely
on the budworm itself. This has proven useful in developing an
understanding of the cyclical nature of the budworm population.
However, to test the effects of various management options it seems
necessary to explicitly include a model which accounts for the changes
in S, the average leaf surface area per tree, as time moves
forward.
The original model discussed in the Nature article proposes a
logistic growth model with a predation term. Their suggested equation
is
dS/dt = p*S*(1  S/L)  e*N
where P is the intrinsic growth rate of S,
L is a capacity limit on leaf surface area and
e is a parameter indicating how the present of budworms
affects tree growth.
You should feel free to incorporate this in a Stella model is you
would like. Unfortunately, the Nature does not provide us
with values for these parameters. So, I propose we examine a simpler
model of tree growth, a model which simply says that S will
grow faster when exposed to fewer budworms and will decline in the
presence of too many. The differential equation for this simple model
of tree growth is:
dS/dt = 0.01*(3000  N).
To implement this model it is necessary to change your Stella
model from one where S is a converter to one where S
is a stock.
Exercises

Create the two species Stella model using the simpler differential
equation for tree growth. Create whatever graphs and/or tables you
feel will be helpful in analyzing the results. Run the model a few
times to see if the output makes intuitive sense to you.

The goals of many private private landholders in the region affected
by spruce budworms is to create large trees and then cut them down.
Larger trees are prefered because they can fetch a particularly
attractive price in the lumber market. Large spruce budworm
populations represent a loss to the landowner because any trees killed
by budworm infestation cannot be sold to lumber mills. Using
information gleened from the model can you suggest a
management scheme for landowners which would allow for the growth of
large trees while keeping the budworm at a somewhat low level?
Possible schemes to consider include use of pesticides, clear cutting
and replanting, (resulting a stands of trees of exactly the same age),
selective cutting, (resulting in trees of varying ages) or leaving the
business altogether.
Make changes in your model which reflect your proposed management
changes and determine whether the the effect is a desired one.
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