A Real World Example

The models we have dealt with so far, the Malthus, logistics and Lotka-Volterra 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 one-half. 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.
- 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.