GREG DWYER

*University of Notre Dame*

Historically, mathematical models in ecology have been used largely to provide qualitative explanations for patterns in nature. A classic example of this approach was the effort to use competition models to explain species diversity (Diamond and Case, 1986). Simple competition models showed that species that utilized the same resource can coexist under the right circumstances (Begon et al.,1996). This theoretical observation, however, leads to much controversy over the general issue of whether competition structures natural communities.

This kind of general statement about nature is arguably of little importance for problems of resource management. Perhaps as a consequence, modeling efforts in many applied fields, especially pest management, have often rejected simple mathematical models in favor of giant simulation models (Onstad, 1988). Simulation models have hundreds of parameters and state variables, take years to construct, and are often so complex that they can take pages to describe. Such models represent the opposite extreme from the simple models used in academic research, in that they attempt to sacrifice understandability for ecological realism.

The last few decades, however, have seen increased interest in applied questions among academic ecologists, and the resulting research has begun to suggest an alternative use for simple mathematical models (Hilborn and Mangel, 1996). Specifically, simple mathematical models can be used as statistical hypotheses much as linear models have been used in classical statistics.

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PROFESSIONAL SOCIETIES and Ecologically Based Pest Management: Proceedings of a Workshop
7
On the Use of Mathematical Models in Ecological Research: Example from Studies of Insect–Baculovirus Interactions
GREG DWYER
University of Notre Dame
Historically, mathematical models in ecology have been used largely to provide qualitative explanations for patterns in nature. A classic example of this approach was the effort to use competition models to explain species diversity (Diamond and Case, 1986). Simple competition models showed that species that utilized the same resource can coexist under the right circumstances (Begon et al.,1996). This theoretical observation, however, leads to much controversy over the general issue of whether competition structures natural communities.
This kind of general statement about nature is arguably of little importance for problems of resource management. Perhaps as a consequence, modeling efforts in many applied fields, especially pest management, have often rejected simple mathematical models in favor of giant simulation models (Onstad, 1988). Simulation models have hundreds of parameters and state variables, take years to construct, and are often so complex that they can take pages to describe. Such models represent the opposite extreme from the simple models used in academic research, in that they attempt to sacrifice understandability for ecological realism.
The last few decades, however, have seen increased interest in applied questions among academic ecologists, and the resulting research has begun to suggest an alternative use for simple mathematical models (Hilborn and Mangel, 1996). Specifically, simple mathematical models can be used as statistical hypotheses much as linear models have been used in classical statistics.

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PROFESSIONAL SOCIETIES and Ecologically Based Pest Management: Proceedings of a Workshop
Moreover, current research suggests that many sets of ecological data cannot statistically justify complex models. That is, although nature may appear to be complicated, real data often cannot prove that more complicated models give a better description than simpler models (Hilborn and Mangel, 1996). Whether this is because nature really is simple, or because our data are noisy, is irrelevant for many practical purposes. The fact is that, if we want useful quantitative descriptions of nature, it is typically the case that we need fewer than 10 parameters.
Current work in ecological modeling thus emphasizes close connections between theory and data, and the use of mathematical models as statistical hypotheses about nature. As a result, models that were once viewed as being of only intellectual interest may well become useful in pest management. To make this point concrete, I will review my own work on a virus disease of a forest pest, the gypsy moth Lymantria dispar.
Ecological models of insect diseases began with a simple model by Anderson and May (1981), which started with a model for human epidemics and added population dynamics of insects and pathogens. Anderson and May used the model to make the general point that pathogens may drive the dynamics of forest insects capable of significant outbreaks such as the larch budmoth, Zeiraphera diniana. Further research on this and other insects has instead suggested first that single-factor explanations for forest insect population dynamics are probably generally insufficient, and second that pathogens are not always important players in the population dynamics of forest insects (Hunter and Dwyer, 1998). Nevertheless, even though the original generalization is too sweeping, features of Anderson and May's model have been useful for understanding insect pathogens.
Specifically, Anderson and May's model assumed that the rate of horizontal transmission of the virus increases linearly with the density of the pathogen. This assumption provided a useful quantitative hypothesis, and it is nonetheless interesting even though data show that it is often incorrect. For example, data for the transmission of the gypsy moth virus reject a linear model but cannot reject a nonlinear model (Dwyer et al., 1997). Additional experiments, however, suggested that this nonlinearity arises because of variability among the host insects in their susceptibility to the virus, and a model that allows for this variability can accurately predict the timing and intensity of virus epidemics (or epizootics) in naturally occurring gypsy moth populations. Surprisingly, the resulting model requires only four parameters.
Although this model arose from efforts to answer questions of basic research, it is beginning to have practical applications. For example, efforts are being made to genetically engineer this and other viruses. Consequently, a question of environmental concern is, “Will engineered virus strains outcompete wild-type strains, thereby altering the ecological balance between host and pathogen?” Because the model can predict epidemics from experimental transmission data, it can be used to assess the risks of releasing engineered strains before any such strains have been released (Dwyer et al., in press). Preliminary work has suggested that at least one deletion mutant of the gypsy moth virus is unlikely to be a superior competitor, and work is now advancing

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PROFESSIONAL SOCIETIES and Ecologically Based Pest Management: Proceedings of a Workshop
to apply the model to assess commercially produced strains of the nuclear polyhedrosis virus of the cabbage pest Trichoplusia ni.
More concretely, gypsy moth populations tend to be very patchily distributed, so that a major challenge for managers is identifying which populations need to be controlled and which are likely to collapse. Because the virus model can be used to predict which populations are likely to have severe virus epidemics, it can assist in identifying which populations are likely to collapse.
These studies demonstrate several advantages of using simple mathematical models. First, compared to the logistic expenses of performing experiments and collecting data, the cost of constructing, simulating, and analyzing models is very low. Second, models can allow us to extrapolate between small-scale field and lab measurements and the dynamics of populations. The gypsy-moth-virus model, for example, uses as input only the initial density and frequency of infection of gypsy moths in the field, and measurements of disease transmission and kill rates from small-scale lab and field experiments. The model can nevertheless predict the timing and intensity of virus epidemics in naturally occurring gypsy moth populations on 3–10 hectare plots with great accuracy across a wide range of densities (Dwyer et al., 1997; Dwyer et al., in press). This ability to extrapolate across scales means that the model can be used to predict the outcome of large-scale releases of engineered viruses from measurements before such releases are carried out. Third, by focusing on simple explanations for what superficially appear to be complex natural phenomena, simple mathematical models provide useful testable hypotheses. Moreover, the success of the gypsy-moth-virus model, which includes only four parameters, suggests that many natural phenomena are simpler than they initially appear.
These advantages of simple models should theoretically be even greater in pest management. This is because questions of ecological research can often be phrased somewhat qualitatively, whereas questions of pest management research are ultimately economic and thus inescapably quantitative. I would therefore argue that the infrequent use of mathematical models in pest management is due to an overemphasis on complex simulation models. In addition to being more difficult to understand, such models are inherently more expensive than the simple models that I advocate here. Complex simulation models are therefore less likely to be tested, and in turn are less likely to be discarded in favor of better models. Hopefully simple mathematical models will eventually come to be as useful in pest management as they are in ecological research.
REFERENCES
Anderson, R.M., and R.M. May. 1981. The population dynamics of microparasites and their invertebrate hosts. Philosophical Transactions of the Royal Society, Series B 291:51–524.
Begon, M., J.L. Harper, and C.R. Townsend. 1996. Ecology: Individuals, Populations, and Communities, 3rd edition. Cambridge, Mass: Blackwell Science.

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PROFESSIONAL SOCIETIES and Ecologically Based Pest Management: Proceedings of a Workshop
Diamond, J., and T.J. Case. 1986. Community Ecology. New York: Harper and Row.
Dwyer, G., J. Dushoff, J.S. Elkinton, and J.P. Burand. In press. Host heterogeneity in susceptibility: Tests using an insect virus . In Virulence Management, U. Dieckmann, J.A.J. Metz, M. Sabelis, and K. Sigmund, eds. Berlin: Springer-Verlag.
Dwyer, G., J.S. Elkinton, and J.P. Buonaccorsi. 1997. Host heterogeneity in susceptibility and the dynamics of infectious disease: Tests of a mathematical model. American Naturalist 150:685–707.
Hilborn, R., and M. Mangel. 1996. The Ecological Detective: Confronting Models with Data. Princeton, NJ: Princeton University Press.
Hunter, A.F., and G. Dwyer. 1998. Outbreaks and interacting factors. Integrative Biology 1:166–177.
Onstad, D.W. 1988. Population-dynamics theory: The roles of analytical, simulation, and supercomputer models. Ecological Modeling 43(1”2):111–124.