Forum

In Defense of Big Ugly Models

]. A. LOGAN

Logan

SINeE THE INTRODUCTION OF MODERN,

high-speed computers, there hasbeen a divergence of approaches to model-ing ecological systems. One path builds onthe historical lineage of mathematical analy-sis pioneered by physicists and appliedmathematicians. The objective of this ap-proach is to build a model amenable to theclassical tools of mathematical analysis.Such models generally have been referred toas theoretical models; for example, as usedby May (1976) in his book, TheoreticalEcology. In an attempt to provide modelsamenable to the tools of mathematical anal-ysis, and also to result in general descriptivepower, theoretical models typically followthe principle of parsimony, which states thatthe modeling process should start with thesimplest possible structure and includecomplexity only when absolutely necessary(Berryman et al. 1990, Royama 1992).

The other path has followed the route ofcomputational power provided by digitalcomputers. In this approach, there is neitherthe need nor the motivation for parsimony,and the objective is to provide a mechanisti-cally detailed representation of the modeledsystem. Models developed through thisapproach generally are referred to assimula-tion models. In contrast to the elegant mod-els that are the goal of theoretical ecology,these biologically explicit models tend to becomplex mathematical descriptions-thebig ugly models that are the topic of thisarticle.

Although there is a continuum thatranges from clearly theoretical models onone hand (e.g., the LotkaNolterra preda-tor-prey equations), to complex simulationmodels of an entire insect life system on theother, the principal of parsimony results inqualitative criteria that serve to identifythese two modeling approaches. In generalterms, theoretical models consist of coupleddifferential or difference equations withonly a few state variables! (typically lessthan 10, perhaps only 1), whereas simula-

IState variables are the defining variables of a mod-eled system and describe the status or state of the systemat any particular point in time. Changes in values of statevariables define the temporal dynamics of the system

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tion models may contain tens or even hun-dreds of state variables. The number of pa-rameters likewise differs between the twomodeling approaches; the number of pa-rameters in a theoretical model is typicallyless than 20 (often as few as 4 or 5), and asimulation model may contain hundreds ofparameters. Interpretation of model param-eters also tends to differentiate the two ap-proaches. The principal of parsimoniousstructure in theoretical models necessitatesthat many system interactions either are ig-nored or their effects are combined, result-ing in abstracted, composite parameters.

In our quantitativerepresentations of ecological

associations, we shouldavoid dichotomous and

antagonistic classificationof quantitative approaches.

After all, a model isonly a model.

The computational intensive, mechanisti-cally detailed descriptions of simulationmodels tend to have parameters withstraightforward biological/ecological inter-pretation. In fact, an underlying princi-pal of the simulation approach is systemdescription in terms of parameter valuesthat can be measured directly. Qualitativecharacteristics that differentiate theoreti-cal models from simulation models arediscussed further in Berryman (1991) andLogan (1989a).

Systems analysis, a formalized procedurefor the study of complex systems, typicallyhas relied up on the construction and analy-sis of mechanistically based computer simu-lation models (Hall & Day 1977). Thisapproach had its origin in the engineeringsciences during the early 1960s. Many of theconcepts of systems analysis were estab-lished in Forrester's (1961) influential publi-cation, Industrial Dynamics. For a bookwith such an unpromising title, IndustrialDynamics (and the principles so lucidly

stated by Forrester) has had a profoundinfluence on the science of insect ecology.Based on the principles outlined in thisbook, the most ambitious ecological re-search program to date was launched underthe auspices of the International BiologicalProgram (IBP).2The IBP represented a mas-sive influx of resources into ecologicalresearch, including an entomological com-ponent that became known as the HuffakerPro;ect (Huffaker 1980). Central to the re-search philosophy of the IPB was the devel-opment of large-scale, computer simulationmodels. The modeling approach of the IBPwas embraced fully by the Huffaker project;of 12 specific objectives, 7 were relateddirectly to the development of simulationmodels (Stone 1989).

At first, the application of simulationmodels was endorsed by the entomologicalcommunity at large. This statement is sup-ported by the profound influence that theHuffaker project had on the concepts of pestmanagement as they evolved during the1970s and early 1980s, including the Adkis-son project (Frisbie & Adkisson 1986),which succeeded the Huffaker project, andthe various big bug programs that weresponsored by the USDA Forest Service (e.g.,Brookes et a!. 1978). Publications on thetopic in influential periodicals, such as theAnnual Review of Entomology (Ruesink1976) and Scimce magazine (Coulman et al.1972), are further evidence of the influencethat systems analysis and simulation modelshave had on pest management in general.The initial infatuation with complex simula-tion models, however, began to wane duringthe 1980s, initiating a trend that has contin-ued to the present. An indication of the ex-tent of this disenchantment is exemplified inrecent articles by A. A. Berryman (1991) andA. M. Liebhold (1994), both of whom arequantitatively oriented insect ecologists.

Berryman compares and contrasts therelative strengths and weaknesses of theo-retical and simulation models. In his analy-sis, the theoretical approach was far

lTotal u.s. funding for the IBP was more than $55million in 1970s dollars U. T. Callahan, personal commu·nication).

AMERICAN ENTOMOI.OGIST • Winter 1994

superior to the simulation approach, notonly for basic ecological inquiry, but also inthe prediction, management, and policy-making arenas. Berryman goes on to discussDouglas-fir tussock moth outbreaks, andcontrasts his theoretical model of this inter-action to a large simulation model. In Berry-man's opinion, the theoretical modelresulted in superior predictive power andsignificant ecological insights that werelacking in the simulation model. By impli-cation, Berryman generalizes this specificexample to characterize the relationship be-tween theoretical models and simulationmodels. In broad terms, Berryman's article isa strong advocacy of abstract theoreticalmodels and an equally strong condemnationof large, complex simulation models. Ascontrasted to Berryman's article, Liebhold'saritcle is focused more narrowly on the man-agement application of specific simulationmodels in forest insect and disease pest man-agement. In his analysis, the simulationapproach is found wanting. Liebhold con-cludes that "there is little evidence that thesesystems models have significantly contribut-ed to solving either applied or theoreticalproblems." Liebhold concludes his articlewith a strong advocacy for starting with asimple model and introducing complexityonly if needed. Once again, the general toneof Liebhold's article is one of advocacy forsimple models and condemnation of largesimulation models.

This article in defense of big ugly modelsis not exactly a rebuttal of either Liebhold orBerryman's articles; in fact, I share many oftheir concerns and agree with much of whatthey have to say. Rather, my intent is to pro-vide a counterpoint to their view. Both Ber-ryman and Liebhold begin with theproposition that simulation models havemade valuable contributions to insect ecolo-gy, but then go on to describe failures oflarge simulation models and raise questionsregarding the basic validity of systems anal-ysis as a modeling approach. In contrast, Iwill first acknowledge that large modelingefforts have not always lived up to their ex-pectations, but will then go on to point outwhere large, complex models have made sig-nificant contributions to basic ecologicalunderstanding and will detail circumstanceswhere this approach is both appropriate andvaluable. In particular, my concern is thatLiebhold's and Berryman's articles could beinterpreted (and may have been intended) asa general condemnation of large simulationmodels and the systems approach. Thiswould be a case of condemning a valuableresearch tool because it has not always ful-fjlled its promise. There is the danger ofthrowing out the wash with the water. I alsotake exception to Berryman's casting of the

AMERICAN ENTOMOl.Ot;lST • Willter 1994

two different modeling approaches as apolemic, with complex simulation modelson one hand versus simple theoretical mod-els on the other. Both modeling approachescan be and, in fact, have been used effective-ly in consort. In my opinion, their concor-dant application holds far greater potentialfor furthering ecological understandingthan either of them applied alone.

In this article, I first will discuss severalexamples in which simulation models pro-vided insights that would be difficult to gainthrough other modeling approaches. Then Iwill discuss possible reasons for the failureof the simulation approach in some pastapplications. Finally, I will consider futureapplications of big ugly models.

Examples of the Power ofSimulation Model Applications

Complexity Versus Simplicity. Central toboth Berryman's and Liebhold's articles isthe inference that somehow simplicity isgood and complexity is bad (the terms aremy choice). This seems a curious notion. No

There is nothing inherentlywrong with complex models,

just as there is nothinginherently correct with

simple models; it is more aquestion of appropriateness.

one will argue that there are indeed complexproblems. Do complex problems necessarilyrequire complex solutions? Perhaps not, aspointed out by Berryman. On the otherhand, some complex problems probably dorequire complex solutions. There is nothinginherently wrong with complex models, justas there is nothing inherently correct withsimple models; it is more a question ofappropriateness. The question should be,what is the most appropriate approach forthe particular problem at hand? A particularstrength of the detailed simulation model isthat it provides the flexibility to test hypoth-eses regarding mechanistic pathways andinteractions. For example, currently we (Lo-gan and Amman 1986, Bentz et al. 1991) aredeveloping a model of mountain pine beetle,Deltdroctonus pOltderosae Hopkins, popu-lation-level response to weather. The currentmodel represents the population response ina mechanistic fashion, with stage-specificdevelopmental rates, temperature-depen-dent recruitment, and stage-specific, cold-temperature mortality. The model has 80 ormore parameters, clearly a complex model

by Berryman's terms. Yet, for a specifiedannual temperature regime, the model willresult in either exponential growth or expo-nential decay of the population. In otherwords, the entire behavior of these 80 ormore parameters could be captured by onesimple parameter, the per-capita instanta-neous rate of population growth! It seemsunavoidable that the principle of parsimonywould demand us to forsake the big uglymodel in favor of the theoretically soundMalthusian model. Unfortunately, by doingso, we would also lose the ability to test forsubtle interactions between seasonality, de-velopmental thresholds, phenology, mortal-ity thresholds, etc., all of which are criticalto address important weather- and climate-related issues such as changing climate sce-narios. Although the elegant theoreticalmodel may capture the behavior of the com-plex simulation faithfully, it does little toimprove our understanding of how seasonalchanges in weather or climate will effectmountain pine beetle populations.

Multiple Causality. There are other in-herent limitations to initiating ecological in-quiry from the principle of parsimony,particularly by the process of using a prioridefined theoretical models as the basis forinterpreting time-series data. The risk is, aswith any inductive argument, that observedpopulation time-series patterns may resultfrom a variety of causes. For example, thecycles that are diagnostic of delayed density-dependent population processes can also re-sult from autocorrelation in an exogenousfactor (Williams & Liebhold 1994). In moregeneral terms, there may be an infinite num-ber of mathematical relationships that areindistinguishable over a limited extent oftheir domain.

Simulation models can add substantiallyto the complex, inductive process that at-tempts to ferret out ecological causality.Mechanistic models that are based on thebest information available can add credibil-ity to a particular interpretation of observedpopulation data. As an example, I devel-oped a mechanistically detailed simulationmodel of the predator-prey interaction be-tween Tetrallychus mcdanieli McGregor,and Metaseiulus occidelltalis (Nesbitt) Lo-gan (1976). This complex simulation modelwas based on data from independent labo-ratory studies and published literature and,as such, represented the current state-of-knowledge involving this interaction. Run-ning the model, which was parameterizedfrom independent data, under prevailingenvironmental conditions, resulted in prey-predator cycles that were similar to the ob-served outbreak cycles. The inference wasthat the current state-of-knowledge was suf-ficient to explain patterns similar to those

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observed in the field. Based on the support-ing evidence from the simulation model, amore analytically tractable theoretical mod-el was used for the analysis of dynamicalsystems properties (Wollkind & Logan1978). This process differs from the type ofinference discussed by Berryman in that theresulting population cycles were an emer-gent property rather than a property derivedfrom a curve-fitting process.

Emergent Properties and HierarchicalStructure. An emergent property is definedby Allen & Starr (1982) as " ... properties ofhigher levels in the system that arc not obvi-ous from the properties of the parts." Anapplication in which the simulation ap-proach resulted in an important ecologicalinsight as an emergent property is providedby Bentz et al. (1991). This work involveddevelopment of a simulation model ofmountain pine beetle phenology construct-ed from individual level, temperature-dependent, developmental data. When theresulting model was run under prevailingannual weather conditions typical formountain pine beetle habitat, a univoltineseasonality emerged from the relationshipsbetween stage-dependent developmentalrates and developmental thresholds. To myknowledge, this result is the only mechanis-tic demonstration for direct control (season-al temperatures alone, Danks 1987) ofunivoltinism in a temperate-zone insect.Direct control of seasonality is an ecologicalinsight, by the way, that has important im-plications for the nondiapausing mountainpine beetle. The important point is that anappropriate seasonality (a population-levelresponse) resulted from the interactions ofstage-specific developmental parameters (anindividual-level process).

The emergence of properties, such as vol-tinism in mountain pine beetle, is a basicproperty of hierarchical structure. Strictlyspeaking, an emergent property can resultonly from estimating parameters at a levellower in the hierarchy than the one of inter-est (Allen & Starr 1982). As such, discoveryof emergent properties is a important poten-tial with most simulation approaches, andone that is not possible when estimating pa-rameters for population-level models frompopulation-time series.

Communication Between Field andQuantitative Ecologists. Irrespective ofwhether one dates the beginning of moderntheoretical ecology from Verhulst's publica-tion of the Logistic equation in 1845, or theindependent rediscovery of that work byPearl and Reed in 1920, the theoretical-modeling approach vastly predates simu-lation modeling and systems analysis.Apparently, theoretical models generated agreat deal of interest from ecologists of the

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day. An indication of this acceptance is Vol-terra's lengthy (38 pages) appendix in Chap-man's (1931) A'limal Ecology with EspecialReference to the Insects, which was perhapsthe most widely used insect ecology text ofthe time. But why, when the theoreticalmodeling approach had predated the simu-lation models by almost SO years, didsimulation modeling and systems analysisstrike such a responsive cord with ecologistsof the 1960s? In my opinion, one importantreason is the highly abstract nature of theo-retical models. The language of highermathematics, and in particular differentialequations, is opaque to most field ecologists.Simplicity, opaqueness, and transparencyare all relative terms. For example, the ele-gant mathematical simplicity of a system ofcoupled differential equations typically isopaque [Q the classically trained field ecolo-gist. Conversely, the mathematically in-elegant representation of an ecological inter-action by a Forrester (1961) flow diagramwill likely be intuitively clear to the sameecologist. Simulation models provide a com-mon ground and overlap between disci-

One of the mostimportant reasons for

the disenchantment withsimulations is that the

approach never came closeto meeting the expectationsthat were generated by its

early proponents.

plines and scientists with little shared intel-lectual experience. If the quantitative scien-tist and the field ecologist can agree on asystems representation of particular ecolog-ical interaction, then there is a good chancethat a more abstract model, amenable tomathematical analysis, can be developedthat also maintains ecological credibility.

Even though many theoreticians havedisdain for big ugly models, an existing sim-ulation can provide the catalyst for furthermodel refinement. The biologist, who feelscomfortable in the simulation environment,can develop a detailed simulation of the sys-tem independently. This description, in turn,can inspire the theoretician to modify themodel further and recast it in a more elegantform. Both gain through the process, or atleast they can. This was clearly the case inthe previously described analysis of theprey-predator relationship between T.mcdaniel; and M. occidellta/lis. This workbegan in the early 1970s with the develop-ment of a complex simulation model of the

interaction (Logan 1976). The simulationmodel provided the inspiration for both thedevelopment of, and a verification checkfor, subsequent theoretical analysis thatspanned the next 15 years (Wollkind et al.1988). One of the best examples from theentomological literature of synergism be-tween simulation and theoretical models isthe well-known analysis of spruce budwormdynamics that occurred at the University ofBritish Columbia during the 1970s (Ludwiget al. 1978). Development of a detailed sim-ulation model proceeded the theoreticalanalysis by several years, and provided asynthesis for the extensive database (Morris1963) on budworm dynamics.

Why the Disenchantment withSimulation Models?

Unrealistic Expectations. If simulationmodeling is such a laudable enterprise (as Icontend it is), why is the approach so vulner-able to criticism by highly qualified insectecologists, such as Berryman and Liebhold?This is a interesting question and its answerprobably has as much to do with humannature as it does with either ecology or math-ematics. One of the most important reasonsfor the disenchantment with simulations isthat the approach never came close to meet-ing the expectations that were generated byits early proponents. The reasons for theseunrealistic expectations were partially a re-sult of the sudden availability of digitalcomputers. In retrospect, the mainframecomputers of the mid-1960s are primitiveindeed, but when compared with the rotaryMonroe calculator that was state-of-the-artfor most ecologists of the time, the capabil-ities seemed unlimited. It really did seempossible that we could recreate nature insidea computer. I agree with Liebhold that, inretrospect, it was a naive expectation. An-other, and perhaps even more important rea·son, is the nature of competitive funding thatwas becoming the dominant means to sup-port research during the 1960s and 1970s. Inthe competitive-funding environment, it isnecessary to paint as optimistic a picture aspossible in order to increase the chances ofsuccess. The more administrative levels thatmust be sold on a research project, the great-er the chance of building false expectations.

Lack of Predictive Power. One of themain points for criticism of simulation mod-els in both Berryman's and Liebhold's arti-cles has been the lack of predictive power. Itis possible that this lack of success at predic-tion results from some basic flaw in the sim-ulation process, as both authors contend. Italso may be an indication of somethingmuch more basic. Recognition of instabili-ties in numerical solutions to differential

AMERICAN ENTOMOLOGIST • Winter 1994

equations goes back at least to the 1920s;however, it is only recently becoming accept-ed that instability of a basically unpredict-able nature (chaos) may be intrinsic toecological structure. There are many rea-sons for expecting insect populations, par-ticularly eruptive species, to exhibit chaoticdynamics. These reasons are discussed inmore detail elsewhere (Logan 1991, Logan& Allen 1992). Suffice it to say that if com-plex nonlinear dynamics are widespread ininsect-population systems, then we need totake a serious look at the whole concept ofprediction. Lack of predictive power may infact not be a criticism at all. To the contrary,it may be a common manifestation of realecological systems and, therefore, anticipat-ed in models that represent their dynamics.

Model Development Strategies. Anotherreason for the unfulfilled promise of somesimulation models is the basic approachthat often has been used to develop largedetailed models. The majority of simulationmodels criticized by Liebhold were devel-oped through a modeling process looselypatterned after the Adaptive EnvironmentalAssessment approach described by Holling(1978). The basis of this process is a seriesof workshops that bring the leading expertsin a field together with computer program-mers and facilitators that are capable ofbridging the gap between field ecologistsand computer programmers. The objectiveof these intensive workshops is to produce aworking systems model. Undoubtedly, thisprocess has made significant contributionstoward an enhanced ecological understand-ing, and several such contributions are doc-umented in the original publication.However, the process also has resulted in atleast several models of questionable value.Liebhold details some of the reasons whythis process has failed. Several additionalreasons come to mind.

First is the principle of model invalida-tion. Holling went to great lengths discuss-ing the concept of model invalidation andstressed the importance of alternative modelstructures. The essential concept of invalida-tion and willingness to discard an existingmodel may be subverted, as Liebhold con-tends, by large simulation models. Theinvestment in money, time, and careers be-comes so great that it becomes increasinglydifficult to discard the model. Developers ofthe model become its advocates rather thanits evaluators.

Second, many of the models criticized byLiebhold were initially developed for theUSDA Forest Service through extramuralcontracts to a consulting company. This ap-proach has the obvious advantage of assem-bling talented modeling expertise to focus ona problem of immediate interest. However, it

AMF.RICAN ENTOMOI.OGIST • Wi1lter 1994

has the liability of abdicating the responsi-bility for model development to those withvested interests that last only to the end of aspecific contract. Rather than a one-shot in-vestment with the model as a goal, experi-ence indicates that the most useful modelsare those that have undergone an evolutionof thought and a continued commitment byecologists whose vested interests are in thesystem rather than the model per se.

Other concerns have resulted from appli-cation of the workshop format used inAdaptive Environmental Assessment toidentify critical processes. In the initial andsubsequent workshops, the general idea is togather the leading experts in a particularproblem area. Then, a variety of techniquesare employed to determine the importantvariables and parameter relationships thatdefine the model. In principle, this approachseems reasonable, and the techniques usedeffectively facilitate communication be-tween workshop participants. In practice,though, it often has resulted in models thatemphasize those variables and relationshipsthat are advocated by the most forceful

There is once again a realneed for research on the

science of modeling. For atleast the past 10 years,

research on modeling per sehas been unfashionable

(and unfundable).

personalities among the workshop partici-pants. The dynamics of the best advocatemay work well in a court of law; however, ithas proven less effective in the course ofscientific investigation.

Additional factors have contributed tothe disenchantment with simulation models.These include the fact that much of the orig-inal motivation for development of simula-tion models was didactic in nature, such asproviding a forum for discussion, evalua-tion of state-of-knowledge, etc. Althoughthese are unquestionably valuable to the sci-entific process, by themselves they are notlikely to result in administrative recognitionor publications. Therefore, many models(perhaps some of those cited by Liebhold)have gained a life of their own. The realvalue as a research tool has been subvertedinto that of a questionable value as a re-search product. Whatever the reasons, it isobviously clear that there is widespread dis-enchantment with large simulation models,even among those like Berryman who havebeen important contributors to large-scale

modeling efforts in the past (Berryman &Pienaar 1974, Raffa & Berryman 1986).

Recommendations

In conclusion, Berryman and Liebholdhave both pointed out deficiencies in severalmajor simulation modeling efforts. We needto recognize that some simulation models,and perhaps some simulation-modeling ap-proaches, have outlived their usefulness. Weshould not live with obsolete models anymore than we should live with any otherobsolete scientific instrument or tool. Seri-ous evaluation of existing models (such asBerryman and Liebhold have provided)need to be made, and those that have out-lived their useful life should be discarded.However, this should not be interpreted as ageneral condemnation of simulation modelsor the systems approach. As with any otherworthwhile scientific endeavor, simulationtechniques and approaches have evolvedover time. Some have worked; others havenot. Attempts to recreate the ecologicalworld, in all its detail and complexity, insidea computer chip generally have not paid off,whereas well-defined and more narrowlyfocused simulations often have. It also isclear that the corporate approach to modelbuilding is easily subverted in practice.

There is once again a real need for re-search on the science of modeling. For atleast the past 10 years, research on modelingper se has been unfashionable (and unfund-able). Topics such as expert systems, individ-ual based modeling approaches, hierarchicaltheory, and self-organizing systems are but asample of ideas that could facilitate develop-ment of ecologically credible simulationmodels. Three research objectives should beto (1) produce more robust techniques forcoping with the complex nature of ecologicalstructure and organization, (2) develop waysto avoid cascading and confounding errorsthat are typical of large simulations, and (3)provide efficient tools that help to reduce theoverhead associated with simulation-modeldevelopment (Logan 1989b).

Although there is no question that theo-retical and simulation models are quitedifferent both philosophically and procedur-ally, I think it is counterproductive to presentthese differences as a polemic, with theoret-ical models on one hand and simulationmodels on the other. In practice, some ques-tions are well suited to simulation approach-es, whereas others are more amenable totheoretical analysis. Simulation models andsystems analysis are particularly powerfultools that can facilitate communication be-tween empirical and theoretical ecologists.Simulation models also clearly have demon-strated their value for testing hypotheses

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regarding the importance (sensitivity) ofmechanistic pathways in ecological struc-ture. Attempting to build a simulation modelis an efficient way to evaluate the depth ofunderstanding regarding a particular eco-logical interaction. Theoretical models, onthe other hand, provide the analytical tracta-bility to test hypotheses regarding questionsof stability, persistence, and dynamicalstructure. In some sense, simulation modelsare well suited to provide insights into theproximate nature of extant, well-definedecological associations, whereas theoreticalmodels are better suited to address questionsof ultimate consequence (a distinction thathas long been recognized [Levins 1966]).What is needed is a recognition of the differ-ences between the two modeling approachesand development of techniques that capital-ize on their particular strengths while sim-ultaneously avoiding ill-suited applicationsthat expose their weaknesses.

Both Liebhold and Berryman extol thevirtues of simple model representation, andfew would disagree with the statement that amodel should be no more complex than isnecessary to represent the ecological processin question. The problem is, however, in de-termining what is important and what isnot. It is easy to cut too fine with Occam'srazor. In my opinion, parsimony serves usbetter as a goal rather than a point of depar-ture. In previous articles (Logan 1982,1987, 1989a), I discussed a procedure thatattempts to combine the descriptive and ex-ploratory power of the simulation approachwith the analytical power of the theoreticalapproach. Plant & Mangle (1987) furtherdeveloped the concept of a concordant mod-eling paradigm. To summarize the compos-ite-modeling approach (Logan 1989a): "Inthis approach, a simulation model is firstdeveloped that captures ecological complex-ity with sufficient resolution to satisfy thegoals of precision and realism. Then,through a systematic process of simplifica-tion and aggregation, progressively moreanalytically tractable models are developed.This finally results in a model amenable tomathematical analysis. Throughout thisprocess, procedures are followed that serveto maintain the link between simplifiedmodels and the more complex models fromwhich they were derived, and thus to theactual ecological system of interest." Theimportant concept is that the simple theoret-ical model is linked procedurally to the de-scriptive, intuitively obvious simulationmodel. The simple model is a resultant prod-uct rather than the point of initial inquiry,in direct opposition to the modeling proce-dure suggested by both Berryman and Lieb-hold. It is important to emphasize theusefulness of both modeling approaches,

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the big ugly model as well as the eleganttheoretical model.

Liebhold and Berryman each end theirarticles with truisms that are difficult todeny. I will end in the same vein: In ourquantitative representations of ecologicalassociations, we should avoid dichotomousand antagonistic classification of quantita-tive approaches. After all, a model is only amodel. The net worth of a model should bemeasured by the insights it provides intoecological processes. Any approach that fa-cilitates this process is valuable and worthyof consideration.

Acknowledgments

First, I thank Sandy Liebhold who providedme with a review copy of his article and for theinteresting discussions we have had on this topic.Secondly, I acknowledge the continuing contribu-tions of my mentor and colleague Alan Berry-man, whose typically provocative style invari-ably provides a catalyst for further discussion. Ialso thank B. J. Bentz, J. C. Allen, and D. Ludwigfor reviewing an early draft of the manuscript.Two anonymous reviewers also made valuablesuggestions. I incorporated many of these sugges-tions in the final version of this article; however,I take sole responsibility for the opinions that areexpressed herein. I also realize that other, perhapsbetter, examples could have been used in manyplaces where I cited my own work. The reason forthis is that I know the motivation behind mychoice of a particular technique or modeling ap·proach, and I can only guess at the motivation ofothers.

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1987. Modeling the pest component of range-land ecosystems. pp. 334-350. [II J. L. Capin-era [ed.], Integrated pest management onrangeland. Westview, Boulder CO.

1989a. Derivation and analysis of compositemodels for insect populations, pp. 278-288.III L. L. McDonald, B.F.J. Manly, J. A. Lock·wood & J. A. Logan [eds.], Estimation andanalysis of insect populations. Springer, NewYork.

1989b. Toward an expert system for pest simu-lation models: lessons learned from applica-tion. pp. 233-243. III N. R. McFarlane [ed.],Progress and prospects in insect control. Br.Crop Prot. Counc. Monogr. 43.

1991. Chaos: much ado about something, pp.1-23.ln J. A. Logan & F.P.Hain [eds.], Chaosand insect ecology. Va. Agric. Exp. Stn. InE.Ser. 91-3.

Logan, J. A. & J. C. Allen. 1992. Nonlinear dy-namics and chaos in insect populations.Annu. Rev. EntomoJ. 37: 455-477.

Logan, J. A. & G. D. Amman. 1986. A distribu-tion model for egg development in mountainpine beetle. Can. Entomol. 118: 361-372.

Ludwig, D., D. D. Jones & C. S. Holling. 1978.Qualitative analysis of insect outbreak sys-tems: the spruce budworm and forest. J.

AMERICAN ENTOMOI.OGIST • Winter 1994

Anim. Ecol. 47: 315-332.May, R. M. 1976. Theoretical ecology: principles

and applications. Saunders, Philadelphia.Morris, R. F. [ed.]. 1963. The dynamics of epi-

demic spruce budworm populations. Mem.Entomol. Soc. Can. 21.

Pearl, R. & L. J. Reed. 1920. On the rate ofgrowth of the population of the United Statessince 1790 and its mathematical representa-tion. Proc. Nat. Acad. Sci. U.S.A. 6: 275-288.

Plant, R. E. & M. Mangel. 1987. Modeling andsimulation in agricultural pest management.SIAM Review 29: 235-261.

Raffa, K. F. & A. A. Berryman. 1986. A mecha-nistic computer model of mountain pine bee-tle populations interactions with lodgepolepine stands and its implications for forestma'nagers. For. Sci. 32: 789-805.

Royama, T. 1992. Analytical population dynam-ics. Chapman & Hall, London.

Ruesink, W. G. 1976. Status of the systems ap-proach to pest management. Annu. Rev. Ento-mol. 21: 27-44.

Stone, N. D. 1989. Knowledge based systems as aunifying paradigm for IPM. pp. 13-24. 111

Proceedings of the National IPM Symposium!Workshop 1989, Las Vegas, NY. Communica-tions Services, N.Y. Agric. Exp. Stn., CornellUniversity, Geneva.

Verhulst, P.F.1845. Recherchs mathematiquessurla loi d'accroissement de la population. Mem.Acad. R. Belg. 18; 1-38 .Cited ;11: G. E.Hutchin-son [1978]. An introduction topopulationecol-ogy. YaleUniversity Press, New Haven, CT.

Volterra, V. 1931. Variations and fluctuations ofthe number of individuals in animal speciesliving together, pp. 409-448. [" R. N. Chap-man. Animal ecology: with especial referenceto insects. McGraw-Hill, New York.

Williams, D. W. & A. M. Liebhold. 1994. Detec-tion of delayed density dependence in ecolog-ical time series: effects of autocorrelation inan exogenous factor. Ecology (in press).

Wollkind, D. J. & J. A. Logan. 1978. Tempera-ture-dependent predator-prey mite ecosystemon apple tree foliage. J. Math. BioI. 6:265-283.

Wollkind, D.J.,J. B. Collins &J. A. Logan. 1988.Metastability analysis in a temperature-de-pendent model system for predator-prey miteoutbreak interactions on fruit trees. Bull.Math. BioI. SO: 379-409. •

Jesse A. Logan is project leader for theMountain Pine Beetle Project, USDA For-est Service, Intermountain Research Sta-tion, Logan Forestry Sciences Laboratory,Logan, UT 84321. He is also a professor inthe Forest Resources Department at UtahState University, and acting director for therecently created Center for Research onDisturbance Ecology, a joint venture be-tween the Intermountain Station and UtahState University. His current research inter-ests are focused on the role of bark beetledisturbance in the structure and function ofwestern coniferous forest ecosystems.

AMF.RICAN ENTOMOl.OGIST • Winter 1994

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