£tOLOGtfllt IIIOIELUIlG

ELSEVIER Ecological Modelling 75/76 (1994) 641-651

Mathematical models and understanding in ecology

Volker G r i m m

Centre for Environmental Research (UFZ), Department of Ecological Modelling (OSA), Postfach 2, 04301 Leipzig, Germany

Abstract

Of all possible mathematical models, only "conceptual" models, i.e., models which are understandable, manageable, and capable of being fully explored, can be of help in attaining an understanding of ecological systems and processes. However, the hitherto promulgated "philosophy" of conceptual mathematical modelling was not in a position to convince empiricists of the possibilities afforded by mathematical models on the one hand, and on the other to prevent theorists from becoming so engrossed in their theories as to neglect the testability of their hypotheses. The goal of this paper is to reveal the shortcom- ings of this "philosophy" of mathematical modelling and to outline a more promising strategy for creating models. It recommends choosing "patterns" actually existing in ecological systems, rather than questions of a general nature, as a point of departure for conceptual models. In this way, the advantages of conceptual models can be utilized without having to relinquish the testability of the hypotheses proposed with the aid of the models.

Key words: Conceptual model; Pattern; Theoretical ecology

I. Introduct ion

The relationship between empiricists and theoreticians in the natural sciences is essentially asymmetrical: empiricists can get along very well without theoreticians, since no empiricist can work seriously without having a more or less explicitly formulated theory in mind regarding the subject he is in the process of studying. A theoretician, however, is one who by definition does no experimental work or field studies. At some time or other in the course of his work, he must interact with empiricists if he does not want to end up completely outside the usual framework of scientific research, which consists of a cycle of observation, hypothesis, predic- tion and experimentation (Salt, 1983). In the case of ecology, though, it seems as if many theoreticians are only able and willing to address other theoreticians, and as if a large portion of the theories are proposed only for their own sake.

0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(94)00026-E

642 V. Grimm/Ecological Modelling 75/76 (1994) 641-651

The reason for this unsatisfactory situation is that the "philosophy" of mathe- matical modelling, which has dominated theoretical ecology for, say, the past 30 years or longer, was not in a position to convince the empiricists of the potential of mathematical models on the one hand, or to prevent theoreticians, engrossed in their theoretical problems, from completely losing sight of the dialogue with empiricists on the other.

In the discussion of the part played by "conceptual" (see below) mathematical models - as is so often the case when discussions become vehement - the positions are becoming perceptibly more unyielding: criticism and rejection take their stand on the one side (e.g., Pielou, 1981; Simberloff, 1981, 1983; Hall, 1988, 1991; Krebs, 1988), and on the other an exaggerated optimism (e.g., Levin, 1981; Roughgarden, 1983; Caswell, 1988; Yodzis, 1989). In my opinion, neither stand- point advances our cause. My goal in this paper is to develop a more differentiated "philosophy" of mathematical modelling. Even though the potential of gaining understanding with the aid of mathematical models far exceeds that of verbal and graphic models, work using mathematical models can develop a momentum of its own which sometimes leads theoreticians into spheres where no empiricist is willing or able to follow.

In the following sections, the definition of "conceptual" models will be briefly presented and the "philosophy" forming the basis for these models will be sketched. Thereafter, I will attempt to show that the strength of mathematical models (in making predictions) is inseparably bound to their weakness (their inherent momentum). As a way out of this dilemma, I will suggest that theoretical ecology relinquish its claim to generality and, with the aid of "pattern-oriented" models, once again become receptive to a dialogue with empiricists.

2. Conceptual mathematical models

This paper deals solely with models which are used for the purpose of gaining understanding. This excludes statistical models as well as complex models of systems. Statistical models serve to discover correlations, trends and patterns in data; they can supply no explanations, however. Complex models of ecological systems are as a rule incomprehensible in the sense that the behaviour of the model cannot be fully explored, so that it often remains unclear which parts of the model determine the results achieved with the model (Mollison, 1986). If an intellectual tool - and mathematical models are nothing other than this (Starfield and Bleloch, 1986) - is to help us gain understanding, then this tool must itself be understandable. In order to be understandable, the behaviour of the model must be capable of being fully explored, and in order to make this possible, the number of variables and parameters employed must be manageable. Models which possess these three mutually dependent attributes (understandable, manageable, able to be fully explored) are designated as "conceptual" models (cf. Wissel, 1989, 1992), or alternatively as "strategic" (May, 1973) or "phenomenological" (Kareiva, 1989) models. Typical representatives of this type are the logistic equation, the models of

V. Grimm /Ecological Modelling 75/76 (1994) 641-651 643

Lotka-Volterra and Nicholson-Bailey and their derivatives, reaction-diffusion equations, etc. (for a different use of the term "conceptual model" see Jorgensen, 1988).

Of course, theoreticians know as well as empiricists that in real ecological systems a great number of variables and parameters can play a part. Yet working with conceptual models corresponds exactly to the approach used by empiricists, who as a rule also limit themselves to some few factors in their experiments and field studies (Caswell, 1988). The desire to understand complex, intricate systems implies precisely this: to simplify them by means of abstraction until they are capable of being assimilated by our limited mental capacities. The strategy of creating conceptual models is thus correct, since we have no other choice. What must be changed is the generally accepted "philosophy" of mathematical mod- elling, i.e., the definition of the limitations and potential of mathematical concep- tual models.

In theoretical ecology, the basic elements of the dominant "philosophy" of mathematical modelling are: (1) conceptual models aid in proposing hypotheses; (2) they supply a "conceptual framework" (May, 1973, p. 11), i.e., they aid in devising concepts; (3) they "sacrifice detail for generality" (Yodzis, 1989, p. 3); (4) they demonstrate the "consequences of what we believe to be true" (Starfield and Bleloch, 1986, p. 3). Until now, many ecologists seem to have given too little consideration to the fact that points (1) through (3) apply to all conceptual models. The only one of these four potentialities of conceptual models which is fully revealed only in mathematical models is point (4). Neither verbal nor graphic models are especially well-suited to demonstrating the consequences of hypothe- ses, i.e., to deducing testable predictions from the hypotheses. In the case of verbal models, it becomes apparent over and over again how difficult it is to apply the rules of logic in words. And graphic models are static, they do not produce any more than has been put into them. Only mathematical models are "animated" (Lotka, 1925). They possess a life of their own, they produce something which cannot be foreseen at their inception.

The capabilities, but also the limits, of mathematical models in ecology are based on just this life of their own. In order to clarify this statement, I will myself use a conceptual model, albeit a graphic one: Fig. 1 is a schematic representation of the cycle consisting of observation, hypothesis, prediction and experiment which is common to the natural sciences (cf. Oeser, 1976). Of course, this schematic representation is oversimplified, the most important element in this case being its cyclical nature, i.e., that at some point in the course of gaining scientific knowl- edge, hypotheses must interact with reality - whether through experimentation or observation. For this to be possible, testable predictions must be deduced from hypotheses (Loehle, 1987). Within the framework of this cycle, conceptual models - of whatever type - help in proposing hypotheses (see small box in Fig. 1). Mathematical models, as has already been pointed out, can do more, however (see large box in Fig. 1): they help in deducing predictions. Not until the model's assumptions have been translated into the formal language of mathematics are their consequences capable of being calculated. The essential point meant to be

644 V. Grimm / Ecological Modelling 75 / 76 (1994) 641-651

l experiment

\

"free-style" modelling ] observation H

prediction

hypothesis

"pattern-oriented" modelling Fig. 1. Schematic diagram of the cyclical gain in knowledge in the natural sciences. The small box shows the domains of all conceptual models, including verbal and graphic models: the proposal of hypotheses. The large box shows the potential of mathematical models: to deduce testable predictions from the hypotheses.

demonstrated in Fig. 1 is: the potential of mathematical conceptual models for producing testable predictions is far greater than the potential of verbal and graphic models.

3. The "state of the art" in creating conceptual models

In view of the potential of mathematical models, it is actually astounding that most models of theoretical ecology make no claim to producing testable predic- tions. They restrict themselves to the other three aspects of creating conceptual models (see above): generating hypotheses, devising concepts and arriving at general assertions. Why is this the case? Let us first take a look at the practice of creating models: when creating models, it is never the case that scientists translate assumptions into equations, then solve the equations and content themselves with the results achieved. Instead, the desire continually arises to modify the equations in order to arrive at different results because the results seem unsatisfactory or simply incomprehensible. This means, however, that the assumptions must be modified, thus beginning anew the model-creating cycle (Fig. 2). Going through this cycle repeatedly can help to uncover logical inconsistencies in the assumptions and thus to arrive at a better hypothesis. This is the best that mathematical models in the small box in Fig. 1 can do: improve the quality of the hypotheses by a formalized "bookkeeping" of the basic assumptions (Poethke et al., 1994, this volume).

V. Grimm/Ecological Modelling 75/76 (1994) 641-651 645

assumptions /" \ mathematical

modifications model

results

Fig. 2. The cycle of model-creating in theoretical ecology. It exhibits an ominous similarity to the cycle in Fig. 1.

The problem is that the model-creating cycle from Fig. 2 bears an ominous resemblance to the cycle of gaining knowledge in the natural sciences (cf. Fig. 1 and 2). If one only keeps moving long enough within the framework of this cycle, it becomes all too tempting to believe that one has already come closer to actually understanding a real ecological system or process. This results in a feeling of complacency with respect to the possible explanations produced by the model- creating cycle and little attention is given to the question of how to find the right one out of all the various possible explanations.

Even worse is the lack of a safeguard mechanism preventing the model-creating cycle from spiralling slowly but surely away from reality and piling up possible explanations which can only be of significance within the scope of theoretical investigations. This is precisely where the equivalence of investigations into empiri- cal and theoretical problems postulated by Caswell (1988) finds its end: empirical investigations are anchored in reality. They may be irrelevant to the understanding of ecological problems, their interpretations can be misleading or false, but they can never make themselves independent of reality to the same extent as mathemat- ical models. The impression is bound to force itself upon empiricists that theoreti- cians are busily constructing their own "best of all possible worlds" (Krebs, 1988, p. 143) in accordance with their own needs - a world in which they, unhampered by irritating data and reluctant empiricists, have total freedom to solve theoretical problems which are absolutely irrelevant to the rest of ecology.

Now it would be quite possible to take the standpoint that it is the theoreticians' problem to decide how they wish to go about their business and the rest of ecology need not bother with their work. But this is not the case, unfortunately. In his provocative article, Simberloff (1981) places the mathematical modellers in the vicinity of faith-healers. This is not fair since, contrary to faith-healers, modellers aim at attaining truth, i.e., understanding - and no less seriously than do empiri- cists. In one point, however, his comparison is very apt: faith-healers satisfy a need (to predict the future), otherwise they would not exist. Theoreticians likewise

646 v. Grimm /Ecological Modelling 75/76 (1994) 641-651

satisfy a need: ecology's immense need for theories, for generally valid explana- tions. Every vehemently postulated ecological theory has an immense effect on the work of empiricists, even if the theory itself is rejected or at least highly disputable.

Let us consider only three examples: the concept of "density dependence", the community matrix models for a complexity-stability hypothesis, and the models related to the concept of niches. In all three cases, a flood of theoretical work gave rise to a flood of empirical work. Assuming that the number of empirical investiga- tions inspired by a model (or family of models) is the measure of success of a model, then these models would indeed have to be considered eminently success- ful. But what is the situation as far as understanding is concerned? The ongoing, heated discussions in all three areas show that, thanks to theoretical work, it may be that we are today able to formulate much more differentiated questions. But we have got stuck at the level of questions. The part played by density dependence, complexity and competition in real ecological systems is still unclear. Is it truly impossible to equip the impressive "man-power" of theoretical ecology with a more promising strategy, so that what comes out of it all in the end will be more than mere questions, namely answers?

Let us now take up again the question posed at the beginning of this section: Why is the potential of mathematical models to produce examinable predictions put to so little use? My own impression is that many theoreticians have become victims of the intrinsic momentum of their models. The very thing which makes mathematical models so attractive and valuable - the fact that they are "animated" objects - turns out to be a trap.

4. The shortcomings in the "philosophy" of creating conceptual models

The reason for the dilemma in which mathematical modelling finds itself is that the phrase mentioned above, " to sacrifice detail for generality", is extremely misleading if not, in many cases, downright wrong. Generally valid theories are what ecologists most fervently desire - and what they have least achieved up to now. All ecological systems have one thing in common: that they are more or less unique in the world. In the first place, a system's abiotic conditions alone can never be found again anywhere else in the world. But the biotic conditions, i.e., the organisms and combination of organisms, are yet more unique by far. The phrase, " to sacrifice detail for generality", raises the understandable desire for generality to a programmatic level and suggestively conveys the idea that the true goal of science - to attain understanding - remains unaffected by this program. The exact natural sciences, such as physics, also employ abstractions to arrive at general theories. But these abstractions are not made at the expense of the theories' testability. In ecology, the level of abstraction concealed behind mathematical models is more far-reaching by many orders of magnitude than in physics (Slobodkin, 1981). The significance of this for ecology is that generality can only be achieved at the expense of testability. We must acknowledge the fact that the

V. Grimm /Ecological Modelling 75/76 (1994) 641-651 647

generality found in many mathematical models is not a virtue in itself, but is bought at the price of testability. In the case of mathematical models, there is a tradeoff between generality and the ability to produce testable predictions.

I suggest emphasizing this tradeoff in a clearer definition of model types: (1) I refer to models with a high level of generality, but which do not make a direct claim to testability (see small box in Fig. 1), as "free-style" models. I do not intend this designation to be derogatory or ironic - scientific study would be impossible without "free-style" theorizing. But the term "free-style" expresses what most models of theoretical ecology in fact are: they are intellectual tools whose effects come to fruition in a realm which is more or less detached from reality. The designation "free-style" is meant to keep not only the creators of the models themselves, but also the consumers of the models from overestimating the signifi- cance of these models. Whoever takes offense at the term "free-style" is welcome to give them a more elegant name such as "metaphors of nature" (Kareiva and Anderson, 1988) or "demonstrations" (Crick, 1988). (2) I refer to models with an intermediate level of generality, which are used for the declared purpose of making testable predictions, as "pat tern-oriented" models (see large box in Fig. 1).

5. "Pattern-oriented" models

The essential difference between "free-style" models and "pat tern-oriented" models is their point of departure: in the case of "free-style" models this is the desire for generality, in "pat tern-oriented" models it is a pattern which can be observed in nature. These two points of departure are not in principle mutually exclusive, but in ecology, there are sadly very few generally observable patterns. I am deliberately leaving the term "pat tern" undefined in order not to narrow it down. Whatever can be discovered - in the way of patterns that are worth explaining - in nature and in the data describing nature, depends on the experi- ence, prior knowledge and skill of each individual. Instead, I will present in what follows some examples intended to illustrate what in my opinion is a suitable point of departure for "pat tern-oriented" models, and what is not.

"Ecology stability", i.e., the persistence of ecological systems (on the term "stability", see Grimm et al., 1992), is not a pattern. Patterns are distinguished by the fact that they are improbable, surprising, and cannot be expected as a result of a chance combination of the effects of various factors (cf. the "risky predictions" called for by Popper; Loehle, 1987). The existence of ecological systems (popula- tions, communities and ecosystems) as units identifiable over long periods of time is obviously not an improbable phenomenon: populations, communities and ecosys- tems have existed for as long as there has been life on earth. Of course we would like to understand why these systems exist, and why they sometimes cease to exist and are supplanted by others. But taking mere existence as a point of departure for mathematical models dealing with this question leads only to possible explana- tions and not to real explanations.

648 V. Grimm /Ecological Modelling 75/76 (1994) 641-651

In contrast, an example of promising patterns are structures in ecological systems arising from self-organization. Jeltsch et al. (1992) choose the oscillating dispersal pattern of a tephritid fly as the point of departure for a model which possesses unusual predictive power for theoretical ecology. Boersma et al. (1991) start by asking which conditions are necessary for two competing species of tree to form a sharp boundary in their areas of distribution when the soil exhibits a continuous gradient in its nutritional content. Further examples of suitable pat- terns are the mosaic structure of central European beech forests (Remmert, 1991; Wissel, 1991) and the widespread natural dieback in monospecific forests in Hawaii (Mueller-Dombois, 1987; Jeltsch, 1992; Jeltsch and Wissel, 1992).

Time series of population size are not a suitable pattern. The same holds true here as for ecological stability. We would very much like to understand which factors determine temporal changes in the size of populations. Granted, we know from the work of May (1976) that abstract, simple dynamic systems can produce not only states of equilibrium, but also oscillations and chaotic behaviour. But it is still not clear how well simple dynamic systems can in fact describe real popula- tions. The search continues for effective methods of taking empirical time series and filtering out the deterministic portion - if such does indeed exist (Crowley, 1992; Turchin and Taylor, 1992) - with but dubious success up to now. The example of time series analyses shows with particular clarity that the claim to generality (creating models of simple dynamic systems) is made at the expense of testability. I personally doubt whether merely staring at time series of whatever number and length (even using the most ingenious statistical methods) can bring us closer to a true understanding of population dynamics (cf. Krebs, 1991).

More promising patterns in population data are in age distributions, size and weight distributions, genetic polymorphisms, etc. In his analysis of a large number of different plant and animal species, Uchmafiski (1985) has discovered recurring patterns. For instance, the asymmetry in the distributions of the final weight (quantified by its skewness) thus tends to increase as a function of the size of the population.

Choosing a pattern - however constituted - from real ecological systems as the point of departure for a model automatically limits the generality of the model while at the same time increasing its predictive power. Another very basic advan- tage of creating "pat tern-oriented" models is that patterns cannot be invented. One must look at data or talk with empiricists in order to get to the pattern. And this is precisely the safeguard mechanism which prevents the creators of models from losing sight of reality and the translation of their theories into reality while occupying themselves with theoretical problems.

In order to preclude misunderstandings, I would like to emphasize expressly that the goal of "pat tern-oriented" models is not the explanation of an altogether specific phenomenon in an altogether specific system. Even if the point of departure is sometimes a specific system (as for example with Jeltsch et al., 1992, see above), it is still a matter of identifying a natural system's general ingredients, which can then lead to the same pattern in other systems. "Pat tern-oriented"

V. Grimm /Ecological Modelling 75/76 (1994) 641-651 649

models are conceptual models. The difference between them and "free-style" models is their point of departure!

6. Conclusions

For the past few years, dissatisfaction with the creation of pure "free-style" models has been spreading - even among theoreticians. Several suggestions have already been made as to how mathematical models can be more effectively utilized for attaining true understanding, i.e., real explanations of ecological processes. Tilman (1987) has postulated "mechanistic models" which, in the case of competi- tion for example, place the actual, biological mechanisms of competition in the centre of attention. Kareiva and Anderson (1988) speak of "tactical" models which, contrary to the "strategic" models, are tailored to particular experimental systems and thus enable the hypotheses to be tested. With these approaches, however, care must be taken that the decisive advantages of creating conceptual models (see above) are not relinquished in favour of a pseudo-realism in the models. If a model includes fifteen parameters instead of five, the increase in approximation to reality usually is still minimal (how many parameters would be necessary to describe reality?), but the possibilities of conceptual models may well be done for. Furthermore, the postulation of mechanistic models does not answer the question regarding what we are to do when we do not yet have any idea what the decisive mechanisms are, and when - as is the rule - no experiments can be conducted.

One aspect of creating models which has hitherto received insufficient discus- sion is the models' point of departure. This determines the character of a model. Many creators of system models consider a system to be a good point of departure. They hope to be able to reproduce their system on a computer (the unrealizable dream of all natural scientists), in order then to perform experiments using this surrogate. If it is a matter of understanding, then this method cannot lead to success (see above). Other theoreticians prefer to take general questions as their point of departure and thus arrive at those models which I call "free-style" models. A great number of solutions to theoretical problems are created in this way, but one may well begin to doubt whether this modelling strategy alone will ever lead to success, i.e., the understanding of real systems. A third strategy, which has until now received too little attention, is to choose a pattern produced by nature as a point of departure. This strategy could finally get the long-awaited dialogue between the theoreticians and empiricists in ecology going. Without this dialogue, theory will be doomed to failure in the end.

I do not deny the important part played by "free-style" theorizing. It would indeed be a "subordinate" (Levin, 1981) conception of theory if testability were the only mark of a model's quality. "Freedom of inquiry" (Levin, 1981) must be guaranteed. As a matter of fact, all people, and especially scientists, work with conceptual models constantly without taking immediate concern for their testabil- ity. This is the manner in which we think. The discontent of empiricists faced with

650 V. Grimm / Ecological Modelling 75 / 76 (1994) 641-651

mathematical, conceptual models is not related to their "conceptual" character, but rather to the fact that they are mathematical models, and therefore objects with a momentum of their own. The greatest biological theory, Darwin's theory of natural selection, would not have been possible without "free-style" theorizing. But Darwin did not work with mathematical models!

It is my belief that "free-style" modelling in ecology should rely more on the power of thinking and less on mathematical formalism. The purpose of this paper was to show that the potential of mathematical models for gaining knowledge in ecology has not yet been fully exploited. It should be possible to utilize the formal logic of mathematics without relinquishing the testability of hypotheses generated by models (Fig. 1).

Acknowledgments

I thank Andreas Huth, Ralf Marsula and especially Christian Wissel for their valuable commentaries on an earlier version of this paper and for stimulating discussions. This work was supported by a grant from the Bundesminister for Forschung und Technik under grant No. 0339286A.

References

Boersma, M., Vanschalk, C.P. and Hogeweg, P., 1991. Nutrient gradients and spatial structure in tropical forests - a model study. Ecol. Model., 55: 219-240.

Caswell, H., 1988. Theory and models: a different perspective. Ecol. Model., 43: 33-44. Crick, F., 1988. What Made Pursuit. Basic Books, New York, NY. Crowley, P.H., 1992. Density dependence, boundedness, and attraction: detecting stability in stochastic

systems. Oecologia, 90: 246-252. Grimm, V., Schmidt, E. and Wissel, C., 1992. On the application of stability concepts in ecology. Ecol.

Model., 63: 143-161. Hall, C.A.S., 1988. An assessment of several of the historically most influential theoretical models used

in ecology and the data provided in their support. Ecol. Model., 43: 5-31. Hall, C.A.S., 1991. An idiosyncratic assessment of the role of mathematical models in environmental

sciences. Environ. Int., 17: 507-517. Jeltsch, F., 1992. Modelle zu natiirlichen Waldsterbeph~inomenen. Doctoral-Thesis, Philipps-Universit~it

Marburg, Germany. Jeltsch, F. and Wissel, C., 1992. Modelling factors which may cause stand-level dieback in forest. In:

R.F. Huettl and D. Mueller-Dombois (Editors), Forest Decline in the Atlantic and Pacific Region. Springer, Berlin, pp. 251-260.

Jeltsch, F., Wissel, C., Eber, S. and Brandl, R., 1992. Oscillating dispersal patterns of tephritid fly populations. Ecol. Model., 60: 63-75.

Jcrgensen, S.E., 1988. Fundamentals of Ecological Modelling. Elsevier, Amsterdam. Kareiva, P.M., 1989. Renewing the dialogue between theory and experiments in population ecology. In:

J. Roughgarden, R.M. May and S.A. Levin (Editors), Perspectives in Ecological Theory. Princeton University Press, Princeton, NJ, pp. 68-88.

Kareiva, P.M. and Anderson, M., 1988. Spatial aspects of species interaction: the wedding of models and experiments. In: A. Hastings (Editor), Community Ecology. Springer, Berlin, pp. 35-50.

Krebs, C.J., 1988. The experimental approach to rodent population dynamics. Oikos 52: 143-149.

V. Grimm / Ecological Modelling 75 / 76 (1994) 641-651 651

Krebs, C.J., 1991. The experimental paradigm and long-term population studies. Ibis, 113 (Suppl. I): 3-8.

Levin, S.A., 1981. The role of theoretical ecology in the description and understanding of populations in heterogeneous environments. Am. Zool., 21: 865-875.

Loehle, C., 1987. Hypothesis testing in ecology: psychological aspects and the importance of theory maturation. Q. Rev. Biol., 62: 397-409.

Lotka, A.J., 1925. Reports of talk. J. Am. Stat. Assoc., 20: 569-570. May, R.M., 1973. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton,

NJ. May, R.M., 1976. Simple mathematical models with very complicated dynamics. Nature, 261: 459-467. Mollison, D., 1986. Modelling biological invasions: chance, explanation, prediction. Phil. Trans. R. Soc.

London B, 314: 675-693. Mueller-Dombois, D., 1987. Natural dieback in forests. BioScience, 37:575 583. Oeser, E., 1976. Wissenschaft und Information - Systematische Grundlagen einer Theorie der Wis-

senschaftsentwicklung (3 Volumes). Oldenbourg, Vienna. Pielou, E.C., 1981. The usefulness of ecological models: a stock-taking. Q. Rev. Biol., 56: 17-31. Poethke, H.J., Oertel, D. and Seitz, A., 1994. Modelling effects of toxicants on pelagic food-webs: many

problems - some solutions. Ecol. Model., 75/76:511-522. Remmert, H., 1991. The mosaic-cycle concept of ecosystems - an overview. In: H. Remmert (Editor),

The Mosaic-Cycle Concept of Ecosystems. Ecological Studies 85. Springer, Berlin, pp. 1-21. Roughgarden, J., 1983. Competition and theory in community ecology. Am. Nat., 122: 583-601. Salt, G.W., 1983. Roles: their limits and responsibilities in ecological and evolutionary research. Am.

Nat., 122: 697-705. Simberloff, D., 1981. The sick science of ecology: symptoms, diagnosis, and prescription. Eidema, 1:

49-54. Simberloff, D., 1983. Competition theory, hypothesis-testing, and other community ecological buz-

zwords. Am. Nat., 122: 626-635. Slobodkin, L.B., 1981. Listening to a symposium - a summary and prospectus. In: M.H. Nitecki

(Editor), Biotic Crises in Ecological and Evolutionary Time. Academic Press, New York, NY, pp. 269-288.

Starfield, A.M. and Bleloch, A.L., 1986. Building Models for Conservation and Wildlife Management. Collier MacMillan, London.

Tilman, D., 1987. The importance of mechanisms of interspecific competition. Am. Nat., 129: 769-74. Turchin, P. and Taylor, A.D., 1992. Complex dynamics in ecological time series. Ecology, 73: 289-305. Uchmafiski, J., 1985. Differentiation and frequency distributions of body weights in plants and animals.

Phil. Trans. R. Soc. London B, 310: 1-75. Wissel, C., 1989. Theoretische Okologie - Eine Einffihrung. Springer, Berlin. Wissel, C., 1991. A model for the mosaic-cycle concept. In: H. Remmert (Editor), The Mosaic-Cycle

Concept of Ecosystems. Ecological Studies 85. Springer, Berlin, pp. 22-45. Wissel, C., 1992. Aims and limits of ecological modelling exemplified by island theory. Ecol. Model., 63:

1-12. Yodzis, P., 1989. Theoretical Ecology. Harper & Row, New York, NY.