Revolution

This document was presented at a Symposium in honour of Peter J. Wangersky held at Dalhousie University in July 1994, but never was written up for publication.

The Revolution in Ecology and Population Dynamics

William Silvert
Habitat Ecology Section
Bedford Institute of Oceanography
P. O. Box 1006
Dartmouth, Nova Scotia
Canada B2Y 4A2

ABSTRACT

The past two decades have been a revolutionary period in theoretical ecology. New mathematical approaches have transformed the discipline. The staid flow charts and differential equations of pioneering ecological modellers have yielded to the bifurcations, strange attractors, and chaotic trajectories of the present day. Underlying these glamorous developments is the fundamental question of whether the new mathematics is reflected in a new kind of biology, or whether the ecological insights of previous decades have stood the test of time. Reevaluation of the papers that P. J. Wangersky wrote about ecology and population dynamics in the 1950's and 1970's suggests that his underlying biological insights remain valid today, even though the tools that we would now use to express these insights have been greatly expanded and refined, and leads to the conclusion that good biology needs brilliant ideas more than powerful techniques.

Introduction

The past few decades have seen dramatic changes in theoretical ecology. In addition to the introduction of new mathematical techniques, even the basic philosophical approach to understanding ecosystem dynamics has changed. This raises the question of whether earlier theoretical studies retain their validity, or whether recent methodological changes have rendered them obsolete. Early research by P. J. Wangersky provides a context for examining this issue by comparing his results with those that might be obtained with current methodology.

Niche Theory

The theory of the niche is a classic example of the development of theoretical concepts in ecology. According to the review by Christiansen and Fenchel (1977), Grinnell introduced the concept in 1917, but the mathematical definition was not developed until several decades later by Hutchinson (1944, 1957). Unfortunately, the mathematics that Hutchinson used turned out to be a difficult and inappropriate choice, so from the viewpoint of a science historian it is interesting to see how ecologists struggled with Hutchinson's definition, and whether they were held back by it or whether they managed to get around the difficulties it raised.

The niche can be thought of as a generalization of range. Every time we observe an organism we can record its spatial coordinates, and after a large number of observations we end up with a cloud of points that can be used to define the organism's range. If we measure not only the spatial coordinates but also environmental variables such as temperature, salinity, and predator concentrations, we can assign each of these variables to a separate axis and plot the points in a multi-dimensional hyperspace. The idea of using hyperspace as a geometric technique for representing environmental data is the basis of Hutchinson's theory of the niche.

The set of all observations of an organism can be represented by a cloud of points in hyperspace, and this cloud of points can be thought of as the observed niche. This is simply a representation of raw data, and although it is an interesting mathematical approach it does not add any scientific content. Where scientific insight and judgement enter the picture is at the stage where we transform this cloud of discrete points into continuous regions (as anyone who learned to draw by connecting the dots in comic books is well aware).

Hutchinson's approach was to define the niche as a set of points in hyperspace at which the species could persist ("exist indefinitely"). Just as the concept of hyperspace is a generalization of real space, this definition of the niche is a generalization of the common term "range". And just as the concept of range is a poor way to represent the distribution of organisms, the idea of a niche as a set loses much of the valuable information contained in a cloud of points (the niche is often referred to as a hypervolume, but this terminology is misleading because the axes have different units so it isn't usually possible to generalize the concept of volume to apply to this sort of hyperspace). All of the information about how the organism is distributed throughout the range/hypervolume is omitted, and the boundaries are poorly defined; a birder would be ill-advised to claim a sighting of an "accidental" just outside its range, and an ecologist should not be surprised to find races or subspecies that can exist just beyond the boundaries of the putative hypervolume defined as their species niche.

Just as the boundaries of a range are not clearly and sharply defined, the boundaries of the niche in hyperspace are, like the boundaries of a real cloud, fuzzy. And once we use the word "fuzzy" a new mathematical tool springs to mind, namely fuzzy set theory. Whereas a regular (the current word is "crisp") set is an all-or-nothing concept -- points are either in the set or out of it -- fuzzy sets permit a variable degree of membership ranging from 0 to 100%. The unrealistically sharp boundary of a crisp set thus can be represented by a broad region over which the membership function gradually falls off to zero.

The use of fuzzy sets to represent the niche is an example of the use of a tool which was not available when the original ecological problem was posed, although other types of mathematical constructs were in use at the time that could be used to represent a cloud of points far better than a set or hypervolume. For example, Levins (1968) reinterpreted the niche as a probability distribution. It is however interesting to see how a Hutchinsonian like Wangersky deals with the original set-theoretic definition.

Wangersky (1972) like Levins adopted the probabilistic approach to defining the niche, concluding that "The representation of the population niche would then consist of a cloud of points in phase space, the more favored parts of the niche being represented by an increased density of points." The population niche thus defined is what I earlier referred to as the observed niche, and although the transition to a probability distribution function is harder than it seems (density means the number of points in a unit hypervolume, but axes in hyperspace have different dimensions so hypervolume is not a well-defined concept), it is clear that the deficiencies of the set-theoretic definition were recognized and that alternative definitions were actively being investigated. However, although both Wangersky and Levins chose probabilistic formulations, these are quite different. Wangersky interpreted the observed niche as a density function describing the distribution of the population in hyperspace, which is a way of deriving the realized niche (a description of where the organisms actually are) but does not tell us anything about the fundamental niche (a description of where the organisms could be), while Levins (1968) describes the niche in terms of "the probability of survival and reproduction", which is more closely related to the fundamental niche. (This is not as great a difference as it might appear, since Wangersky's paper dealt with evolution and in time the realized niche should evolve to cover the fundamental niche.)

The issue under consideration here is whether the absence of modern methods posed a major obstacle to the development of niche theory. It was widely recognized that Hutchinson's definition of the niche as a set of points, a hypervolume in hyperspace, was seriously flawed (although not for the reason given by Wangersky, that "The set-theory statement is not amenable to mathematical manipulation, and comparison of populations or of niches is difficult."; the mathematics of sets is very easy to work with, but leads to ecologically untenable results). An alternative approach that provided a better way to represent the niche as a cloud of points was sought, and since probability distribution functions are relatively well-known mathematical constructs, they were proposed as alternative formulations.

If this work were being done today, the use of fuzzy sets would probably be a better choice for describing the niche. Unlike the crisp sets used by Hutchinson, they permit differentiation between ideal and marginal conditions within the niche. Probability distributions can be interpreted as fuzzy sets, but the greater generality of fuzzy set theory permits the membership function to be defined without reference to probability, and, as was pointed out previously, interpreting a cloud of points as a probability distribution in a hyperspace where the axes have different units and scale factors is a questionable proposition (the situation is analogous to trying to compare the densities of organisms in tropical and polar regions on a Mercator map, aggravated by the fact that the dimensions of the axes in hyperspace are incommensurate). However, in terms of providing a formalism for describing the niche, the various alternatives to set theory are probably equally efficacious. For example, they resolve the problem represented by Figure 1; this shows the hypothetical niches of two species along an environmental gradient, where the two functions can be thought of as fuzzy membership functions or probability distributions or any other measure of environmental suitability. From a set-theoretic point of view these two species have exactly the same niche, namely the same hypervolume in which they can persist, but no matter what formal interpretation we assign to the functions plotted in Figure 1, it is clear that one end of the environmental gradient is more suitable for one of the species while the other is preferable for the other.

Figure 1. Hypothetical niches of two populations along an environmental gradient. The curves show fuzzy membership functions as described in the text for two species which are differently distributed within the same region of one-dimensional hyperspace.

Population Dynamics

Wangersky's work in population dynamics (Wangersky and Cunningham 1957; Wangersky 1978) provides a deeper perspective on the development of theoretical ecology, since not only were modern tools lacking, but the basic philosophy of population modelling at the time was different from what it is today, at least in some sectors of the scientific community. The emphasis was on predictive models that gave clear answers to practical problems, not on models that improved our understanding and described qualitative features of complex systems.

Perhaps the best part of these papers is the refreshing observation that " the situations described by the equations are so far from those observed in natural populations that the validity of this whole approach has often been questioned. In many cases the behavior of laboratory populations grown under carefully controlled conditions has been quite different from that predicted by theory." (Wangersky and Cunningham 1957, although often attributed to Murphy). Anyone who spends much time looking at ecological models is certain to note the dramatic differences between ecosystems as described in the language of ecologists and ecosystems described by the language of mathematics. Wangersky (1978) devotes several pages to this curious phenomenon, although perhaps he understates the view held by many members of what he refers to as "the sub-class of mathematical modellers" that many of their experimental colleagues, given a choice between a model that is correct and one that is simple, would choose the simpler model through some curious misinterpretation of Ockham's razor (Pauly 1994).

This is certainly true of the field of population biology, and in the context of the present paper it is interesting to read on the eve of the appearance of personal computers that "... one of the advantages of the relatively simple logistic equation is that its constants can easily be manipulated, with the aid of a modern pocket-sized programmable calculator, to simulate anything ..." Wangersky (1978). Today we are (or should be) unconstrained by computational concerns, but the following sentence, "... let us examine models of the next level of complexity (and possibly reality)" does not seem to have struck a resonant chord. In any case, the issue addressed by Wangersky's papers in population dynamics is that the variability of real populations, whether observed in the field or in the laboratory, is very different from that predicted by many of the most widely used models.

Wangersky and Cunningham (1957) addressed this issue in terms of time lags and demonstrated that differential equations with lags exhibit a rich variety of trajectories which can offer the possibility of explaining the variability of natural populations. Wangersky (1978) reviewed these results on time lags and discussed other possible modifications of the logistic and Lotka-Volterra equations that would make them behave more realistically. From a modern perspective we see these papers as marking the discovery of dynamical systems theory by ecologists, and the pendulum has swung to the point where population models that do not exhibit chaotic behaviour are boring.

The significance of this concern with variability can be appreciated if one looks at the history of the Ricker (1954) equation for fish populations, x_t+1 = x_texp(- x_t). Although this equation has been around for a long time and has been widely used, it was only with the intrusion of chaos theory into ecology that it seems to have been noted that for large values of the solutions exhibit remarkable dynamical properties, including chaotic trajectories (May 1976). It appears reasonable to suggest that ecologists did not concern themselves with these properties for the simple reason that they were not interested in unpredictable and irreproducible behaviour. Wangersky (1978) writes "Models are too often considered simply as predictors," and clearly a chaotic model would not be acceptable in that context.

The noteworthy feature of the Wangersky and Cunningham (1957) paper is therefore not the detailed calculation of dynamical behaviours of differential equations with time lags, but rather the recognition that a model was needed that reflected the variability of the systems being modelled, and not simply their approach to some hypothetical stable equilibrium. The tools they used to explore variability were generalizations of the kinds of models used in population dynamics at the time, but the underlying approach was farsighted and addresses a problem which is still significant and relevant almost forty years later.

Summary

The lack of good methods certainly makes the task of the theoretical ecologist difficult, but the examples discussed above indicate that a shortage of good mathematical methods need not be an insurmountable obstacle to the development of good theories. The deficiencies of the set-theoretic definition of the niche were widely recognized, and although the alternative definition of the niche in terms of probability densities (Wangersky 1972) is not rigorously valid and is hence not as good a definition as one based on fuzzy set theory, it does provide a much better alternative and is a far better descriptive tool than a set or hypervolume. In the case of population dynamics the important aspect is that one needs to focus on variability. Today we have good tools for studying variability with different kinds of differential and difference equations, but the work of Wangersky and Cunningham (1957) and Wangersky (1978) show that once the basic issue is identified, some method can probably be found for dealing with it. The end result may lack the elegance that more sophisticated methods can offer, but this is of little practical concern.

In some fields, such as astronomy and high-energy physics, it is difficult to imagine how current issues could be addressed without modern research tools. In theoretical ecology this appears not always to be the case; important theoretical concepts can be investigated and developed with tools that are primitive or even in some respects improperly applied. This does not mean that new and powerful mathematical techniques are not useful in ecology, but it does suggest that at least for some of the more inspired thinkers in the field, the absence of powerful and rigorous formalism need not be an insurmountable obstacle to scientific progress.

References

Christiansen, F. B., and T. M. Fenchel. 1977. Theories of Populations in Biological Communities. Springer-Verlag, Berlin.

Hutchinson, G. E. 1944. Limnological studies in Connecticut. Ecology 25:3-26.

Hutchinson, G. E. 1957. Concluding remarks. Cold Spring Symposia on Quantitative Biology 22:415-427.

Levins, R. 1968. Toward an evolutionary theory of the niche. In Evolution and Environment, E. T. Drake ed. p. 325-340.

May, Robert M. 1976. Models for single populations. In Theoretical Ecology, R. M. May, ed. p. 4-25.

Pauly, D. 1994. Resharpening Ockham's razor. Naga, the ICLARM Newsletter, Apr. 1994, p. 6-7.

Ricker, W. E. 1954. Stock and recruitment. J. Fish. Res. Board Canada 11:559-623.

Wangersky, P. J. 1972. Evolution and the niche concept. Trans. Conn. Acad. Sci. 44:367-376.

Wangersky, P. J. 1978. Lotka-Volterra population models. Ann. Rev. Ecol. Syst. 9:189-218.

Wangersky, P. J., and W. J. Cunningham. 1957. Time lags in population models. Cold Spring Symposia on Quantitative Biology 22:329-338.

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