Quality as a characteristic of ecological models

Ecological Modelling, 55 (1991) 161-174

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Quality as a characteristic of ecological models Boris Zeide Department of Forest Resources, University of Arkansas at Monticello, Arkansas Agricultural Experiment Station, Monticello, AR 71655, USA (Accepted 3 October 1990)

ABSTRACT Zeide, B., 1991. Quality as a characteristic of ecological models. Ecol. Modelling, 55: 161-174. Trade-offs between model characteristics (accuracy, generality, robustness, and so on) should not be accepted as inevitable. Good models manage to combine desirable features, and such a combination constitutes the quality of a model. A good model should reflect the overwhelming importance of reproductive effort to living beings. The quality of a model can also be judged by the constancy of its parameters and the absence of patterns in residuals. A novel (or rather long-forgotten) approach to model construction is proposed. It calls for the simultaneous formulation of two idealized extreme models of the same process prior to quantification of the central tendency. Examples and implications of this approach are discussed.

A N E G L E C T E D F E A T U R E OF MODELS

Models are characterized by accuracy, generality, complexity, testability, robustness, flexibility, and many other attributes. It is commonly believed (Levins, 1966; Sharpe, 1990) that 'good' characteristics cannot be combined in one model: accurate models lack generality, simple models are not realistic, and so on. The view that such trade-offs are inevitable was eloquently expressed by Levins (1966) who outlined several strategies for model building in population biology. These strategies are distinguished by 'sacrifices' among generality, realism, and precision. The view that these sacrifices cannot be avoided is accepted in ecological (Kareiva, 1989) and forestry (Sharpe, 1990) modelling.

This report is approved for publication by the Director, University of Arkansas Agricultural Experiment Station. 0304-3800/91/$03.50

© 1991 - Elsevier Science Publishers B.V.

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Levins' strategies imply and even sanction a greater sacrifice, the advancement of science. The trade-off between model characteristics is a hallmark of mediocre, expendable models. These 'strategically' built sacrificial models are irrelevant to scientific progress. The most pertinent feature of a model is its quality. Although this feature is missing from even the longest lists of model characteristics, it should never be sacrificed.

Quality of modelling A good model does not trade accuracy for generality or vice versa but encompasses both. Understanding is advanced by models that are accurate and, at the same time, are simple and general. If a model can be depicted as a point in space with axes representing increasing accuracy, generality, robustness, etc., then model quality can be quantified as the distance between this point and the origin. Plants and animals often resort to trade-offs and sacrifices. These activities alleviate deficiencies of light, water, nutrients, and other factors. They promote fitness and contribute to the survival of the organism. These trade-offs should not be confused with those in models in which trade-offs reflect the inability to grasp reality. It is not necessary to describe deficiencies of nutrients with deficient models. Contrary to Levins' (1966, p. 422) belief that "this cannot be done", models that excel in many of the above aspects do exist. Levins' own contributions, aside from his philosophical digressions, may testify to this. Although rare at any given period, these superior models constitute the majority, if not all, of models that have shaped our knowledge. In the process of natural selection of models, only those of exceptional quality find a place in the genealogy of science. The model of the solar system by Copernicus and Kepler, to use a familiar example, is more accurate, simple, and meaningful than that of Ptolemy. Sacrifice and trade-offs can and should be avoided in modelling.

Models and reality Models are sometimes viewed as necessarily inferior to reality. Again, Levins (1966, p. 430) expressed this view best: "All models leave out a lot and are in that sense false, incomplete, inadequate." This view is inadequate because of the intimate connection between a model and the process of data acquisition. Without a model, we simply cannot observe or collect data. If models were indeed false, so would be the data. We would be caught in a vicious circle between false models and misleading data. To show that models can be superior to reality, consider, for instance, the

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process of uninhibited cell division, represented by the sequence 2, 4, 8, 1 6 . . . It could be modeled by a high degree polynomial. A polynomial with the number of parameters equal to the number of data points would match the sequence exactly, even when the numbers deviate from doubling. Given these deviations, the exponential function, the only sensible model of the process, will be less accurate. Still, we would prefer not reality, exactly copied by the polynomial, but the exponent even though its predictions would be at variance with actual observations. The proper function, in this case the exponential, screens random noise and presents nature in its refined form. Precisely because this function, in Levin' words, "leaves out a lot" (in this instance, noise), it is correct and adequate, while the indiscriminating polynomial is irrelevant. The work of a scientist, as Plato taught right at the start of this profession, is a constant struggle to filter out chance deviations and penetrate b e y o n d appearance. The Latin source of our word 'understand' (intelligere) means to read what is inside a thing (intus legere). With all due respect for facts, there are many situations when, given contradiction between facts and a model, one would say so much the worse for the facts. Although it might sound contradictory, this is an expression of the highest regard for facts and not contempt for them. At every turn of model construction, we are dealing with two uneven groups of facts: those few at hand, and the total sum of knowledge accumulated by humans. If a model is derived from basic axioms that e m b o d y this sum of facts, then we might trust the model rather than particular facts. Another dichotomy of modelling deals with two different faces of reality, the past and the future. Growth models aim at predicting growth 5, 10, 20 and more years into the future. Certainly, actual future growth will be a more accurate mirror of itself than that produced b y a model. But the future is inaccessible. The existing data describing growth of trees in the past represent another part of reality. Can we use data from one stand to predict future growth of another stand? We can, but not directly. Usually, we analyze the data to determine to what degree the soil, location, stand history and other factors are similar to those of the investigated stand. To facilitate the analysis and reduce the information to a manageable number of variables, we fit the data to a model. Even though there is some justification for viewing models as false and inadequate, in our work we prefer them to the reality, which is either inaccessible or only remotely relevant. To sum up, quality is the chief characteristic of models. It should be neither traded away nor sacrificed. G o o d models manage to combine desirable features and, in some respects, are preferable to reality. These models, not raw data, make sense out of the world around us. G o o d models

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describe and predict reality better than those 'strategically' built models. Yet, the main difference among models is that quality models contribute to the understanding of nature while others do not. CRITERIA OF QUALITY Superior models, such as Mendel's genetic combinatorics, attract fresh minds by their logic, coherence, and beauty ( in short, by their quality) even when empirical evidence is weak or missing. The question arises as to whether there are more tangible criteria of quality besides this nebulous attraction to fresh minds and beauty. The fact that an irrelevant formula (for example, a polynomial) can describe a process more accurately (that is, with a smaller standard error or sum of squared residuals) than the proper function demonstrates that routine statistics are not always a reliable criterion of quality. Several criteria of wide applicability are discussed below. These criteria of quality can be divided into two groups. One group, illustrated by the first of the following criteria (maximization of reproductive effort), addresses coherence or reasonableness of models. The second group deals with the correspondence between model and reality.

Maximization of reproductioe effort At first glance, as a general criterion of ecological modelling, the maximization of reproductive effort looks odd. For m a n y organisms the direct manifestations of this effort are negligible. For example, in trees, seed production is trifling, periodicity is irregular, and measurements are not reliable. Currently, only a few models reflect reproduction in an auxiliary submodel. There is a general, if tacit, consensus that the effect of reproduction on tree and forest growth can be safely disregarded. Thus, the index of the most recent and largest collection of works on forest growth modelling (with 38 contributions) contains no entries on reproductive effort or seed production (Dixon et al., 1990). For humans interested in the production of wood, oxygen, or clean water, this consensus is expected. For trees, however, the items we value (foliage, branches, stems) are merely means of maximizing reproduction. Models neglecting the overwhelming significance of reproduction reflect our vested interests rather than those of the tree. In a model, plant growth and all other activities should be presented as a way to maximize reproductive effort. A model attempting to describe and explain biological phenomena from the plant's standpoint is likely to be more meaningful and successful than one based on our values. Instead of a mere description of growth or mortality, any forest

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process or pattern should be viewed as a contribution to the ultimate goal of tree life. A focus on maximizing reproductive effeort would not necessarily make models more complicated. The estimation of reproductive effort does not require counting every seed and can be done with information already at hand. This effort is equivalent to the difference between the efforts expended for growth and the returns in the form of assimilates. The parameters of a model should be calculated so as to maximize this difference. Such a model would be more efficient because the condition of reproductive m a x i m u m is equivalent to knowing one parameter of the model. This approach sometimes leads to novel and unexpected results. For example, plant growth is usually pictured as a smooth curve that can be differentiated at any point. The smoothness of this curve is a secondary p h e n o m e n o n that masks a break along the curve. This break indicates the onset of reproduction. As was proven mathematically, in a predictable environment maximal reproductive effort occurs only with the complete switchover from vegetative growth to seed production (Cohen, 1971; Insarov, 1975). The lack of environmental predictability smooths the transition. Actual growth curves result from a compromise between maximization and acceleration of reproductive effort. When growth is considered as a way of maximizing reproductive effort, growth curves can provide much more information than is realized. They can reveal the intensity of competition responsible for timing of the transition, the expected longevity of plants, and the degree of environmental predictability. Assuming that this degree is constant on a given site, the smoothness of a growth curve can tell us about the ability of various plant species with similar phenology to anticipate changes in the environment. The application of the described criterion is simple. When considering a model, we should check t o see if it is designed to maximize reproductive effort. A fuller presentation of this approach to modelling, along with m a n y applications, is given in Semevsky and Semenov (1982, reviewed in Zeide, 1986). This approach is based on the optimality principle, which is a generalization of the theory of natural selection.

Constancy of parameters A model is supposed to explain a process and expose its essential features. One of these features is the change of the dependent variable with respect to independent variables. The predicted values must change to match the modeled variable. Another complementary feature uncovered by a model is the constancy of parameters that govern the change of variables. We study not the change,

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but its repetitive patterns. A change without a pattern is beyond science. A model presents the means to discover these patterns. By tracing the apparent change, a model reveals hidden invariance. Therefore, parameter constancy might be a more reliable measure of quality than minimum of squared errors. This constancy can be tested by comparing the values of the parameters calculated at different segments of the model's domain. The above example of cell division can be used to demonstrate this procedure. Given two segments of, for instance, five points, the exponent will be less accurate on each of the segments than a fifth-degree polynomial. However, unlike the polynomial, the parameters of the exponent will remain practically the same on both segments.

Lack of patterns in the residuals Reality, as presented by raw data, contains two opposite elements: meaningful pattern and misleading noise. The quality of a model is determined by its ability to separate these elements of reality. Technically, this aspect of quality can be judged by the absence of patterns in the residuals. This is possible when the model exactly fits the underlying pattern (not the data). It seems that the total amount of order is invariant. When order is present in the residuals, it is lacking in the model and vice versa.

Least squares (with qualifications) Under certain conditions, conventional statistics based on least squares can be of help when judging a model's quality. When compared models have the same number of parameters, the more accurate one would be of higher quality. Least squares also are appropriate for describing quality of models when they are used for extrapolation rather than for interpolation. For example, in the case of cell division, an exponent will predict the future number of dividing cells more accurately (in terms of least squares) than a polynomial of any degree. HOW CAN WE CONSTRUCT

A GOOD

MODEL?

Although there is no rational answer to this question, experience gained through oscillations from one model (or 'paradigm') to another can provide certain hints for model construction.

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Failure of the refined and consistent The separation of reality into a comprehensible pattern and irrational noise might be misleading because we do not know where the pattern is hidden and whether it is unique. Too often we tend to assume uniqueness. Given a relationship between variables, which might be visualized as a scatter of dots, we immediately and almost instinctively tend to draw a single line representing the central trend. This line is presumed to eliminate accidental variation and convey the essential information hidden in the data. The same is true in more complicated situations that cannot be reduced to a cloud of dots. We tend to concentrate on the unique essence represented by the mean and believe in a single explanation of a given event at the expense of the entire range of possibilities. It is useful to define the same approach in negative terms: we are reluctant to admit the coexistence of opposing explanations and allow contradictions within a single entity. Facing a diversity in reality, we respond by separating the opposites either in space (by dichotomizing objects or producing conflicting contemporaneous theories) or in time (by holding contradictory theories consecutively). We try to keep a given event uniform and free from contraditions, either ordered or disordered, continuous or discrete, animate or inanimate, deterministic or stochastic, constant or variable, yes or no, black or white, good or evil, and so on. An appeal to formulate alternative hypotheses (Platt, 1964) does not contradict this approach because the intention is to weed out all but the best. Models produced in this spirit might be consistent and logical, only when they remain within the realm of the selected facts and propositions. Exorcised from inside the realm, contradictions sooner or later appear outside when the model is confronted with a wider reality. At this point we discard the model, often in favor of the opposite explanation. This second generation model highlights a different group of facts and neglects the rest. When sooner or later its opposition, the third generation model, appears, it is likely to resemble the forgotten first generation.

Example. Evolution of a major ecological concept, plant c o m m u n i t y and, in particular, our view of forest stands, shows that the above description of scientific progress is hardly an exaggeration. At the beginning of this century, the prescientific view of forests as wild and chaotic was replaced by the opposite image, the Clementsian doctrine of orderly and predictable succession. The change in vegetation was assumed to be caused primarily by autogenic, density-dependent factors and was directed towards an equilibrium climax formation. This climax formation was described as " a n

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organic entity" and " a n organism" with a highly ordered arrangement of plants (Clements, 1916). Half a century later this doctrine was abandoned in favor of the opposite model which is similar to the prescientific perception of forests. The changes are considered disorderly and unpredictable, and too frequent to allow equilibrium. Plants exist in a turbulent environment, rather than coexist with each other. "This local unpredictability is globally the most predictable aspect of the system, and may be the single most important factor in accounting for the survival of m a n y species" (Levin and Paine, 1974, p. 2744). Consequently, community structure is viewed as " m e r e l y a fortuitous juxtaposition of plant individuals" (Gleason, 1936, p. 44). Naturally, this extreme view meets, in turn, objections that " t h e pendulum may swing too far toward catastrophic recycling at the expense of concepts which involve long-term autogenic development" (Bormann and Likens, 1979, p. 661). Currently, a more balanced view of pattern and process in the plant community is gaining ground. This view attention to "causes which make for order and those that tend to upset it. Both sets of causes must be appreciated" (Watt 1947, p.2). Two extremes instead of one mean: Straddle rather than pinpoint

Experience gained from biology, as well as from other branches of science, indicates that no single proposition is fully satisfactory and that our thoughts tend to oscillate between opposites. In our attempts to pinpoint a problem, we too often miss the point. A more reliable approach would be to straddle the problem before pinpointing it. After so m a n y years, we can take a step beyond Plato's Theory of Forms. If indeed we can learn from our experience, we should not waste our efforts in meandering from one extreme to another. Instead, it makes sense to take the direct course and simultaneously formulate two idealized extreme models of the same process prior to quantification of the central tendency. Returning to the scatter of dots, this would mean that two lines representing extreme cases are drawn before the medial one. Examples. This approach (which can be traced back to the pre-Socratics, in particular Heraclitus) is often applied in ecology when we talk about density-dependent and density-independent processes, predictable versus unpredictable environments, r-and K-selection, territiorial or group behavior, biotic potential and environmental resistance, and the like. Not all scientists considered these extremes to be mutually exclusive. Sometimes, they were viewed as end points of a continuum of possibilities for plant or animal performance.

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Each growth equation (the logistics, Gompertz, von Bertalanffy, Chapman-Richards, and others) incarnates the outlined approach. All of them describe growth as the conflict between two opposing components and can be presented as the difference of two terms (Zeide, 1989). Generally, the positive term expresses the intrinsic tendency toward exponential increase while the negative one represents the environmental resistance. Let's consider, as an example of the outlined approach, the construction of a model describing individual tree survival. At one extreme, it can be assumed that tree size in a given stand varies at r a n d o m and is independent of tree position and past interaction with neighbors. Based on this assumption, it was proven that when the number of trees per unit area decreases from NO to N 1, the survival probability of a tree, Pi, is a power function of its normalized size, or rank r, (Zeide and Semevsky, 1972): p,=r, v (1) where v is the average number of 'victories' won by each surviving tree. This exponent (v) can be calculated as the ratio of the number of trees that died to the number of those that survived during a certain period: (2)

v = ( N O - N a) / N a

At the other extreme, we can assume that variation in tree size results exclusively from the interaction between trees. Consequently, there is a strict correspondence between degree of suppression and tree size. In this case only the largest and, therefore, least suppressed trees will survive: (~ P~=

when when

O
(3)

m
where m = ( N O - N 1 ) / N o indicates the highest rank and the proportion of trees that died (mortality). Note the simplicity (as well as generality and accuracy) of the expressions for both extremes: they contain no regression coefficients. Actual tree size results from both past suppression and r a n d o m events. Therefore, on a plane with the axes representing tree rank and survival probability, the observed tree survival lies in the field bounded by the lines described by equations (1) and (3) (Fig. 1). In addition to the perimeter of possible solutions, these equations provide the following boundary conditions that specify the position of the line depicting the expected actual survival. Because the extreme lines intersect at a specific point (r~ = m, Pi = mr), the expected line will pass through this point. The second derivative of this line will be zero at the same point, while the first derivative will be zero when r~ = 1. The integral of the expected line (the area below it) must be equal to N 1 / N o. Thus, the consideration of imaginary ideal cases, which seemed quite

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1.0-

Interaction~ ~ i

0.8 0.6 Actual~/ lff'/ Survival~ 0.4 Random /~'~l

~

0.2 ~

0

. . . . .

0.2

..1

0.4 m=0.6

0.8

1.0

RANKS, r i Fig. 1. Extreme cases and actual survival of trees.

remote from reality, proves to be helpful in formulating a realistic model describing the field of possibilities as well as a line depicting expected tree survival.

Are extremes imaginary? One conceivable drawback of the above modelling approach is that it defies parsimony and multiplies entities: we search for two (extremes) within one (phenomenon). This approach would be more appealing if could be shown that the extremes are not merely a figment of our imagination or temporary scaffolds to be disposed of after model building. We can touch the plant of mean mass (or near it) in a given population. Can the extremes, ensuing from our assumptions, be made as tangible? This question has interested people for a long time. It was the main fifth-century (B.C.) controversy between the Ionians, who maintained that things were mixtures of opposites apprehended by sense, and the Eleatics, who denied the reality of opposites (Cornford, 1960). Despite its long history, I did not know of a clear answer to this question until a small desert plant, Gymnarrhena micrantha Desf. (Compositae), provided me with one (Zeide, 1978). We can envision two ways in which plants could maximize their reproductive effort. As was mentioned above, in a predictable environment reproductive effort is maximized when a plant is doing one thing at a time. The plant grows and accumulates resources first. Then, the plant stops growth completely and switches to flower and fruit production. This strategy is called

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optimistic, because it makes sense only if there is confidence that nothing will interrupt the plant's natural life span. In a totally unpredictable environment, where a plant can perish at any moment, selective advantage would be gained by resorting to the opposite (pessimistic) strategy: immediate allocation, at the expense of fruit quantity, of a portion of its resources to reproduction, in order to assure that at least some fruit would be produced prior to a sudden death. The outlined strategies are incompatible: accelerated fruiting can be achieved only at the expense of quantity, while maximum yield necessitates a delay in reproduction. Yet, because the real environment is neither predictable nor totally unpredictable, plants are faced with the problem of reconciling the irreconcilable. However reasonable they are, these speculations about a plant's quandary are of questionable utility because, in the actual fruit production, both strategies are mixed and their existence is impossible to prove or disprove. It is indeed impossible to ascertain these strategies when dealing with typical plants that bear fruits on one kind. G. micrantha differs from most plants in its ability to produce two kinds of fruit (both sexual), called amphicarphy. They differ drastically: one is aerial, and the other is subterranean. Aerial fruits are light and numerous (up to 1500). They are produced after vegetative growth has ceased and are dispersed by the wind. The plants that did not reach a certain size do not produce these fruits. Thus, the production of these fruits follows the optimistic strategy. This optimism is supported by the much larger subterranean fruits which do not disperse. They secure the place occupied by the mother plant for its offspring. This adaptation is of great advantage to annual plants in deserts where only a tiny proportion of the entire area is suitable for existence. It is not surprising that in precarious desert conditions, the majority of G. micrantha plants (92%) originate from subterranean fruits. The number of fruits never exceeds five (three is the average) and is independent from plant size. Their mass, however, slowly increases with size (and age). These facts indicate that the production of subterranean fruits starts almost at the beginning of the plant's life and peters out. In other words, the production of subterranean fruits clearly conforms to the pessimistic strategy. This solution to the reproductive delemma has made G. micrantha highly successful in the Negev desert of Israel (where I studied it). Contrary to the c o m m o n belief that all organisms "must reach some compromise between the two extremes" of r- and K-strategies (Pianka, 1978, p. 122), G. micrantha shows that a plant can be perfect as both r- and K-strategists. The same solution sheds some light on the question of the reality of opposites. After handling thousands of these fruits that so neatly embody the conceived strategies, I am inclined to side with the Ionians and believe that there is some substance behind extremes.

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Fractal geometry and unity of opposites We used to think that spatial dimensions were whole numbers. Equally strong was the conviction that length, area, and volume of measured objects do not change with the unit of measurement. The spell of Plato's teaching and its crowning achievement, Euclidean geometry, was so powerful that only recently have we realized that this is not true for natural objects such as coastlines or tree crowns. Their size increases predictably when measured with smaller units that reveal finer details of a line or a surface. Their dimensions are fractional numbers unique for each object. Still, the integer spatial dimensions of classical geometry are not lost in the multitude of newly discovered fractal dimensions. They play the special role of extreme or limiting cases of fractal dimensions. Dimensions of natural lines are greater than one and smaller than two. The excess over one shows how convoluted the line is. These lines are hybrids between Euclidean lines and surfaces, just as classical surfaces and volumes serve as the ideal extremes of actual surfaces. Designed to be the ultimate embodiment of the ideal form (approximated by the population mean), it is ironic that Euclidean geometry has been put into service to place the two rows of buoys that mark the channel for the observable. This demonstrates that actual events can be viewed, not as defective copies of Plato's Forms or deviations from the central tendency, but as sound compromises, perfect products of opposing entities. Mandelbrot's (1983, p. 160) prediction that " t h e repeatedly scalloped surfaces of many large trees can be represented by scaling fractals of dimension D between 2 and 3" was shown to be true. For ten coniferous species, the dimensions varied from 2.21 for intolerant ponderosa pine (Pinus ponderosa) to 2.81 for very tolerant hemlock (Tsuga heterophylla) (Zeide, 1990). A tree crown cannot be described in the terms of classical geometry because it is neither a three-dimensional volume, nor a two-dimensional surface. Rather, the crown is a fractal, a crossbreed of surface and volume, formed by a convoluted and disjunctive photosynthetic layer. Fractal geometry has been successful in modelling natural objects because, besides spatial dimensions, it brings together a number of other opposites: chaos and order, repetition and novelty, accidental and essential, reductionism and holism, ideal and real.

Why two opposites? It is possible to imagine more than two opposites and organize our thoughts into triads, tetrads, etc. In my work, however, I found that these possibilities are less convincing and not as deep. After some reflection,

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multiple opposites can often be reduced to two. "Two is the smallest and simplest number that gets off the ground" (Quine, 1987, p. 55). While discussing a related type of modelling, alphabetic and numeric notation, Quine (p. 225) talks about "the miracle of binary notation." The next best notation to binary is Quine's description of this miracle in English: "Everything worth saying, and everything else as well, can be said with two characters." Actually, our discussion is not limited to the binary opposition. The two reproductive strategies are the means to a single end, maximum reproduction. The two extreme models of self-thinning (equations 1 and 3) describe two aspects of one process, tree survival. These means and ends are interwoven in the resulting model that brings together the submodels of the extremes. The outlined approach to modelling is not so much dualism or monism as it is a combination of both. The perseverance of both views provides another example of utility to unite opposites within a single model.

Advantages of modelling from extremes There are several advantages to modelling based on extremes. Consideration of two coexisting opposites promotes understanding of the resulting phenomenon. It is often much easier to describe two idealized situations than a single actual one. Mutual consideration of two extreme models usually provides substantial information (such as initial values, position of the inflection point, and other boundary conditions) for modelling the central tendency. Variation becomes bounded so that actual events can be viewed as participants in both extremes, instead of just being random deviations. The focus on extremes, rather than the mean, alerts us to the emergence of a new trend that can be visualized as branching from the original. The major advantage on which the above mentioned points are based is that many biological phenomena actually result from the conbined action of two opposing factors or groups of factors. Ultimately, all change in ecosystems results from the conflict between infinity implicit in multiplicative reproduction and the limit imposed by finite space. Therefore, it is only natural that good models reflect this conflict and its resolution. The outlined approach is not an unerring recipe for model building. It is not easy to uncover simultaneous extremes and incorporate them into a model. Ecological modelling will always be a challenge to our creativity. ACKNOWLEDGMENTS

I am grateful for valuable comments made by John L. Greene, Daniel J. Leduc, Lynne C. Thompson, and Suzanne Wiley. The cooperation of Fedor N. Semevsky is highly appreciated.

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