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The hierarchical organization of the aquatic ecosystem: an outline how reductionism and holism may be reconciled
Ecological Modelling, 66 (1993) 81 - 100
8!
Elsevier Science Publishers B.V., Amsterdam
The hierarchical organization of the aquatic ecosystem: an outline how reductionism and holism may be reconciled Claudia Pahl-Wostl Swiss Federal Institute of Technology, Ziirich, Institute fbr Aquatic Sciences, DObendorf Switzerland (Received 5 November 1991; accepted 5 June 1992)
ABSTRACT Pahl-Wostl, C., 1993. The hierarchical organization of the aquatic ecosystem: an outline how reductionism and holism may be reconciled. Ecol. Modelling, 66: 81-100. The aquatic ecosystem is viewed from the perspective of a hierarchical system of positive feedback cycles. The system's behaviour is governed by the dualism between cooperative phenomena at the level of the ecological network as a whole and processes at the level of its components. The former are characterized by the network's spatiotemporal organization; the latter are characterized by what is here referred to as the spatiotemporal niche, determined mainly by body-weight. Accordingly, the hierarchical structure of the system cannot be simply based on body-weight in a reductionist fashion, but must be interpreted from a holistic point of view taking into account the feedback structure of the whole network. The perspective outlined is that of a self-organizing system operating far from its equilibrium state. Possible consequences associated with such a view are suggested for research and management decisions. For instance the focus may be shifted from the investigation of single processes of control to questions dealing with the system's potential for spatiotemporal self-organization and its associated adaptability.
INTRODUCTION
The investigation of ecological processes over a range of spatial and temporal scales has revealed the importance of spatiotemporal variability for ecosystem function. Such findings cast doubt on the relevance of many Correspondence to: C. Pahl-Wostl, Swiss Federal Institute of Technology, Ziirich, Institute for Aquatic Sciences, Ueberlandstrasse 133, CH-8600 Diibendorf, Switzerland. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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established ecological principles, such as competitive exclusion, which are based on the assumption of a state of equilibrium. These principles lose their significance in a constantly changing environment. However, new theoretical principles have yet to be formulated. Unfortunately, the wealth of information provided by the detailed investigation of the heterogeneity of ecological processes in space and time threatens to overwhelm us. The important question arising here is whether such variability represents a source for stochasticity and randomness large enough to render all efforts in the search for ordered patterns futile. If so, we shall have to talk about heterogeneity, about fluctuations caused by the influence of stochastic environmental forcing on a random assembly of species. We may, however, adopt a different viewpoint and talk about the spatiotemporal organization, manifested in patterns in space and time, which results from cooperative behaviour at the level of the system as a whole. In the following ! shall attempt to provide evidence for the latter point of view, namely for the presence of a spatiotemporal hierarchical organization. The approach chosen is based on some hypotheses regarding whole system performance, viz.: • ecological networks represent hierarchical dynamic structures of positive feedback cycles; • the spatiotemporal patterns resulting from the dualistic interplay between the interactions amongst species and constraints at the macroscopic level of the ecological network are indicative of whole system performance; • spatiotemporai organization and the associated functional changes can be quantified using tools from information theory. The importance of cooperative behaviour at the community level has been emphasized recently by Cousins (1990). He suggested a new food web entity, the ecotrophic module, delimited in space and time by the scales determined by the social group of the largest predator. The necessity of properly defined units at the community level is evident, and my contribution in this paper may also be seen along these lines. Many problems arise from the intrinsic vagueness of the concept of the ecosystem. In contrast to Cousins, who decided to base his definition solely on the biota, I also consider it important to include abiotic components such as the detrital pool. Most of the processes of nutrient recycling, which involve the detrital pool, are an essential part of the communication network and feedback structure and are shaped and influenced by the biota. I argue here that the main contribution to organization derives from positive feedback, of which nutrient recycling constitutes one major source. The importance of this translates through all trophic levels, as was pointed out by Northcote (1988) and further supported by Vanni and Findlay (1990). They provided evidence for the significant contribution of fish to
H I E R A R C H I C A L O R G A N I Z A T I O N OF THE A Q U A T I C ECOSYSTEM
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nutrient recycling. Another review by Bianchi et al. (1989) highlighted the importance of positive feedback associated with grazing in different types of ecosystems: e.g. the productivity of coral reefs may be ten times higher in coral reefs grazed by the sea urchin Diadema antillarium than those without grazing. O d u m and Biever (1984) emphasized the importance of mycorrhizae for nutrient recycling in nutrient-poor forest ecosystems. The network approach suggests itself as the appropriate theoretical framework for describing such effects. In contrast to most applications of network analysis, which are limited to the description of static network properties, I would like to focus on the dynamics of the network in space and time, This paper constitutes an attempt to outline the various sources of spatiotemporal organization in the ecological network, their importance for ecosystem function, and their relationship to the overall hierarchical structure of the whole system. THE HIERARCHICAL O R G A N I Z A T I O N OF THE ECOLOGICAL NETWORK
Lately, the hierarchical structure of ecological systems has been receiving increasing attention (e.g. O'Neill et al., 1986). Traditionally, hierarchies are viewed as fully nested, with higher levels representing assemblies of lower-level units. The standard example is provided by the sequence c e i l - o r g a n i s m - p o p u l a t i o n - c o m m u n i t y . More recent is the growing emphasis on spatial and temporal scales and the associated hierarchy of functional couplings. In aquatic systems, these investigations have been motivated mainly by the desire to match the scales of biological and physical processes (for a short review see e.g. Powell, 1989). I wish to elaborate further on this concept of scales by looking more closely at the spatiotemporal hierarchy within the ecological network. The basis for the spatiotemporal scaling behaviour is provided by the energy transfer along a gradient of increasing body weight as being characteristic for the energy transfer in aquatic systems. The food chain shown in Fig. 1 was chosen as a simplified approximation of such a flow structure. Each trophic compartment represents an aggregation of species over a range of weight-classes with weight increasing from compartment 0 to compartment n. Fig. 1A depicts the flow of energy and Fig. 1B the corresponding flow of nutrients. Such a distinction between the two types of flow is made in order to emphasize that in most cases recycling is more important for nutrients than for energy, It is striking to realize that most feedback cycles include the detrital pool, which is far too often neglected by ecologists. The feedback structure will be dicussed in more detail later. First I should like to devote my attention to the elucidation of the
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A)
energy flow
Kmn-X in - 1 ~ (I -k)m oXo
(1-k)m iX1
(I-klmnXn
B) n u t r i e n t f l o w
E
NUTRIENT POOL
t
Fig. 1. Linear chain of (A) energy and (B) nutrient transfers, m denotes the metabolic rate and k the transfer efficiency. spatiotemporal hierarchical structure implicit in such weight-dependent energy transfers. To this end it is important to remember that the spatiotemporal characteristics of an organism are closely related to its weight. A variety of physiological and ecological properties scale as a power law with weight. The first of these allometric relationships to be observed were those between the weight, W, of an organism and its physiological properties (for a review see Peters, 1983). For instance, the rate of respiration, r, varies as a power function of W: r = f l W -~
(1)
the values for y range from 0.25 to 0.3 for a large variety of organisms of different weight and taxa. The temporal scale of response may be represented by the metabolic rate. It corresponds to the organism's perception of its environment and to its speed of adaptation to environmental changes. In addition, allometric relationships have been detected for ecologically relevant properties such as generation time and home range. Hence, weight determines what is referred to here as the "spatiotemporal niche", charac-
HIERARCHICAL ORGANIZATION OF THE AQUATIC ECOSYSTEM
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terized by: the range in space and time - represented by the home range. which may be equated with the "spatial radius of activity" and the generation time, which may be equated with the "temporal radius of activity". The use of a spatiotemporal niche as elementary characterization is further supported by the allometric scaling observed for the number of species in a logarithmic weight-class, as well as for the population density (May, 1986; Dickie et al., 1987). Such observations lead to the conclusion that the spatiotemporal characteristics are a major determinant for speciation. (A logarithmic weight-class is defined as a class where the ratio of the weight range of the two neighbouring classes is constant. In general the ratio is chosen equal to two, corresponding to a doubling in weight with each successive class.) Such a spatiotemporal niche may be occupied by an assemblage of species. The distribution of the species' abundance reflects their ability to use and to adapt to certain spatiotemporal patterns in their environment. These patterns may derive from variations of the external factors of the physical environment, or they may be internally generated by species interactions, or, most often, they may represent a combination of both. In loose analogy to the concept of centrifugal organization for plant communities (e.g. Keddy and MacLellan, 1990), a type of log-normal distribution in abundance can be expected. In the following I shall use the aggregated assemblages of species within a spatiotemporal niche as the smallest units of the network, and I shall call them "ecounits". The distribution of taxonomic species across spatiotemporal niches is discussed in more detail in Pahl-Wostl (1993). The scaling with body weight of the properties characterizing the spatiotemporal niche confers a hierarchical spatiotemporal structure upon the food chain. The compartments in our model operate on different scales, and each compartment represents an aggregation of ecounits over a range in time and space. To resolve this aggregated structure, the scaling properties in our model have to be defined. As already mentioned, population density scales allometrically with body-weight. In this respect it is of interest that the distribution of biomass across the weight spectrum in aquatic systems shows a remarkable regularity, in that the biomass contained in a weightclass is approximately the same over a whole range of weight-classes. The abundance (the number of individuals in a class) seems to decrease in proportion to the increase in body-weight (e.g. Sheldon et al., 1972; Sprules and Munawar, 1986). A more general case is introduced into the model, namely that the abundance is proportional to body-weight to the power of - a , thereby treating the constancy of the biomass corresponding to a = 1 as one special case.
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The basic assumptions of the model are: (1) A level i of the food chain represents a logarithmic weight class with a m e a n weight of W,. and a total biomass of X i, i = 0, 1. . . . . n. (2) The numerical abundance N~ is proportional to Wi-% Hence, the biomass X i = N,.Wi is related to weight according to: X ~ W ~l - ~
(2)
If a = l then X 0 = X l-- ... = X , . (3) Energy balance: (f
d X i dXi ~ 0
(integrated over a temporal and spatial period that exceeds the temporal and spatial range of activity for the longest living species in the system). (4) The transfer efficiency, k, and the weight ratio of consumer to producer, q (q >> 1), are assumed to be the same for all levels. The change in weight as a function of level is d e t e r m i n e d by q: W,. = qW~_~ ~
W,. = q ' W o
(3)
The metabolic rate, mi, is assumed to scale according to eq. (1): mi = q-Ymi-i
= mi = q-iYmo
(4)
For a given input of energy at the base of the food chain, higher trophic levels must have a reduced metabolic cost to compensate for the decrease in available energy, if eq. (2) holds: mix i = kmi_lXi_
1
H e n c e k may be derived from eqs. (2), (3) and (4): k = q¢l-,,-~)
(5)
where k = transfer efficiency, q = ratio of W, to W,._~, a = allometric exponent of the scaling of a b u n d a n c e (cf. eq. 2), y = allometric exponent of the scaling of respiration (cf. eq. 1). The scaling behaviour of the biomass distribution puts a constraint on the efficiency if the energy balance is to be maintained. An even distribution of biomass among weight-classes is not a prerequisite to achieving energy balance. However, such regular scaling has interesting consequences for the structure of the flows within the network. As we shall see in the following, it can be interpreted in terms of a dynamic self-similar structure. To resolve the aggregated c o m p a r t m e n t s into ecounits we still need the scaling relation of the range of the spatiotemporal niches.
H I E R A R C H I ( ' A L O R G A N I Z A I I O N O F T H E AQ)UA I I C EC()Ssr STEM
~7
The generation time, T,, corresponding to the temporal radius of activity is assumed to scale inversely with the metabolic activity, rni: L
~
q
"v
To
(6)
It is reasonable to assume that the number of ecounits as a function of weight, E , , depends on the width of the temporal niche as quantified by 7,: E i =
q - ' ~ E i i ~ E cz I'V - 7
(7a)
If in addition the d e p e n d e n c e on space is taken into account: Ei = q . . . . ~q-i~.Eo = q - ~ l +~)i~,Ei ) ~ E cz W -~L + ' ~
(7b)
where o- represents the effective spatial dimensionality, o- must not necessarily represent a Euclidean dimension, but may instead be fractal. It will d e p e n d on the structure of the habitat and the way in which the organisms perceive their environment, or may have a value of around 2 for man,,' organisms in a benthic habitat and one more closely to 3 in the pelagial. In the following I will continue with a one-dimensional scaling of the ecounits as a function of the level, and T, is chosen as being representative for the range of an ecounit. The use of such a one-dimensional scaling is motivated by the fact that temporal scales seem to be associated with biological processes, whereas spatial scales seem to be associated with physical processes (e.g. Powell, 1989). Each c o m p a r t m e n t in the food-chain of Fig. 1 may now be resolved into ecounits as shown in Fig. 2. The decline in the n u m b e r of ecounits along the food chain d e p e n d s on q. The scaling in Fig. 2 (q = 17.5 and k = 0.5) was chosen to facilitate the graphical representation of the hierarchical levels of the food chain. It would be much steeper for realistic values of q (around 1000) and k (around 0.1). As indicated by the dashed lines, a modular structure can be discerned, the basic building blocks of which are defined next. In Fig. 2 the structure of the feedback cycles was neglected for the sake of preserving a simple representation of the hierarchy. However, as pointed out before, feedback cycles are of utmost importance for the processes of self-organiziation. A cyclic feedback unit as shown in Fig. 3 is therefore suggested as the basic modular building block. The consumer ecounit of level i interacts directly only with the ecounits in the level immediately below it. Due to the hierarchical structure, however, it also indirectly influences the aggregated assemblages of ecounits of levels 0 to i - 1, aggregated over the range in space and time determined by the consumer in level i. The detrital pool is common to the whole system. The modules are characterized by their scale in time and in space, a corresponding internal organization, and their response to environmental
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range in space and time
f
I
flow of energy
Fig. 2. Schematic drawing of the model in Fig. 1 resolved into ecounits. The size of the ecounits refers to their spatiotemporal extension, not to any magnitude of the standing stocks. The dashed lines indicate the hierarchical modular structure. See text for further explanations.
factors. T h e resulting m o d u l a r h i e r a r c h i c a l s t r u c t u r e o f f e e d b a c k loops is d e p i c t e d in Fig. 4 A with level 0 c o m p r i s i n g the p r i m a r y p r o d u c e r s in the f o r e g r o u n d . T h e spatial a n d t e m p o r a l scales i n c r e a s e f r o m level to level as
aggregated ecounits level 0 to i-I
¢on$ kll~let" level i
ill -) I COrn m o i l d e t r i t a l pool
I~ I
I Fig. 3. Basic module of the ecological network. Each module is characterized by a certain spatiotemporal scale. The detrital pool is common to the whole system.
HIERARCHICAL
ORGANIZATION
A
OF THE AQUA.TIC ECOSYSTEM
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t
A) t\ \ \
\
~ix'
\
\\
\
x
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\\ \
S
~
\\
\\
\\
't
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\
\
A ~" \
x
\x. \\
\
I
\\\\
\
v
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log [
B)
log s
r log
t
Fig. 4. (A) Modular hierarchical structure of the ecological network with level 0 comprising the primary producers in the foreground. The spatial and temporal scales increase from level to level as illustrated by the change in scale of the coordinate axes. The change in the size of the arrows represents the decrease in flow intensity at each level. Feedback cycles are indicated by the arrows emerging from the sides of the compartments. (B) Intensity of the corresponding feedback cycles as a function of space and time scales. With increasing module size, intensity decreases and the range extends over larger areas of space and time.
illustrated by the c h a n g e in scale o f the c o o r d i n a t e axes. T h e f e e d b a c k cycles are i n d i c a t e d by the a r r o w s e m e r g i n g f r o m the sides o f t h e c o m p a r t m e n t s . T h e c h a n g e in the size o f the arrows r e p r e s e n t s t h e d e c r e a s e in intensity o f t h e f l o w s with i n c r e a s i n g level. T h e intensity is d e t e r m i n e d by the m e t a b o l i c rate c h a n g i n g with i n c r e a s i n g t r o p h i c level a c c o r d i n g to (4).
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Hence, the flow intensity is inversely proportional to the corresponding space and time scales, as shown in Fig. 4B. However, it should be noted that in case of equal biomass the total amount of matter, Fi, transferred over a certain feedback loop is the same when averaged over space and time. F~ may be derived from eqs. 2, 4 and 6: Fi =
T i m i X i = m o T o X = const, for c~ = 1.
The origin of the observed regularities in behaviour is still unknown. An explanation is attempted by focussing on the question of how a network's spatiotemporal organization and cooperative behaviour may be effected through the interactions of the c o m p o n e n t parts. SOURCES OF SPATIOTEMPORAL ORGANIZATION To quantify the degree of what is here called spatiotemporal organization, a m e t h o d based on information theory has been developed (PahlWostl, 1990, 1991). The space- a n d / o r time-dependent average mutual information quantifies the reduction in redundancy of functionally equivalent interactions in a network due to a segregation of activity along the dimensions of space a n d / o r time. Definitions and details of the mathematical expressions are given in the Appendix. The macroscopic spatiotemporal organization of an ecosystem arises from interactions within and from interactions across levels of spatiotemporally similar components. Organization within a level may emerge from resource partitioning as achieved by a segregation of activity of functionally redundant units along the dimensions of time a n d / o r space. In a recent paper (Pahl-Wostl, 1990) the importance of spatiotemporal resource partitioning was emphasized for species coexistence and for the efficiency of nutrient utilization. The network in Fig. 5 may serve as an illustrative example. Two functionally equivalent producers, e.g. two species of algae, both share the same nutrient pool D and are preyed upon by the same consumer K. Fig. 5A depicts the spatially and temporally averaged network structure. However, as shown in Fig. 5B, due to resource partitioning the producers may become separated along a dimension, d, of either time or space. The seasonal succession of algal species may serve as an example of temporal organization. An example of spatial organization is provided by the occurrence of different algal communities as a function of depth in the stratified water column. The second type of spatiotemporal organization across levels may emerge from pulsed consumption, where the processes of production and consumption are segregated in time a n d / o r space as shown in Fig. 5C
HIERARCHICAL O R G A N I Z A T I O N OF THE AQU.kTIC ECOSYSTEM
0[
A)
"-.
B)
d
/ / /
d
consumption
Fig. 5. Network consisting of two producers, P1 and P2, and one consumer K. (A) averaged flow structure; (B) segregation of the two producers along a dimension, d (C) segregation of production and consumption along adimension, d. d may represent the dimension of time or a dimension of space.
(Richardson and Odum, 1981; Pahl-Wostl, 1991). One may think of the daily vertical migration of zooplankton as an example, where both temporal and spatial segregation are present. In the case of pulsed consumption the system switches between two different functional states. Such behaviour contrasts with resource partitioning where the functional structure of the overall system does not change. However, the same functional niche is occupied by different species at different intervals of time or space. I have studied the changes in organization associated with these patterns (Pahl-Wostl, 1991) using reaction-diffusion models to describe the coupled
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biological and physical processes. When the transfer rate from producer to consumer crosses a critical threshold the equilibrium state becomes unstable, leading to the onset of oscillatory behaviour in space and time. Hence, the degree of organization depends on system-inherent parameters and geometrical constraints - this endows the system with remarkable self-regulatory properties. In contrast to the chance patterns emerging from physical and chemical processes we have to remember that biological systems have evolved. The scales of their response and the patterns subsequently produced can be expected to be the product of selection by long term evolution. To sum up, the analysis of spatiotemporal organization suggests that certain patterns may be favoured over others due to the associated functional changes and feedback effects expressed in the network of the whole system. One has to distinguish between the interactions within a spatiotemporal level of functionally equivalent species and the interactions between levels, which serve as couplings of the system across scales. These concepts may be applied to the food chain model discussed in the preceding paragraph. It is assumed that resource partitioning and segregation of the processes of production and consumption translate through all hierarchical levels. The corresponding behaviour of the spatiotemporal organization, I,~ (see also Appendix), can be derived as shown in Fig. 6. Its is represented as a function of the interval of resolution along the dimensions of space and time. The contribution of each dimension would depend on the details of the pattern involved. The resolution was successively
5.5-
4.G
£, 3.5
2.5
1.5 o
log
of
resolution
interval
Fig. 6. l,s as determined for the hierarchical network represented in Fig. 4A as a function of the logarithm of the resolution interval.
HIERARCHICAL
ORGANIZATION
OF THE AQUATIC
ECOSYSTEM
Q3
increased choosing always the scale of the next hierarchical level in the model. The spatiotemporal segregation of feedback flows results in a substantial decrease in redundancy and increase in organization. The uniform slope is caused by the assumed self-similar structure of the network. Changes would indicate breaks in self-similaritT. It is a difficult task to separate the influence of the spatial and temporal dimensions. Generally they are interdependent (see also Pahl-Wostl, 1991) and the elucidation of the interrelationships still requires much effort. As mentioned above, one difference resides in the fact that the influence of physical processes may dominate space and that of biological processes may dominate time. Here we touch on another important aspect, namely that of the segregation of exogenous from endogenous sources of organization. As in the case of space and time, we face the problem that both sources of organization are interdependent. The approach presented here provides a theoretical quantitative framework for dealing with such questions. As shown in the Appendix, the various contributions may be quantified by splitting up the average mutual information into terms related to inputs, internal exchanges, exports and dissipations, respectively. A pictorial representation is provided as well to assist those readers unfamiliar with aspects of information theory referred to here. As always when one starts off with simple models, one has to realize that reality, is far more complex. The chosen food-chain model depicts only the extreme case of separate trophic levels with the biomass at each level being centered within a narrow weight range. A more realistic outline is sketched
energy
transfer
dW
I slope
._ ) I
q { I
), size W
Fig. 7. Schematic drawing of the energy transfers in a food web as a function of weight. The lines representing the various trophic levels may show considerable overlap.
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in Fig. 7. The lines represent the levels as a function of the number of transfers from the energy's point of entry into the system. The spatiotemporal organization is characterized by the slope and the weight range, dW, of such a level, and the p r e d a t o r - p r e y ratio, q. These properties may be different in the various levels, and they may change as a function of space a n d / o r of time. Does such variability imply that all the derivations based on this simple model are invalidated? It is argued not to be the case because a start from a simple conceptual framework is important and complexity may be introduced step by step. Seasonal changes, for example, may be interpreted as resonance p h e n o m e n a in terms of the oscillation frequencies of the various feedback cycles. CONCLUDING REMARKS
The spatiotemporal organization as defined in the context of this paper represents a property of the network as a whole. On the one hand, the organizational pattern depends on the spatiotemporal characteristics of the species. On the other hand, the structure of the network imposes constraints on an organism by virtue of its being imbedded in a certain environment. The mutual relationship between the macroscopic level of the network and the level of the components is illustrated in Fig. 8. Discussed in detail, however, the situation is not as simple as presented in this figure. The components are not uniform in their spatiotemporal behaviour and one has to deal with a hierarchy of cascading interactions. Analogously to the slaving principle advocated by Haken (1983) one may think of a hierarchy of instabilities governing system behaviour. This principle applies to dynamic systems close to a bifurcation point where the behaviour is
macroscopic n e t w o r k s t r u c t u r e s p a t i o t e m p o r a l or! a n i z a t i o n
,I
I
I
J
J I, 1
level of the ecounits s p a t i o t e m p o r a ] niches
Fig. 8. Mutual interaction between the macroscopic level of the ecological network (as characterized by the spatiotemporal organization) and the level of the ecounits (as characterized by the spatiotemporal niche).
HIERARCHICAL
ORGANIZATION
Of 7 T H E A Q U . - \ T I ( EC'OS~tSFES, I
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dominated by the slowest mode, the so-called order parameter. This allows long-range correlations in space and time, which in our case may be stabilized by the very. fact of the system being hierarchical. At the risk of being repetitive 1 wish to emphasize once more that even when talking about the dominance of order parameters one has to keep in mind that these order parameters arise from the interactions among the components of the systems through processes of self-organization. Along this line of reasoning the dichotomy between top-down and bottom-up control converges to a mutual and inseparable dependence of both factors. Neither a purely reductionist approach nor a merely holistic perspective is sufficient to encompass the intrinsic nature of the system's behaviour. At the organism level it is easily accepted that growth and development constitute cooperative phenomena. Body-weight imposes well-known constraints on the behaviour of the components. Adopting this point of view at the ecosystem level is not as straightforward. However, the striking regularity observed in the behaviour of many properties of the ecological system suggest strongly that this description also holds true in this case. Arguments in favour of autonomous cooperative behaviour are corroborated by a variety of experimental findings. Sprules and Munawar (1986) reported breaks in allometric scaling of the plankton weight spectra as a consequence of perturbations. A slope of - l, corresponding to an equal biomass distribution in steady state and little variability, characterized oligotrophic, undisturbed ecosystems. In these systems we may expect a high degree of endogenous spatiotemporal organization due to the high degree of autonomy of the system. Another example is provided by the p h e n o m e n o n of eutrophication where the system is driven by external inputs and its autonomy is greatly affected. The controlling influence of positive feedback via nutrient recycling is diminished, and a major potential for endogenous organization is reduced. Hence, two properties are suggested as whole system descriptors:
Autonomy: ratio of internally generated to externally imposed organization. In the context of this paper, autonomy is equivalent to the independence of the system from external influences, expressed in terms of input of energy and resources (see also Appendix).
Adaptability: ability of the system to respond to environmental changes without decline in organizational characteristics. It is argued here that an increase in spatiotemporal organization is connected with an increase in autonomy and efficiency. Such a development should also coincide with an increase in food web complexity and
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food chain length. Support for this argument is given by experimental findings summarized recently by Briand and Cohen (1987). They provided statistical evidence for the influence of the environment on food chains by showing that food chains are longer in 3- than in 2-dimensional habitats. The dimensionality of the habitat is positively correlated with the opportunity for spatial organization leading to an increase in the number of spatiotemporal niches and the diversity of the ecounits. In this largely phenomenological approach I have tried to provide a theoretical basis within a framework of empirical facts for what could become a potent tool for exploring and understanding food web and ecosystem patterns. In order to establish a firm base and to be able to substantially challenge current thinking, much additional work is needed, some of which is already in progress. The most interesting developments may include: • Investigation of the relationship between the spatiotemporal organization and the adaptability of a system. • A general definition of the compartments of an ecological ne~'ork based on the concept of the spatiotemporal niche. • Determination of the hierarchical level of an external disturbance in terms of its energy content and spatiotemporal pattern. • Quantification of the spatiotemporal information provided by the environment and the consequences for the internal organization of the system. Such knowledge would be very valuable for management decisions that should be taken in the light of preserving a system's potential for adaptation and self-organization instead of attempting to impose external control strategies. To end with I should like to establish a link to the ecology of terrestrial systems by hypothesizing that regular biomass distributions, similar to those in aquatic ecosystems, would be found in terrestrial systems as well if the biomass were to be grouped into logarithmic classes, not of body-weight, but of metabolic turnover time (cf. Pahl-Wostl, 1993). We have to keep in mind that plants are modular organisms. Because of their long lifetimes (e.g. trees) they show a highly flexible hierarchical internal structure. Steele (1985) pointed out the difference in the characteristics of the spectrum of environmental variability in the aquatic and terrestrial environment. Aquatic systems show red-noise, terrestrial systems predominantly whitenoise spectra. Hence, one would expect major differences in the characteristics of spatiotemporal organization. A comparative analysis of terrestrial and aquatic ecosystems, focusing on these aspects, might provide new insights into the function and organization of ecosystems. The approach presented in this paper offers some, albeit quite controversial and tentative, ideas and a quantitative tool to cope with this task.
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ACKaNOWLEDGEMENTS T h i s essay has u n d e r g o n e n u m e r o u s a n d s u b s t a n t i a l revisions since the first draft. A m o n g o t h e r s , D . M . L i v i n g s t o n e , U. G a e d k e , R. U l a n o w i c z , S. C o u s i n s a n d L. S t o n e p r o v i d e d h e l p f u l c o m m e n t s w h i c h a i d e d in the d e v e l o p m e n t of the p r e s e n t v e r s i o n . REFERENCES Bianchi, J.. Jones, C. and Shachak, M., 1989. Positive feedback of consumer population density on resource supply. TREE, 4: 234-238. Briand, F. and Cohen J., 1987. Environmental correlates of food chain length. Science, 238: 956-960. Cousins, S., 1990. Countable ecosystems deriving from a new food web entity. Oikos, 57: 270-275. Dickie, L., Kerr, S. and Boudreau, P., 1987. Size-dependent processes underlying regularities in ecosystem structure. Ecol. Monogr., 57: 233-250. Haken, H., 1983. Synergetics. Springer, Berlin. Keddy, P. and MacLellan, P., 1990. Centrifugal organization in forests. Oikos, 59: 75-84. May, R., 1986. The search for patterns in the balance of nature: advances and retreats. Ecology, 67: 1115-1126. Northcote, T.G., 1988. Fish in the structure and function of freshwater ecosystems: a "top-down" view. Can. J. Fish. Aquat. Sci., 45: 361-379. Odum, E. and Biever, L., 1984. Resource quality, mutualism, and energy partitioning in food chains. Am. Nat., 124: 360-376. O'Neill, R., DeAnge[is, D., Allen, T.F. and Waide, J., 1986. A Hierarchical Perspective of Ecosystems. Princeton University Press, Princeton, NJ. Pahl-Wostl, C., 1990. Temporal organization: a new perspective on the ecological network. Oikos, 58: 293-305. Pahl-Wostl, C., 1991. Information theoretical analysis of functional temporal and spatial organization in flow networks. Math. Comp. Model., 16: 35-52. Pahl-Wostl, C., 1993. Foodwebs and ecological networks across spatial and temporal scales. Oikos, in press. Peters, R., 1983. The Implications of Body Size. Cambridge University Press, Cambridge, UK. Powell, T., 1989. Physical and biological scales of variability in lakes, estuaries, and the coastal ocean, in: J. Roughgarden, R.M. May and S.A. Levin (Editors), Perspectives in Ecological Theory. Princeton University Press, N J, pp. 157-176. Richardson, J. and Odum, H., 1981. in: W.J. Mitsch, R.W. Bosserman and J.M. Klopatek (Editors), Energy and Ecological Modelling. Elsevier, New York, pp. 641-647. Sheldon, R., Prakash, A. and Sutcliffe, W., 1972. The size distribution of particles in the ocean. Limnol. Oceanogr., 17: 327-340. Sprules, W. and Munawar, M., 1986. Plankton size spectra in relation to ecosystem productivity, size, and perturbation. Can. J. Fish. Aquat. Sci., 43: 1789-1794. Steele, J., 1985. A comparison of terrestrial and marine ecosystems. Nature, 313: 355-358. Ulanowicz, R., 1986. Growth and Development: Ecosystems Phenomenology. Springer, New York. Vanni, M. and Findlay, D., 1990. Trophic cascades and phytoplankton community structure. Ecology, 71: 921-937.
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APPENDIX
Definitions of expressions used in the text (for further details see Pahl-Wostl, 1990, 1991 and Ulanowicz, 1986). All expressions are defined in a network of flows of energy and matter between the compartments representing standing stocks.
A. Definitions of some general expressions as used in this paper Growth: increase in total system throughput T. Organization: deviation of the observed network structure from a randomly connected network, where flows are statistically distributed among compartments. The degree of organization may be quantified by the average mutual information I of the ecological network. Decelopment: increase in organization as quantified by L Function: defined within the topology of the ecological network. Functionally equivalent compartments may be aggregated without a change in organization (e.g. P1 and P2 in Fig. 5A). Redundancy: multiplicity of functionally equivalent pathways. High redundancy implies low organization and vice versa. Spatiotemporal organization: reduction in redundancy increase in average mutual information upon resolving the structure of flows along the dimensions of space and time (e.g. Fig. 5B).
B. Definition and calculation of specific network properties T: total system throughput n
n+2
r
s
T= Z E Z Z Zjik' j=0
i=1
k=l
l=1
I: average mutual information in the space and time averaged network I=
Y~ -~-log j=0
i=1
.
.
It: average mutual information in the time resolved, space averaged network
I,--
E £
log
H I E R A R C H I C A L O R G A N I Z A T I O N OF THE AQUAT1C ECOSYSTEM
Its:
Q0
average mutual information in the time and space resolved network
= j = 0 i = l k=lZ=l T
log
Tji . Tj.ktTikt
where: Tii: flow from c o m p a r t m e n t j to i; Tick: flow from j to i during time interval k; ~ikt: flow from j to i during time interval k and space i n t e r a t l. A point denotes summation over the corresponding index. It can be shown that: It,. > I t > I > O. This description refers to an open system where separate compartments are defined to account for the exogenous transfers. C o m p a r t m e n t 0 serves as source for the exogenous inputs into the system, compartment n + 1 serves as sink to accept the exogenous exports, compartment n + 2 receives the dissipative losses (in the case of energy flows). The average mutual information is expressed in "bits" if logarithms to the base 2 are used.
C. Decomposition of the ac'erage mutual information into terms stemming from inputs (Io) , internal exchanges (Ii), exports (I E) and dissipation (I D) I = I o + I I + IE + ID where
io
~_. T°i
( T°iT
= ,=1 7 - l o g
( <,r
,,-- jf:= l i =fl v- log Is =
j=1
T
j=l
T
log
Tj.T.(~ ÷ ~)
Tj.T.(n + 2)
I t and Its may be d e c o m p o s e d accordingly into 1,0 and /,s0... The difference b e t w e e n the various terms gives the contribution of the temporal (e.g. I,o-I o) and spatiotemporal (e.g. I,o-I o) structure of the various flows. To assist those readers unfamiliar with information theory, it is helpful to illustrate the meaning of the terms defined using Venn diagrams as shown in Fig. 9. In this representation, the uncertainty in a particular
100
C, PAHL-WOSTL
A)
B)
C)
Fig. 9. Venn diagrams representation of concepts derived from information theory. See text for further explanations.
attribute is represented by a circle. Thus, Fig. 9A depicts the respective uncertainties of inputs and outputs in a network. The dotted areas represent the external exchanges. To facilitate the representation the distinction between dissipative flows and exports was neglected. The knowledge of the network configuration provides the mutual information I as represented by the shaded overlap region in Fig. 9B. The amount of overlap is a measure for the specificity of the flows in the network. The black region corresponds to the contribution of the inputs I 0. The smaller the black region the higher is the redundancy associated with the inputs which means that the inputs are distributed over many network compartments. As shown in Fig. 9C, another circle represents the uncertainty associated with the set of all time intervals in the network where the flows are resolved along the dimension of time. The resulting information I t is depicted by the total shaded region. The black region corresponds to the difference I t o - I o, corresponding to the reduction in redundancy of the inputs obtained by resolving their temporal variation.
8!
Elsevier Science Publishers B.V., Amsterdam
The hierarchical organization of the aquatic ecosystem: an outline how reductionism and holism may be reconciled Claudia Pahl-Wostl Swiss Federal Institute of Technology, Ziirich, Institute fbr Aquatic Sciences, DObendorf Switzerland (Received 5 November 1991; accepted 5 June 1992)
ABSTRACT Pahl-Wostl, C., 1993. The hierarchical organization of the aquatic ecosystem: an outline how reductionism and holism may be reconciled. Ecol. Modelling, 66: 81-100. The aquatic ecosystem is viewed from the perspective of a hierarchical system of positive feedback cycles. The system's behaviour is governed by the dualism between cooperative phenomena at the level of the ecological network as a whole and processes at the level of its components. The former are characterized by the network's spatiotemporal organization; the latter are characterized by what is here referred to as the spatiotemporal niche, determined mainly by body-weight. Accordingly, the hierarchical structure of the system cannot be simply based on body-weight in a reductionist fashion, but must be interpreted from a holistic point of view taking into account the feedback structure of the whole network. The perspective outlined is that of a self-organizing system operating far from its equilibrium state. Possible consequences associated with such a view are suggested for research and management decisions. For instance the focus may be shifted from the investigation of single processes of control to questions dealing with the system's potential for spatiotemporal self-organization and its associated adaptability.
INTRODUCTION
The investigation of ecological processes over a range of spatial and temporal scales has revealed the importance of spatiotemporal variability for ecosystem function. Such findings cast doubt on the relevance of many Correspondence to: C. Pahl-Wostl, Swiss Federal Institute of Technology, Ziirich, Institute for Aquatic Sciences, Ueberlandstrasse 133, CH-8600 Diibendorf, Switzerland. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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established ecological principles, such as competitive exclusion, which are based on the assumption of a state of equilibrium. These principles lose their significance in a constantly changing environment. However, new theoretical principles have yet to be formulated. Unfortunately, the wealth of information provided by the detailed investigation of the heterogeneity of ecological processes in space and time threatens to overwhelm us. The important question arising here is whether such variability represents a source for stochasticity and randomness large enough to render all efforts in the search for ordered patterns futile. If so, we shall have to talk about heterogeneity, about fluctuations caused by the influence of stochastic environmental forcing on a random assembly of species. We may, however, adopt a different viewpoint and talk about the spatiotemporal organization, manifested in patterns in space and time, which results from cooperative behaviour at the level of the system as a whole. In the following ! shall attempt to provide evidence for the latter point of view, namely for the presence of a spatiotemporal hierarchical organization. The approach chosen is based on some hypotheses regarding whole system performance, viz.: • ecological networks represent hierarchical dynamic structures of positive feedback cycles; • the spatiotemporal patterns resulting from the dualistic interplay between the interactions amongst species and constraints at the macroscopic level of the ecological network are indicative of whole system performance; • spatiotemporai organization and the associated functional changes can be quantified using tools from information theory. The importance of cooperative behaviour at the community level has been emphasized recently by Cousins (1990). He suggested a new food web entity, the ecotrophic module, delimited in space and time by the scales determined by the social group of the largest predator. The necessity of properly defined units at the community level is evident, and my contribution in this paper may also be seen along these lines. Many problems arise from the intrinsic vagueness of the concept of the ecosystem. In contrast to Cousins, who decided to base his definition solely on the biota, I also consider it important to include abiotic components such as the detrital pool. Most of the processes of nutrient recycling, which involve the detrital pool, are an essential part of the communication network and feedback structure and are shaped and influenced by the biota. I argue here that the main contribution to organization derives from positive feedback, of which nutrient recycling constitutes one major source. The importance of this translates through all trophic levels, as was pointed out by Northcote (1988) and further supported by Vanni and Findlay (1990). They provided evidence for the significant contribution of fish to
H I E R A R C H I C A L O R G A N I Z A T I O N OF THE A Q U A T I C ECOSYSTEM
~3
nutrient recycling. Another review by Bianchi et al. (1989) highlighted the importance of positive feedback associated with grazing in different types of ecosystems: e.g. the productivity of coral reefs may be ten times higher in coral reefs grazed by the sea urchin Diadema antillarium than those without grazing. O d u m and Biever (1984) emphasized the importance of mycorrhizae for nutrient recycling in nutrient-poor forest ecosystems. The network approach suggests itself as the appropriate theoretical framework for describing such effects. In contrast to most applications of network analysis, which are limited to the description of static network properties, I would like to focus on the dynamics of the network in space and time, This paper constitutes an attempt to outline the various sources of spatiotemporal organization in the ecological network, their importance for ecosystem function, and their relationship to the overall hierarchical structure of the whole system. THE HIERARCHICAL O R G A N I Z A T I O N OF THE ECOLOGICAL NETWORK
Lately, the hierarchical structure of ecological systems has been receiving increasing attention (e.g. O'Neill et al., 1986). Traditionally, hierarchies are viewed as fully nested, with higher levels representing assemblies of lower-level units. The standard example is provided by the sequence c e i l - o r g a n i s m - p o p u l a t i o n - c o m m u n i t y . More recent is the growing emphasis on spatial and temporal scales and the associated hierarchy of functional couplings. In aquatic systems, these investigations have been motivated mainly by the desire to match the scales of biological and physical processes (for a short review see e.g. Powell, 1989). I wish to elaborate further on this concept of scales by looking more closely at the spatiotemporal hierarchy within the ecological network. The basis for the spatiotemporal scaling behaviour is provided by the energy transfer along a gradient of increasing body weight as being characteristic for the energy transfer in aquatic systems. The food chain shown in Fig. 1 was chosen as a simplified approximation of such a flow structure. Each trophic compartment represents an aggregation of species over a range of weight-classes with weight increasing from compartment 0 to compartment n. Fig. 1A depicts the flow of energy and Fig. 1B the corresponding flow of nutrients. Such a distinction between the two types of flow is made in order to emphasize that in most cases recycling is more important for nutrients than for energy, It is striking to realize that most feedback cycles include the detrital pool, which is far too often neglected by ecologists. The feedback structure will be dicussed in more detail later. First I should like to devote my attention to the elucidation of the
84
C. PAHL-WOSTL
A)
energy flow
Kmn-X in - 1 ~ (I -k)m oXo
(1-k)m iX1
(I-klmnXn
B) n u t r i e n t f l o w
E
NUTRIENT POOL
t
Fig. 1. Linear chain of (A) energy and (B) nutrient transfers, m denotes the metabolic rate and k the transfer efficiency. spatiotemporal hierarchical structure implicit in such weight-dependent energy transfers. To this end it is important to remember that the spatiotemporal characteristics of an organism are closely related to its weight. A variety of physiological and ecological properties scale as a power law with weight. The first of these allometric relationships to be observed were those between the weight, W, of an organism and its physiological properties (for a review see Peters, 1983). For instance, the rate of respiration, r, varies as a power function of W: r = f l W -~
(1)
the values for y range from 0.25 to 0.3 for a large variety of organisms of different weight and taxa. The temporal scale of response may be represented by the metabolic rate. It corresponds to the organism's perception of its environment and to its speed of adaptation to environmental changes. In addition, allometric relationships have been detected for ecologically relevant properties such as generation time and home range. Hence, weight determines what is referred to here as the "spatiotemporal niche", charac-
HIERARCHICAL ORGANIZATION OF THE AQUATIC ECOSYSTEM
85
terized by: the range in space and time - represented by the home range. which may be equated with the "spatial radius of activity" and the generation time, which may be equated with the "temporal radius of activity". The use of a spatiotemporal niche as elementary characterization is further supported by the allometric scaling observed for the number of species in a logarithmic weight-class, as well as for the population density (May, 1986; Dickie et al., 1987). Such observations lead to the conclusion that the spatiotemporal characteristics are a major determinant for speciation. (A logarithmic weight-class is defined as a class where the ratio of the weight range of the two neighbouring classes is constant. In general the ratio is chosen equal to two, corresponding to a doubling in weight with each successive class.) Such a spatiotemporal niche may be occupied by an assemblage of species. The distribution of the species' abundance reflects their ability to use and to adapt to certain spatiotemporal patterns in their environment. These patterns may derive from variations of the external factors of the physical environment, or they may be internally generated by species interactions, or, most often, they may represent a combination of both. In loose analogy to the concept of centrifugal organization for plant communities (e.g. Keddy and MacLellan, 1990), a type of log-normal distribution in abundance can be expected. In the following I shall use the aggregated assemblages of species within a spatiotemporal niche as the smallest units of the network, and I shall call them "ecounits". The distribution of taxonomic species across spatiotemporal niches is discussed in more detail in Pahl-Wostl (1993). The scaling with body weight of the properties characterizing the spatiotemporal niche confers a hierarchical spatiotemporal structure upon the food chain. The compartments in our model operate on different scales, and each compartment represents an aggregation of ecounits over a range in time and space. To resolve this aggregated structure, the scaling properties in our model have to be defined. As already mentioned, population density scales allometrically with body-weight. In this respect it is of interest that the distribution of biomass across the weight spectrum in aquatic systems shows a remarkable regularity, in that the biomass contained in a weightclass is approximately the same over a whole range of weight-classes. The abundance (the number of individuals in a class) seems to decrease in proportion to the increase in body-weight (e.g. Sheldon et al., 1972; Sprules and Munawar, 1986). A more general case is introduced into the model, namely that the abundance is proportional to body-weight to the power of - a , thereby treating the constancy of the biomass corresponding to a = 1 as one special case.
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C. P ~HL-WOSTL
The basic assumptions of the model are: (1) A level i of the food chain represents a logarithmic weight class with a m e a n weight of W,. and a total biomass of X i, i = 0, 1. . . . . n. (2) The numerical abundance N~ is proportional to Wi-% Hence, the biomass X i = N,.Wi is related to weight according to: X ~ W ~l - ~
(2)
If a = l then X 0 = X l-- ... = X , . (3) Energy balance: (f
d X i dXi ~ 0
(integrated over a temporal and spatial period that exceeds the temporal and spatial range of activity for the longest living species in the system). (4) The transfer efficiency, k, and the weight ratio of consumer to producer, q (q >> 1), are assumed to be the same for all levels. The change in weight as a function of level is d e t e r m i n e d by q: W,. = qW~_~ ~
W,. = q ' W o
(3)
The metabolic rate, mi, is assumed to scale according to eq. (1): mi = q-Ymi-i
= mi = q-iYmo
(4)
For a given input of energy at the base of the food chain, higher trophic levels must have a reduced metabolic cost to compensate for the decrease in available energy, if eq. (2) holds: mix i = kmi_lXi_
1
H e n c e k may be derived from eqs. (2), (3) and (4): k = q¢l-,,-~)
(5)
where k = transfer efficiency, q = ratio of W, to W,._~, a = allometric exponent of the scaling of a b u n d a n c e (cf. eq. 2), y = allometric exponent of the scaling of respiration (cf. eq. 1). The scaling behaviour of the biomass distribution puts a constraint on the efficiency if the energy balance is to be maintained. An even distribution of biomass among weight-classes is not a prerequisite to achieving energy balance. However, such regular scaling has interesting consequences for the structure of the flows within the network. As we shall see in the following, it can be interpreted in terms of a dynamic self-similar structure. To resolve the aggregated c o m p a r t m e n t s into ecounits we still need the scaling relation of the range of the spatiotemporal niches.
H I E R A R C H I ( ' A L O R G A N I Z A I I O N O F T H E AQ)UA I I C EC()Ssr STEM
~7
The generation time, T,, corresponding to the temporal radius of activity is assumed to scale inversely with the metabolic activity, rni: L
~
q
"v
To
(6)
It is reasonable to assume that the number of ecounits as a function of weight, E , , depends on the width of the temporal niche as quantified by 7,: E i =
q - ' ~ E i i ~ E cz I'V - 7
(7a)
If in addition the d e p e n d e n c e on space is taken into account: Ei = q . . . . ~q-i~.Eo = q - ~ l +~)i~,Ei ) ~ E cz W -~L + ' ~
(7b)
where o- represents the effective spatial dimensionality, o- must not necessarily represent a Euclidean dimension, but may instead be fractal. It will d e p e n d on the structure of the habitat and the way in which the organisms perceive their environment, or may have a value of around 2 for man,,' organisms in a benthic habitat and one more closely to 3 in the pelagial. In the following I will continue with a one-dimensional scaling of the ecounits as a function of the level, and T, is chosen as being representative for the range of an ecounit. The use of such a one-dimensional scaling is motivated by the fact that temporal scales seem to be associated with biological processes, whereas spatial scales seem to be associated with physical processes (e.g. Powell, 1989). Each c o m p a r t m e n t in the food-chain of Fig. 1 may now be resolved into ecounits as shown in Fig. 2. The decline in the n u m b e r of ecounits along the food chain d e p e n d s on q. The scaling in Fig. 2 (q = 17.5 and k = 0.5) was chosen to facilitate the graphical representation of the hierarchical levels of the food chain. It would be much steeper for realistic values of q (around 1000) and k (around 0.1). As indicated by the dashed lines, a modular structure can be discerned, the basic building blocks of which are defined next. In Fig. 2 the structure of the feedback cycles was neglected for the sake of preserving a simple representation of the hierarchy. However, as pointed out before, feedback cycles are of utmost importance for the processes of self-organiziation. A cyclic feedback unit as shown in Fig. 3 is therefore suggested as the basic modular building block. The consumer ecounit of level i interacts directly only with the ecounits in the level immediately below it. Due to the hierarchical structure, however, it also indirectly influences the aggregated assemblages of ecounits of levels 0 to i - 1, aggregated over the range in space and time determined by the consumer in level i. The detrital pool is common to the whole system. The modules are characterized by their scale in time and in space, a corresponding internal organization, and their response to environmental
88
C, PAHL-WOSTL
range in space and time
f
I
flow of energy
Fig. 2. Schematic drawing of the model in Fig. 1 resolved into ecounits. The size of the ecounits refers to their spatiotemporal extension, not to any magnitude of the standing stocks. The dashed lines indicate the hierarchical modular structure. See text for further explanations.
factors. T h e resulting m o d u l a r h i e r a r c h i c a l s t r u c t u r e o f f e e d b a c k loops is d e p i c t e d in Fig. 4 A with level 0 c o m p r i s i n g the p r i m a r y p r o d u c e r s in the f o r e g r o u n d . T h e spatial a n d t e m p o r a l scales i n c r e a s e f r o m level to level as
aggregated ecounits level 0 to i-I
¢on$ kll~let" level i
ill -) I COrn m o i l d e t r i t a l pool
I~ I
I Fig. 3. Basic module of the ecological network. Each module is characterized by a certain spatiotemporal scale. The detrital pool is common to the whole system.
HIERARCHICAL
ORGANIZATION
A
OF THE AQUA.TIC ECOSYSTEM
~9
t
A) t\ \ \
\
~ix'
\
\\
\
x
\
\\ \
S
~
\\
\\
\\
't
\\
\
\ \
\
\ \\ \\ \
\
\
A ~" \
x
\x. \\
\
I
\\\\
\
v
\
log [
B)
log s
r log
t
Fig. 4. (A) Modular hierarchical structure of the ecological network with level 0 comprising the primary producers in the foreground. The spatial and temporal scales increase from level to level as illustrated by the change in scale of the coordinate axes. The change in the size of the arrows represents the decrease in flow intensity at each level. Feedback cycles are indicated by the arrows emerging from the sides of the compartments. (B) Intensity of the corresponding feedback cycles as a function of space and time scales. With increasing module size, intensity decreases and the range extends over larger areas of space and time.
illustrated by the c h a n g e in scale o f the c o o r d i n a t e axes. T h e f e e d b a c k cycles are i n d i c a t e d by the a r r o w s e m e r g i n g f r o m the sides o f t h e c o m p a r t m e n t s . T h e c h a n g e in the size o f the arrows r e p r e s e n t s t h e d e c r e a s e in intensity o f t h e f l o w s with i n c r e a s i n g level. T h e intensity is d e t e r m i n e d by the m e t a b o l i c rate c h a n g i n g with i n c r e a s i n g t r o p h i c level a c c o r d i n g to (4).
90
C. PAHL-"~ OSTL
Hence, the flow intensity is inversely proportional to the corresponding space and time scales, as shown in Fig. 4B. However, it should be noted that in case of equal biomass the total amount of matter, Fi, transferred over a certain feedback loop is the same when averaged over space and time. F~ may be derived from eqs. 2, 4 and 6: Fi =
T i m i X i = m o T o X = const, for c~ = 1.
The origin of the observed regularities in behaviour is still unknown. An explanation is attempted by focussing on the question of how a network's spatiotemporal organization and cooperative behaviour may be effected through the interactions of the c o m p o n e n t parts. SOURCES OF SPATIOTEMPORAL ORGANIZATION To quantify the degree of what is here called spatiotemporal organization, a m e t h o d based on information theory has been developed (PahlWostl, 1990, 1991). The space- a n d / o r time-dependent average mutual information quantifies the reduction in redundancy of functionally equivalent interactions in a network due to a segregation of activity along the dimensions of space a n d / o r time. Definitions and details of the mathematical expressions are given in the Appendix. The macroscopic spatiotemporal organization of an ecosystem arises from interactions within and from interactions across levels of spatiotemporally similar components. Organization within a level may emerge from resource partitioning as achieved by a segregation of activity of functionally redundant units along the dimensions of time a n d / o r space. In a recent paper (Pahl-Wostl, 1990) the importance of spatiotemporal resource partitioning was emphasized for species coexistence and for the efficiency of nutrient utilization. The network in Fig. 5 may serve as an illustrative example. Two functionally equivalent producers, e.g. two species of algae, both share the same nutrient pool D and are preyed upon by the same consumer K. Fig. 5A depicts the spatially and temporally averaged network structure. However, as shown in Fig. 5B, due to resource partitioning the producers may become separated along a dimension, d, of either time or space. The seasonal succession of algal species may serve as an example of temporal organization. An example of spatial organization is provided by the occurrence of different algal communities as a function of depth in the stratified water column. The second type of spatiotemporal organization across levels may emerge from pulsed consumption, where the processes of production and consumption are segregated in time a n d / o r space as shown in Fig. 5C
HIERARCHICAL O R G A N I Z A T I O N OF THE AQU.kTIC ECOSYSTEM
0[
A)
"-.
B)
d
/ / /
d
consumption
Fig. 5. Network consisting of two producers, P1 and P2, and one consumer K. (A) averaged flow structure; (B) segregation of the two producers along a dimension, d (C) segregation of production and consumption along adimension, d. d may represent the dimension of time or a dimension of space.
(Richardson and Odum, 1981; Pahl-Wostl, 1991). One may think of the daily vertical migration of zooplankton as an example, where both temporal and spatial segregation are present. In the case of pulsed consumption the system switches between two different functional states. Such behaviour contrasts with resource partitioning where the functional structure of the overall system does not change. However, the same functional niche is occupied by different species at different intervals of time or space. I have studied the changes in organization associated with these patterns (Pahl-Wostl, 1991) using reaction-diffusion models to describe the coupled
92
c. PAHL-WOSTL
biological and physical processes. When the transfer rate from producer to consumer crosses a critical threshold the equilibrium state becomes unstable, leading to the onset of oscillatory behaviour in space and time. Hence, the degree of organization depends on system-inherent parameters and geometrical constraints - this endows the system with remarkable self-regulatory properties. In contrast to the chance patterns emerging from physical and chemical processes we have to remember that biological systems have evolved. The scales of their response and the patterns subsequently produced can be expected to be the product of selection by long term evolution. To sum up, the analysis of spatiotemporal organization suggests that certain patterns may be favoured over others due to the associated functional changes and feedback effects expressed in the network of the whole system. One has to distinguish between the interactions within a spatiotemporal level of functionally equivalent species and the interactions between levels, which serve as couplings of the system across scales. These concepts may be applied to the food chain model discussed in the preceding paragraph. It is assumed that resource partitioning and segregation of the processes of production and consumption translate through all hierarchical levels. The corresponding behaviour of the spatiotemporal organization, I,~ (see also Appendix), can be derived as shown in Fig. 6. Its is represented as a function of the interval of resolution along the dimensions of space and time. The contribution of each dimension would depend on the details of the pattern involved. The resolution was successively
5.5-
4.G
£, 3.5
2.5
1.5 o
log
of
resolution
interval
Fig. 6. l,s as determined for the hierarchical network represented in Fig. 4A as a function of the logarithm of the resolution interval.
HIERARCHICAL
ORGANIZATION
OF THE AQUATIC
ECOSYSTEM
Q3
increased choosing always the scale of the next hierarchical level in the model. The spatiotemporal segregation of feedback flows results in a substantial decrease in redundancy and increase in organization. The uniform slope is caused by the assumed self-similar structure of the network. Changes would indicate breaks in self-similaritT. It is a difficult task to separate the influence of the spatial and temporal dimensions. Generally they are interdependent (see also Pahl-Wostl, 1991) and the elucidation of the interrelationships still requires much effort. As mentioned above, one difference resides in the fact that the influence of physical processes may dominate space and that of biological processes may dominate time. Here we touch on another important aspect, namely that of the segregation of exogenous from endogenous sources of organization. As in the case of space and time, we face the problem that both sources of organization are interdependent. The approach presented here provides a theoretical quantitative framework for dealing with such questions. As shown in the Appendix, the various contributions may be quantified by splitting up the average mutual information into terms related to inputs, internal exchanges, exports and dissipations, respectively. A pictorial representation is provided as well to assist those readers unfamiliar with aspects of information theory referred to here. As always when one starts off with simple models, one has to realize that reality, is far more complex. The chosen food-chain model depicts only the extreme case of separate trophic levels with the biomass at each level being centered within a narrow weight range. A more realistic outline is sketched
energy
transfer
dW
I slope
._ ) I
q { I
), size W
Fig. 7. Schematic drawing of the energy transfers in a food web as a function of weight. The lines representing the various trophic levels may show considerable overlap.
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in Fig. 7. The lines represent the levels as a function of the number of transfers from the energy's point of entry into the system. The spatiotemporal organization is characterized by the slope and the weight range, dW, of such a level, and the p r e d a t o r - p r e y ratio, q. These properties may be different in the various levels, and they may change as a function of space a n d / o r of time. Does such variability imply that all the derivations based on this simple model are invalidated? It is argued not to be the case because a start from a simple conceptual framework is important and complexity may be introduced step by step. Seasonal changes, for example, may be interpreted as resonance p h e n o m e n a in terms of the oscillation frequencies of the various feedback cycles. CONCLUDING REMARKS
The spatiotemporal organization as defined in the context of this paper represents a property of the network as a whole. On the one hand, the organizational pattern depends on the spatiotemporal characteristics of the species. On the other hand, the structure of the network imposes constraints on an organism by virtue of its being imbedded in a certain environment. The mutual relationship between the macroscopic level of the network and the level of the components is illustrated in Fig. 8. Discussed in detail, however, the situation is not as simple as presented in this figure. The components are not uniform in their spatiotemporal behaviour and one has to deal with a hierarchy of cascading interactions. Analogously to the slaving principle advocated by Haken (1983) one may think of a hierarchy of instabilities governing system behaviour. This principle applies to dynamic systems close to a bifurcation point where the behaviour is
macroscopic n e t w o r k s t r u c t u r e s p a t i o t e m p o r a l or! a n i z a t i o n
,I
I
I
J
J I, 1
level of the ecounits s p a t i o t e m p o r a ] niches
Fig. 8. Mutual interaction between the macroscopic level of the ecological network (as characterized by the spatiotemporal organization) and the level of the ecounits (as characterized by the spatiotemporal niche).
HIERARCHICAL
ORGANIZATION
Of 7 T H E A Q U . - \ T I ( EC'OS~tSFES, I
~45
dominated by the slowest mode, the so-called order parameter. This allows long-range correlations in space and time, which in our case may be stabilized by the very. fact of the system being hierarchical. At the risk of being repetitive 1 wish to emphasize once more that even when talking about the dominance of order parameters one has to keep in mind that these order parameters arise from the interactions among the components of the systems through processes of self-organization. Along this line of reasoning the dichotomy between top-down and bottom-up control converges to a mutual and inseparable dependence of both factors. Neither a purely reductionist approach nor a merely holistic perspective is sufficient to encompass the intrinsic nature of the system's behaviour. At the organism level it is easily accepted that growth and development constitute cooperative phenomena. Body-weight imposes well-known constraints on the behaviour of the components. Adopting this point of view at the ecosystem level is not as straightforward. However, the striking regularity observed in the behaviour of many properties of the ecological system suggest strongly that this description also holds true in this case. Arguments in favour of autonomous cooperative behaviour are corroborated by a variety of experimental findings. Sprules and Munawar (1986) reported breaks in allometric scaling of the plankton weight spectra as a consequence of perturbations. A slope of - l, corresponding to an equal biomass distribution in steady state and little variability, characterized oligotrophic, undisturbed ecosystems. In these systems we may expect a high degree of endogenous spatiotemporal organization due to the high degree of autonomy of the system. Another example is provided by the p h e n o m e n o n of eutrophication where the system is driven by external inputs and its autonomy is greatly affected. The controlling influence of positive feedback via nutrient recycling is diminished, and a major potential for endogenous organization is reduced. Hence, two properties are suggested as whole system descriptors:
Autonomy: ratio of internally generated to externally imposed organization. In the context of this paper, autonomy is equivalent to the independence of the system from external influences, expressed in terms of input of energy and resources (see also Appendix).
Adaptability: ability of the system to respond to environmental changes without decline in organizational characteristics. It is argued here that an increase in spatiotemporal organization is connected with an increase in autonomy and efficiency. Such a development should also coincide with an increase in food web complexity and
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food chain length. Support for this argument is given by experimental findings summarized recently by Briand and Cohen (1987). They provided statistical evidence for the influence of the environment on food chains by showing that food chains are longer in 3- than in 2-dimensional habitats. The dimensionality of the habitat is positively correlated with the opportunity for spatial organization leading to an increase in the number of spatiotemporal niches and the diversity of the ecounits. In this largely phenomenological approach I have tried to provide a theoretical basis within a framework of empirical facts for what could become a potent tool for exploring and understanding food web and ecosystem patterns. In order to establish a firm base and to be able to substantially challenge current thinking, much additional work is needed, some of which is already in progress. The most interesting developments may include: • Investigation of the relationship between the spatiotemporal organization and the adaptability of a system. • A general definition of the compartments of an ecological ne~'ork based on the concept of the spatiotemporal niche. • Determination of the hierarchical level of an external disturbance in terms of its energy content and spatiotemporal pattern. • Quantification of the spatiotemporal information provided by the environment and the consequences for the internal organization of the system. Such knowledge would be very valuable for management decisions that should be taken in the light of preserving a system's potential for adaptation and self-organization instead of attempting to impose external control strategies. To end with I should like to establish a link to the ecology of terrestrial systems by hypothesizing that regular biomass distributions, similar to those in aquatic ecosystems, would be found in terrestrial systems as well if the biomass were to be grouped into logarithmic classes, not of body-weight, but of metabolic turnover time (cf. Pahl-Wostl, 1993). We have to keep in mind that plants are modular organisms. Because of their long lifetimes (e.g. trees) they show a highly flexible hierarchical internal structure. Steele (1985) pointed out the difference in the characteristics of the spectrum of environmental variability in the aquatic and terrestrial environment. Aquatic systems show red-noise, terrestrial systems predominantly whitenoise spectra. Hence, one would expect major differences in the characteristics of spatiotemporal organization. A comparative analysis of terrestrial and aquatic ecosystems, focusing on these aspects, might provide new insights into the function and organization of ecosystems. The approach presented in this paper offers some, albeit quite controversial and tentative, ideas and a quantitative tool to cope with this task.
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97
ACKaNOWLEDGEMENTS T h i s essay has u n d e r g o n e n u m e r o u s a n d s u b s t a n t i a l revisions since the first draft. A m o n g o t h e r s , D . M . L i v i n g s t o n e , U. G a e d k e , R. U l a n o w i c z , S. C o u s i n s a n d L. S t o n e p r o v i d e d h e l p f u l c o m m e n t s w h i c h a i d e d in the d e v e l o p m e n t of the p r e s e n t v e r s i o n . REFERENCES Bianchi, J.. Jones, C. and Shachak, M., 1989. Positive feedback of consumer population density on resource supply. TREE, 4: 234-238. Briand, F. and Cohen J., 1987. Environmental correlates of food chain length. Science, 238: 956-960. Cousins, S., 1990. Countable ecosystems deriving from a new food web entity. Oikos, 57: 270-275. Dickie, L., Kerr, S. and Boudreau, P., 1987. Size-dependent processes underlying regularities in ecosystem structure. Ecol. Monogr., 57: 233-250. Haken, H., 1983. Synergetics. Springer, Berlin. Keddy, P. and MacLellan, P., 1990. Centrifugal organization in forests. Oikos, 59: 75-84. May, R., 1986. The search for patterns in the balance of nature: advances and retreats. Ecology, 67: 1115-1126. Northcote, T.G., 1988. Fish in the structure and function of freshwater ecosystems: a "top-down" view. Can. J. Fish. Aquat. Sci., 45: 361-379. Odum, E. and Biever, L., 1984. Resource quality, mutualism, and energy partitioning in food chains. Am. Nat., 124: 360-376. O'Neill, R., DeAnge[is, D., Allen, T.F. and Waide, J., 1986. A Hierarchical Perspective of Ecosystems. Princeton University Press, Princeton, NJ. Pahl-Wostl, C., 1990. Temporal organization: a new perspective on the ecological network. Oikos, 58: 293-305. Pahl-Wostl, C., 1991. Information theoretical analysis of functional temporal and spatial organization in flow networks. Math. Comp. Model., 16: 35-52. Pahl-Wostl, C., 1993. Foodwebs and ecological networks across spatial and temporal scales. Oikos, in press. Peters, R., 1983. The Implications of Body Size. Cambridge University Press, Cambridge, UK. Powell, T., 1989. Physical and biological scales of variability in lakes, estuaries, and the coastal ocean, in: J. Roughgarden, R.M. May and S.A. Levin (Editors), Perspectives in Ecological Theory. Princeton University Press, N J, pp. 157-176. Richardson, J. and Odum, H., 1981. in: W.J. Mitsch, R.W. Bosserman and J.M. Klopatek (Editors), Energy and Ecological Modelling. Elsevier, New York, pp. 641-647. Sheldon, R., Prakash, A. and Sutcliffe, W., 1972. The size distribution of particles in the ocean. Limnol. Oceanogr., 17: 327-340. Sprules, W. and Munawar, M., 1986. Plankton size spectra in relation to ecosystem productivity, size, and perturbation. Can. J. Fish. Aquat. Sci., 43: 1789-1794. Steele, J., 1985. A comparison of terrestrial and marine ecosystems. Nature, 313: 355-358. Ulanowicz, R., 1986. Growth and Development: Ecosystems Phenomenology. Springer, New York. Vanni, M. and Findlay, D., 1990. Trophic cascades and phytoplankton community structure. Ecology, 71: 921-937.
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APPENDIX
Definitions of expressions used in the text (for further details see Pahl-Wostl, 1990, 1991 and Ulanowicz, 1986). All expressions are defined in a network of flows of energy and matter between the compartments representing standing stocks.
A. Definitions of some general expressions as used in this paper Growth: increase in total system throughput T. Organization: deviation of the observed network structure from a randomly connected network, where flows are statistically distributed among compartments. The degree of organization may be quantified by the average mutual information I of the ecological network. Decelopment: increase in organization as quantified by L Function: defined within the topology of the ecological network. Functionally equivalent compartments may be aggregated without a change in organization (e.g. P1 and P2 in Fig. 5A). Redundancy: multiplicity of functionally equivalent pathways. High redundancy implies low organization and vice versa. Spatiotemporal organization: reduction in redundancy increase in average mutual information upon resolving the structure of flows along the dimensions of space and time (e.g. Fig. 5B).
B. Definition and calculation of specific network properties T: total system throughput n
n+2
r
s
T= Z E Z Z Zjik' j=0
i=1
k=l
l=1
I: average mutual information in the space and time averaged network I=
Y~ -~-log j=0
i=1
.
.
It: average mutual information in the time resolved, space averaged network
I,--
E £
log
H I E R A R C H I C A L O R G A N I Z A T I O N OF THE AQUAT1C ECOSYSTEM
Its:
Q0
average mutual information in the time and space resolved network
= j = 0 i = l k=lZ=l T
log
Tji . Tj.ktTikt
where: Tii: flow from c o m p a r t m e n t j to i; Tick: flow from j to i during time interval k; ~ikt: flow from j to i during time interval k and space i n t e r a t l. A point denotes summation over the corresponding index. It can be shown that: It,. > I t > I > O. This description refers to an open system where separate compartments are defined to account for the exogenous transfers. C o m p a r t m e n t 0 serves as source for the exogenous inputs into the system, compartment n + 1 serves as sink to accept the exogenous exports, compartment n + 2 receives the dissipative losses (in the case of energy flows). The average mutual information is expressed in "bits" if logarithms to the base 2 are used.
C. Decomposition of the ac'erage mutual information into terms stemming from inputs (Io) , internal exchanges (Ii), exports (I E) and dissipation (I D) I = I o + I I + IE + ID where
io
~_. T°i
( T°iT
= ,=1 7 - l o g
( <,r
,,-- jf:= l i =fl v- log Is =
j=1
T
j=l
T
log
Tj.T.(~ ÷ ~)
Tj.T.(n + 2)
I t and Its may be d e c o m p o s e d accordingly into 1,0 and /,s0... The difference b e t w e e n the various terms gives the contribution of the temporal (e.g. I,o-I o) and spatiotemporal (e.g. I,o-I o) structure of the various flows. To assist those readers unfamiliar with information theory, it is helpful to illustrate the meaning of the terms defined using Venn diagrams as shown in Fig. 9. In this representation, the uncertainty in a particular
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A)
B)
C)
Fig. 9. Venn diagrams representation of concepts derived from information theory. See text for further explanations.
attribute is represented by a circle. Thus, Fig. 9A depicts the respective uncertainties of inputs and outputs in a network. The dotted areas represent the external exchanges. To facilitate the representation the distinction between dissipative flows and exports was neglected. The knowledge of the network configuration provides the mutual information I as represented by the shaded overlap region in Fig. 9B. The amount of overlap is a measure for the specificity of the flows in the network. The black region corresponds to the contribution of the inputs I 0. The smaller the black region the higher is the redundancy associated with the inputs which means that the inputs are distributed over many network compartments. As shown in Fig. 9C, another circle represents the uncertainty associated with the set of all time intervals in the network where the flows are resolved along the dimension of time. The resulting information I t is depicted by the total shaded region. The black region corresponds to the difference I t o - I o, corresponding to the reduction in redundancy of the inputs obtained by resolving their temporal variation.