Ecological Modelling 108 (1998) 3 – 21
Gradients in ecological systems
Felix Müller
Ecology Centre of the Uni6ersity of Kiel, Central Department for Ecosystem Research, Schauenburger Straße 112,
D-24116 Kiel, Germany
Accepted 17 March 1998
Abstract
In this paper, the potentials of a holistic and hierarchical gradient approach to ecosystem analysis and ecosystem
theory are discussed, using some examples from the ecosystem research project in the Bornhöved Lakes Region
(Northern Germany). In the gradient concept, which originates in the thermodynamic non-equilibrium principle,
structural ecosystem properties are comprehended as concentration gradients in space and time. They build up
potentials to carry out mechanical work, chemical reactions, or biological interactions. Ecosystem function is defined
as the general characteristic of the systems’ gradients dynamics. The gradient concept is theoretically discussed as an
integrating tool for the aspects of thermodynamics, self-organization, and hierarchy theory. It helps to avoid
inadequate reductions from holistic data sets to non-representative theoretical variables. Also, it can be used as an
indicator to test theoretical hypotheses which are often based on non-measureable variables, and it may improve the
cooperation between theoreticians and empirical ecologists. The necessary interfaces between this strategy and
important ecosystem theoretical ideas are briefly described in this text. As an illustration, three aspects of the gradient
concept are presented in empirical case studies: spatial, temporal, and functional gradients. In these examples, the
gradient systems of nature-near ecosystems are compared with those of stressed ecosystems. On this basis, the
applicability of the gradient concept in theory and practice is discussed. © 1998 Elsevier Science B.V. All rights
reserved.
Keywords: Ecosystem theory; Thermodynamics; Self-organization; Ecological hierarchies; Spatial gradients; Temporal
gradients; Functional gradients
1. Introduction: gradients in ecological concepts
One of the central problems in current systems
ecology arises from the integration of structural
and functional ecosystem characteristics, which is,
however, a basic necessity for the development of
a holistic ecosystem comprehension. These two
poles, ecosystem structures and functions, usually
have been investigated separately on the basis of
distinct questions, strategies, methods, and scales.
While the structural approach has mostly been
based upon statically oriented, descriptive con-
0304-3800/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved.
PII S0304-3800(98)00015-5
F. Müller / Ecological Modelling 108 (1998) 3–21
4
cepts, such as spatial diversity and heterogeneity
of specific systems’ elements, functional research
has often been connected with the concepts of
flows, balances, movements, and dynamics, trying
to explain and summarize the interactions between the (structural) elements of ecological systems. In making attempts to unify these strategies,
many recent textbooks and papers have focused
on energetic values (Odum, 1983a,b; Jörgensen,
1992; Hall, 1995; Odum, 1995), thermodynamic
concepts (Morowitz, 1968; Brooks and Wiley,
1986; Wicken, 1987; Herendeen, 1989; Jörgensen,
1992; Nielsen, 1992; Schneider and Kay, 1994a,b,
Jörgensen, 1996a,b; Nielsen, submitted; Svirezhev,
submitted), network theories (Finn, 1976; Patten,
1985; Ulanowicz, 1986; Higashi and Burns, 1991;
Patten, 1992, 1995, Patten and Jörgensen, 1996),
and deduced or systems analytical strategies
(Margalef, 1968; Allen and Starr, 1982; Holling,
1986; O’Neill et al., 1986; Salthe, 1993;
Straskraba, 1995; Pahl-Wostl, 1996) to explain the
interactions between structural and functional
ecosystem characteristics.
Adopting these ideas, it often turned out that
they are either too restrictive, reducing the basic
holistic concepts to incomplete, non-representative variable sets1, or too abstract to be directly
coupled with and proofed by measured data. As a
consequence, many of the integrative concepts are
totally dependent on models which in some cases
can hardly be verified or validated. Therefore,
also the testability of the basic theoretical hypotheses may be reduced. For example both, the
state variable exergy (Mejer and Jörgensen, 1979;
Jörgensen, 1992), which is a well-suited theoretical
variable to describe ecosystems as wholes, or the
well-known and often used systems attribute ascendency (Ulanowicz, 1986, 1990), are completely
based upon the results of models. Both attributes
cannot be measured. Thus, in spite of theoretical
evidence, and even in spite of the potentially
satisfying validity of the models used, there will
always be empirical doubts about the significance
of these conceptually convincing variables.
1
E.g. the reduction of ecosystem dynamics to energy flows,
neglecting attributes such as the qualities of nutrient transfers
or community characteristics.
Hence, we need a reliable level of translation
and indication, that enables us to test the theoretical hypotheses on the base of empirical data. The
corresponding indicators might also be helpful to
improve the cooperation between empirical ecologists and theoreticians. In the following text a
rather simple, integrative concept is proposed,
which originates in the fundamental thermodynamic ideas of Schneider and Kay (1994a) and
Jörgensen (1992) on the one hand and in the
potentials of practical ecosystem research on the
other (Bormann and Likens, 1979; Ellenberg et
al., 1986; Likens, 1987; Hörmann et al., 1992;
Ulrich, 1992; Breckling and Müller, 1996; Fränzle
et al., 1996): Structures as well as functions are
understood as systems of interacting gradients
that result in different quantitative states of ecological variables. From this point-of-view it is
possible to observe spatial distributions and biocoenotic, energetic or materialistic disequilibria
(structures) as patterns of concentration gradients.
With the emergence of such gradients, on all
biological scales, appropriate potentials are built
up. The ecological agents achieve the ability to
carry out physical or chemical work when gradients are built up. The responsible gradients can be
defined by the directions, steepnesses, and
amounts of concentration slopes, as vectors of the
potentials’ changes, or by the temporal differences
of potential changes at a certain place. Gradients
thus symbolize the spatial, functional, or temporal
differences of structures or energetic and material
units in ecological systems or subsystems. All
flows and changes consequently are considered as
reactions which build up or degrade structural
gradients. The general form of a transfer equation
for ecosystems J= X/R with ecological flux (J);
ecological gradient (X); resistence (material constant) (R) demonstrates this significance of gradients: They are the driving forces of all ecological
processes, and ecosystem dynamics is completely
dependent on the development of the systems’
gradient patterns. Ecosystem function is therefore
understood as a general characteristic of the dynamics of the ecosystems’ gradients.
This approach will be discussed from different
points-of-view in the following paper: to start
with, different ideas from thermodynamics and
F. Müller / Ecological Modelling 108 (1998) 3–21
5
synergetics are used to derive the gradient principle, and to test the compatibility of gradient
methods with other ecosystem theoretical concepts. In a second part, some types of gradients
are introduced and exemplified, and at the end of
this paper, the gradient approach is discussed
critically. The examples will also be used to show
the anthropogenic impairment of ecological gradients, leading to the question whether the gradient
approach might also be useful in practical environmental management.
1.1. Gradients as thermodynamic objects
The most general hypothesis of the introduced
concept is that gradients are consequences of all
dissipative self-organized procedures. Whenever in
an open system, structures are created they can be
made visible and characterized by the gradients
the system builds up. For example the temperature patterns in Bénard cells (Haken, 1983; Kay,
1983; Schneider and Kay, 1994a; Müller et al.,
1996b) as well as the chemical colour arrangements of the Belouzov – Zhabotinskii reaction
(Müller et al., 1996a) can be described and understood only on the basis of their internal gradient
dynamics. Also if we look at the simple model in
Fig. 1 it becomes clear, that the very general
thermodynamic behaviour is coupled with the dynamics of concentration gradients. In a system
state near thermodynamic equilibrium (boxes in
the middle of Fig. 1), many properties, such as a
low heterogeneity (Kolasa and Picket, 1991), a
small amount of order (Haken, 1983), a high
entropy state (Jörgensen, 1992), and a low degree
of organization (Salthe, 1985) are correlated with
the absence of gradients. In the two corresponding states, explicit gradients are visible, implying a
change of the other mentioned system features, as
well. The most fascinating transitions are interrelated with the changes between the states of the
boxes in the center of Fig. 1 and the structurized
distribution patterns below: Dissipative self-organization is able to build up new structures although there are no directing inputs or
regulations from the outside. The system exhibits
a spontaneous creation of macroscopic order
from microscopic disorder. Thus, dissipative selforganization is a formation of gradients from an
Fig. 1. A simplified scheme on the interrelations between
thermodynamic developments and the emergence of gradients:
The figure shows a simple model of two types of gas molecules
that are seperated into distinct chambers. When these boxes are
opened, diffusive processes occur and the molecules are distributed homogeneously (boxes in the middle). This is a state
near thermodynamic equilibrium. In dissipative systems, self-organized sequences of processes are able to establish a new order
(boxes below) which can easily be characterized by the gradients
that are provided by the distribution patterns of the elements.
environment which displays no gradients at all.
Representing the results of self-organizing processes, gradients in many cases are emergent
properties.
1.2. Gradients as emergent features of
self-organization
Following this concept, it can be stated that in
open systems the imported exergy2, which is grad2
The solar radiation.
F. Müller / Ecological Modelling 108 (1998) 3–21
6
ually degraded3 and transformed into entropy due
to networks of irreversible reactions4, gives cause
to the formation of structural and functional gradients. Such self-organized concentration differences come into existence if the systems are open
in a thermodynamic sense, if the subsystems are
able to interact (e.g. by flows and storages of
energy, matter, or information), if there are internal, mutual, and cooperative control mechanisms,
and if the system moves into a symmetry breaking
organization (Ebeling, 1989).
With an increasing degree of gradient formation the system will move away from thermodynamic equilibrium. It will build up hierarchies of
gradients, including patterns of internal self-control, and thus a certain metastability (resilience) of
the structure will occur as a part of an irreversible, historical development.
The close connection between these synergetic
systems features and gradients has been formulated by Schneider and Kay (1994a) as the nonequilibrium principle which claims the following:
if an open system is moved away from thermodynamic equilibrium by the application of a flow of
exergy, it will build up as much dissipative structure as possible to reduce the effect of the applied
exergy gradient. In other words, living systems are
degrading and utilizing externally applied gradients by the self-organized formation of a hierarchy of nested, internal gradients. This hierarchy
fundamentally enables the system to cope with the
external gradients. Without a system of mutually
adapted, synergetic transfer steps of a whole exergy degradation staircase (Müller, 1996; Kappen
et al., submitted), the applied exergy could not be
utilized at all, as the gradient would be too strong
to be operated. Therefore, a high number of small
gradients which are actively interacting in an interdependently adapted degradation network is
necessary.
How do these ideas influence concepts of
ecosystem development? The non-equilibrium
principle states that the number of structural sub-
3
By the plants’ production and the following consumption
processes in the food webs.
4
By respiration, transpiration, or nutrient loss.
systems that take part in the exergy degradation
will rise with the amount and duration of the
energy imported. Therefore, the flow diversity will
increase as well as the total system’s throughput.
This implies an increasing total entropy production, although the specific entropy production,
which is produced by the single tranfer steps, will
decrease. As a consequence of the growing structurization (amount of gradients), exergy storage is
rising at the same time.
1.3. Gradients as integrati6e ecosystem theoretical
objects
This thermodynamically based principle is compatible with many other approaches of ecosystem
theory. Self-organization can be understood as the
processing of autonomous gradient constructions,
and thus gradients can be interpreted as emergent
properties of ecological systems (Müller et al.,
1996b). Their creation and their continuous extension, which is a general feature of successional
dynamics, enhances the exergy storage of evolving
ecosystems, in biomass as well as in structure and
information (Jörgensen and Mejer, 1979, 1981;
Jörgensen, 1992). These stores effect growing potentials for transfer, and consequently, the exergy
flows in ecological systems are increasing
(Schneider and Kay, 1994a) in parallel with a
growing number and extensity of gradients. As
the flow densities are increasing, the system is also
producing and exporting a growing amount of
entropy (Brooks and Wiley, 1986) due to a higher
energetic demand for the maintenance of the
achieved gradient system (respiration), as well as
unavoidable losses at the interfaces between the
single transfer steps (Kappen et al., submitted). In
evolving biological networks, also systems attributes such as network homogenization, network amplification, network synergism (Patten,
1992), ascendency (Ulanowicz, 1986, 1990; Weber
et al., 1989), power or emergy (Odum, 1983a,
1995) can also be connected with gradient principles if we consider the flows in the foodwebs as
degrading reactions between energetic or nutritional gradients (gradient dissipation according to
Schneider and Kay (1994a)). Some examples in
the second chapter of this paper may be used as
indications for the reliability of these hypotheses.
F. Müller / Ecological Modelling 108 (1998) 3–21
The dynamic aspect of the described interrelations between different ecosystem theoretical approaches in the last chapter comes near to the
basic ideas of succession theories (Odum, 1969;
Breckling, 1993). Thus, ecosystem development
can also be understood as a continuous processing
of interrelated gradients. The dynamics connected
with these processes can be described with the
methods and variables of dynamic concepts, such
as resilience (Holling, 1986), stability (Fränzle,
1978; Grimm et al., 1992), chaos (Markus, 1992),
or catastrophy (Bendoricchio et al., 1994). Consequently, it will be possible to use structural and
functional gradients as indicators for the general
ideas of very different theoretical approaches, in
particular as tools for the description of ecosystems. Some examples will be discussed in section
2, trying to illuminate the question whether gradients really are appropriate tools for ecosystem
theory.
A further question is whether gradients can also
be used to investigate problems of ecological control mechanisms. Therefore, in the following
chapter the interrelationships between gradients,
hierarchies and cybernetics are discussed briefly.
1.4. Gradients as characteristics of ecological
hierarchies
One theoretical approach which can be used to
mediate structural and functional aspects is hierarchy theory (O’Neill et al., 1986; Allen and
Hoekstra, 1992; Allen and Starr, 1982) which is
strictly interrelated with control theory and cybernetics (Straskraba, 1995) on the one hand, and
with thermodynamics and information theory on
the other (Müller, 1992; Müller and Nielsen,
1996). If we convert some central ideas of hierarchy theory to the concept of ecological gradients, the following hypotheses, which will be
discussed on the basis of the following examples,
can be formulated.
“ Gradients are basic structural characteristics of
holons (Allen and Starr, 1982), representing
functionally autonomous entities (which are
built up by inferior gradients) as well as subsystems of superior organizational units.
7
“
Ecosystems thus are organized by an ensemble
of gradients, that are interacting through
highly interrelated processes of gradient construction and gradient dissipation.
“ An ecological hierarchy therefore can be comprehended as a partly ordered set of gradients
which are interrelated by asymmetric
interactions.
“ Within this hierarchy, those gradients which
are coupled with large spatial extents coordinate the small-scaled gradients, functioning as
constraints that limit the degrees of freedom of
the inferior hierarchical levels, which form the
systems’ biotic potentials.
“ Such spatial characteristics are interrelated
with the temporal features of the holons: Constraining gradients comprise of slowly changing
processes (and gradients) while the constrained
holons usually change rapidly.
“ Therefore, the scale of a gradient also determines its functionality as a step in the whole
systems’ control pattern.
The following examples will be used to prove
these hypotheses and to elucidate the thermodynamic and synergetic ideas reported above. The
central questions concerning this point are as
follows:
“ Can gradients in fact be used to integrate aspects of ecosystem structures and functions
into a holistic picture?
“ Does the gradient approach have the potential
to be used as a compatible integration level for
the variety of ecosystem theoretic concepts?
“ Which are potential consequences for ecosystem theory, ecological modelling and environmental management?
2. Some types of ecological gradients
All data of the following examples have been
measured and interpreted as parts of the project
Ecosystem Research in the Bornhöved Lakes District, which focuses on a connected system of
watersheds in Northern Germany (Hörmann et
al., 1992; Müller et al., 1996c). Within these catchments a main research area of about 60 ha has
been investigated through comprehensive time se-
8
F. Müller / Ecological Modelling 108 (1998) 3–21
Fig. 2. The main research area at Lake Belau within the Bornhöved Lakes District. The figure focuses on two spatial grid schemes
which have been investigated intensively (Reiche et al., unpublished). They are situated in a 100 years old beech forest and in a crop
field ecosystem which was used for intensive maize cropping. Further examples in this paper originate in data from the Southern
field ecosystem, in the Northern wet grassland area, and the alder break which is placed at the lake side of the Northern forest zone.
ries analyses and spatial examinations of various
factors on various scales since 1988 (Fig. 2). The
data used in the following descriptions have been
measured by numerous colleagues, and the results
and methods have been documented in many
different theses and articles. A (non-representa-
tive) selection of papers relevant to the presented
gradient data refers to literature from Beyer et al.
(1993), Dibbern (1994), Dilly (1994), Eschenbach
(1995), Hörmann et al. (1992), Irmler (1994),
Kappen et al. (submitted), Kerinnes (1994), Kluge
et al. (1994), Kutsch (1994), Piotrowski (1991),
F. Müller / Ecological Modelling 108 (1998) 3–21
Reiche et al. (unpublished), Schrautzer et al.
(1996), and others.
2.1. Spatial gradients
The spatial structure provides one of the basic
prerequisites for the development and consolidation of the ecological potentials in an ecosystem.
The spatial patterns determine the potential display of community structures, flow rates, and
turnover activities. Thus, many spatial attributes
take high influence in the characteristics of the
environmental constraints envelope (O’Neill et al.,
1989) of an ecosystem, whereby emerging constraints limit the degree of freedom for many
short-term-dominated processes. Spatial structures usually are documented in maps when different classes of attribute values are designed for a
certain area. The differences between the spatial
parameter values of a certain transect within the
mapped area can easily be comprehended as a
gradient. As has been discussed before, these gradients symbolize the concentration profiles of a
certain state variable or system feature within a
certain area. Hierarchy theory hypothesizes that
these gradients can be arranged in a sequence
which distinguishes slowly changing structures
with broad extents on high levels5 from processes,
structures and gradients with high temporal dynamics that are assigned to small spatial
extensions6.
In Fig. 3 some spatial gradients of soil characteristica from the investigated beech forest (Fig. 2)
are delineated. The relative size of the gradients
(vertical axis) has been calculated on the basis of
the quartil ranges of the whole respective data
sets, while the horizontal axis, which symbolizes
the spatial extensions of the parameters, shows
the statistical extent distance from semi-variogram
plots (Reiche et al., unpublished). In the figure,
we can find spatial gradients on three distinct
scales.
5
The basic geological, geomorphological, pedological, or
climatological gradients.
6
Nutrient, hydrological, or microbial gradients.
9
(i) In the central part of the upper half of the
figure, the small gradients between tree trunks and
clearings are sketched. These potentials are
formed as a consequence of the water and nutrient uptake of the tree roots, the exsudations of
the root systems, and the differences of deposited
inputs via stemflow and several throughfall paths.
The small gradients increase and develop, directly
influenced by the growth of the trees. When the
trees fall, the respective gradients will function as
potentials for dissipating processes, forming a
habitat for many different saprophagous organisms. The gradients will be degraded slowly as a
consequence of mineralizing and diffusive organismic activities, while new gradients will be built
up in the neighbourhood due to the influence of a
new generation of the dominating tree species.
(ii) In the upper part of the figure, we can also
find high gradients of the 10 m× 10 m grid
scheme. They have been formed from the multitude of small gradients, such as the concentration
profiles between trunks and clearings, on the one
hand. On the other hand there are enormous edge
effects and—above all—anthropogenic influences
which are effective on this scale. They predominately originate in the influences of the neighbouring field (gradients on the right side) and the
central forest track (gradients on the left side),
which has mainly effected the calcium concentrations and the soil pH in the last 100 years, since it
has been maintained.
(iii) The lower part of the figure demonstrates
broad scale gradients (e.g. influences of the relief
and the soil texture) which cannot be detected in
the 10 m× 10 m grid scheme because they are
relevant and effective on a different scale. Besides
these extremely slowly changing constraints, again
the track as well as the lateral fertilizer inputs are
dominant border conditions of the soil gradient
patterns. For example, it can be deduced that the
fertilizer inputs function as chemical buffers at the
forest
edges,
preserving
relatively
high
saprophagous activities. Consequently, these soils
are storing smaller amounts of organic carbon
(SOC) than those in the middle of the forest.
Here, the inputs from ammonia fertilization and
acid precipitations are effective, reducing the
saprophagous activity which leads to an accumu-
10
F. Müller / Ecological Modelling 108 (1998) 3–21
Fig. 3.
F. Müller / Ecological Modelling 108 (1998) 3–21
lation of phosphorous and SOC. Again the gradients of Ca and protons (H + ) are very interesting.
They are directly dependent on a marling activity,
exactly 100 years ago, which has lead to an extraordinary steep, linear historical gradient of Ca with
a breadth of only 2 m. This gradient has been
degraded enormously, as we can see from the extent
on the 50 m×50 m grid scheme. It has become
effective in a distance of more than 125 m from the
original source. As there are no abiotic lateral
transport processes in this forest, only one procedure can be responsible for this degradation: the
root uptake of Ca by the beeches and the subsequent distribution of the cation by litter fall. Thus,
in fact we can recognize both, the dissipative
function of ecosystems, leveling existing gradients,
and the self-organized construction of small internal gradients. Furthermore, the example shows that
human influences, such as acid precipitation or
fertilization can provide enormous gradients in
nature-near systems, which can become dominant
constraints, thus overburdoning the natural gradient structures by far.
But this is only one example of human influences
on natural gradient patterns. Fig. 4 demonstrates
another form of stressed gradient patterns. While
in the forest above all the indirect effects of human
activities are most important, the landuse practice
on the neighbouring field gives rise to an extreme
degradation of structures and potentials. Taking
into account that 100 years ago the forest and the
field from Fig. 4 have formed one identical ecosystem with an identical history, the directly manmade differences become very clear: The gradients
of Ca as well as those of SOC have completely
dissapeared. For other parameters the same development could be found. Moreover, this result
demonstrates that modern agriculture in fact destroys the biotic and the abiotic heterogeneity of
11
ecosystems, and thus intensive agricultural landuse
also obstructs the potentials of gradient emergence
for long time periods.
2.2. Temporal gradients
The discussion of spatial gradient patterns in the
forest ecosystem has shown that the spatial characteristics are always connected with the temporal
features of a variable or gradient. From the pointof-view of hierarchy theory this means that the
global constraints, which change slowly in noncatastrophic times without rare events, also function as controls for those variables and gradients
which show high dynamics in smaller spaces. Thus
different variables and gradients must be characterized by their temporal features as well as the spatial
properties. Their temporal gradients can be defined
as the differences (slopes) of the values of a state
variable within a defined temporal duration. If
these gradients, which coincide with many wellknown statistical measures from time-series analysis, are small, the system will be highly buffered.
The temporal variance will be predominantly influenced by long-term processes, and the parameter
values will be functioning as constraints for the
development of other systems elements. There will
be a well developed and significant output environ
(Patten, 1992). If the gradients are high, there will
be a small buffer capacity, the constraining influences will be high in quantity and variety, and the
effects on other elements of the whole system will
be relatively small. The input environ will dominate
the output environ.
For a discussion of these hypotheses Figs. 5 and
6 show a hydrological example, consisting of the
dynamics of the water levels in Lake Belau, the
groundwater level at a place near the lake’s shore
(alder wetland, about 15 m distant from the lake),
Fig. 3. Spatial gradients of different soil parameters in the upper soil layer of a beech forest in the main research area of the
Bornhöved Lakes Region. The depicted soil data have been sampled in two different grid schemes on the base of mixed samples
from nine strictly defined soil cores in each case. The upper part of the figure shows the gradients of a 10 m ×10 m grid scheme,
while the lower sketch bases upon a 50 m × 50 m sampling pattern. In the upper part, some data from a 1 m × 1 m grid scheme
(in the space between tree tops and clearings) are included. The gradients, which are displayed as triangles, have been calculated on
the basis of the extents from semi-variogram analyses (horizontal axis) and the relative percentual portions of the ranges between
the first and the third quartil range of the whole data sets (vertical axis). The numbers in the lower figure refer to (1) a marled forest
track; (2) fertilizer dust inputs from neighbouring fields; (3) ammonia inputs from slurry fertilization; (4) atmospheric inputs after
log-distance transports; (5) gradient center around the forest track; (6) gradient center around the forest edge.
12
F. Müller / Ecological Modelling 108 (1998) 3–21
Fig. 4. A comparison of the concentration gradients of (a) calcium and (b) soil organic matter in the upper soil layers of two directly
neighbouring test areas with different landuse schemes but identical long term soil development on the base of spatial data on a
10× 10 m grid scheme. Data from Dibbern (1994), Kerinnes (1994) and Hari (unpublished).
and the groundwater dynamics in the central part
of the lake’s Western watershed (about 400 m
distance from the lake). The long term curve as well
as the hourly data from July 1990 show that the
highest fluctuations take place in the alder wetland.
The hydrology of this ecosystem is influenced by (i)
the lake water table; (ii) the groundwater input
transported from the distant areas of the watershed; (iii) the rapidly reacting regional water dynamics of the peaty soil; (iv) the high transpiration
dynamics of the alders. The small vertical distance
of the groundwater level from the soil surface
enhances the enormous fluctuations. These temporal changes are much smaller in the lake itself,
because this system is buffered by its size, the total
surrounding wetland areas, and the balance between inflow and discharge of the river. This
big-area-control is related to slower reaction times.
And if we finally look at the groundwater in the
catchment area, the whole buffering capacity of the
vegetation, soils and the unsatured zones becomes
noticeable. Nevertheless, the catchment dynamics
represent the constraints for the other processes
described. The gradients in the landscape scale
determine the inputs into the lake (which is also
influenced by the hydrological processes in the
communicating watersheds), and the gradients in
both subsystems are operating as constraints for
the temporal gradient dynamics in the wetlands.
Thus, what we can detect (Fig. 6) is that the
constraining processes do not only operate with
distinct spatial extents but also with very different
temporal characteristics. The examples also show
that temporal gradients, spatial potentials and
F. Müller / Ecological Modelling 108 (1998) 3–21
13
Fig. 5. Water level dynamics of groundwater and lake on different scales.The temporal development of the groundwater table in
different distances from Lake Belau and the dynamics of the lake surface are shown for the periods (i) from 1990 to 1994 in a daily
resolution and (ii) in an hourly resolution for July 1990. Data from Kluge (1993) and Kluge (unpublished).
control mechanisms are in fact correlated in the
way it has been supposed in context with the
hierarchy hypotheses.
In order to remain in the water cycle, the
question for direct human influences on the temporal features of state variables and systems elements will be illustrated by another example from
wetland hydrology. Fig. 7 shows the groundwater
level development of two neighbouring wet grassland ecosystems. One of them is extensively used
by mowing (A: Calthion), while the second system
is intensively grazed by cattle (B: Lolio-Potentillion). These different landuse systems provoke
very distinct communities, soil properties (e.g.
bulk densities or solid volumes), and groundwater
amplitudes. Due to the high degree of soil compaction, in the disturbed Lolio-Potentillion small
scaled processes have a much higher influence on
the temporal gradients than in the more healthy
Calthion. The hydrological buffer capacity has
been reduced enormously, and therefore the temporal gradient characteristics of the two systems
show well recognizeable differences.
2.3. Functional gradients
The functional aspect has been mentioned several times before, because the hierarchical approach implies the idea that the hierarchy of
spatial and temporal gradients provides a basic
structure for the constraints and control mechanisms in the ecosystems. As this idea has been
discussed in other papers (Müller, 1992; Kappen
et al., submitted; Müller, 1996), in the following
paragraph, the focus will be on the comprehension of functional ecosystem units as systems of
gradients, utilizing the example of the carbon
cycle in two adjacent ecosystems (Kappen et al.,
submitted). Fig. 8 shows the well-known form of
the terrestrial carbon cycle in its upper part. Here,
we can understand all arrows as flows and all
boxes as storages. Thus, a transfer to the gradient
principles is very simple: the lower part shows the
carbon pools as gradients, exhibiting potentials
for a degradation of the energy which has been
incorporated into the system with primary
production.
14
F. Müller / Ecological Modelling 108 (1998) 3–21
Fig. 6
F. Müller / Ecological Modelling 108 (1998) 3–21
If we utilize the unfolded sketch in Fig. 8 and
introduce the measured annual values from two
different ecosystems, we obtain the gradient patterns of Fig. 9. The gradients are distributed
within a large quantitative range, reaching from 1
kg C/ha (invertebrate biomass of the maize field)
up to a sum of 422 t C/ha in the SOC of the alder
wetland ecosystem. In all cases the gradients,
which of course could be differentiated in a much
higher degree, have been created by the ecological
process sequences of the systems themselves. They
form a solid basis for the long term existence of
the corresponding ecosystems. Another interesting
aspect results in the fact that the different gradients are operating on different spatio-temporal
scales. For example we find (i) annual fluxes, such
as the respiration terms, the carbon amount in
litter or the production gradients of the different
subsystems, and (ii) long term stores like the
wood of the alder or the soil organic carbon
which in some cases has turnover rates of more
than 1000 years. A third property which has to be
mentioned here, is that the saprophagous degradation pathways of carbon and energy provide a
very high throughflow.
In a comparison of the two ecosystems, some
extreme differences in the gradient patterns become visible, which are direct consequences of the
landuse activities. For example in the maize field
ecosystem we can find anthropospheric flows, resulting in fertilizer inputs and yield outputs. The
management of the field ecosystem, which is successfully orientated to a maximization of net primary production, has effected relatively small
long term storage capacities (gradients of SOC,
wood or litter). Also the community oriented
parameters, such as the biomass of invertebrates,
the number of species, and the carbon flows
through the food web are much smaller in the
intensively used ecosystem. Furthermore, the car-
15
bon (and energy) dissipation which is necessary to
maintain the structural gradients, of the field is
much smaller than in the nature-near ecosystem.
Summarizing, the agricultural landuse also effects
a decrease of functional gradients in ecological
systems.
3. Conclusions
The three case studies have shown that gradients are system’s attributes which can be operated
in empirical studies without difficulties. Taking
spatial, temporal and functional gradients into
account will even enable practical ecologists to
find new (informative and ordering) aspects in the
collected data sets of ecosystem studies. It has
also been shown that gradients can be well used
to describe general (practical and theoretical) features of ecosystems. Furthermore, there is no
doubt that gradients provide the fundamentals of
ecological potentials, and that they therefore function as driving forces for all ecological processes,
integrating structural and functional aspects. This
is another reason why gradients should be accepted as important ecosystem features and why
their applicability should be tested with more
emphasis in future. Especially the fact that the
empirical and methodological aspects as well as
theoretical considerations may identify gradients
as interesting and manageable objects, qualifies
them as integrative indicators and mediating levels for an improved transaction between practical
and theoretical ecology.
3.1. Gradients and ecosystem theory
From the theoretical aspect, many questions
may be left open, but many advantages of the
gradient concept seem to be worth mentioning.
Fig. 6. Some results from a wavelet analysis of the groundwater level time series (A) under the central western catchment area of
Lake Belau (distance from lake about 400 m) and (B) under the alder wetland (distance from lake about 20 m) and the dynamics
of the original data. The wavelet analysis has been carried out by Clemen (1998, this volume) this volume and Li et al. (submitted).
The figures show decompositions of the wavelet signals, representing the relative correspondence between groundwater dynamics
and two specific, dominant scales. After a comprehensive wavelet characterization of the signals, the two temporal scales of 364 days
(in the highly buffered case in the central catchement) and 24 h (in the short term dominated alder wetland) turned out to have most
significant influences on the composition of the two time series. Thus both systems provide extremely different temporal gradients.
16
F. Müller / Ecological Modelling 108 (1998) 3–21
Fig. 7. Temporal ground water characteristics and soil properties of two related ecosystems with different landuse. Data from Trepl
(unpublished) and Schrautzer et al. (1996).
One extraordinary qualification of gradients arises
from the fact that they are direct results of self-organizing processes on the one hand, and suppositions for the further development of ecosystems
on the other. The thermodynamic non-equi-
librium principle demonstrates that the formation
of gradients and the increasing gradient structurization are fundamental units for the understanding of ecosystem dynamics and for the unification
of these ideas with the theories of self-organiza-
F. Müller / Ecological Modelling 108 (1998) 3–21
tion and emergence. Including these concepts into
the general ecosystem comprehension might lead
to an understanding of ecosystems as models of
dissipative, self-organizing structures that consist
of living and non-living components, which are
interacting on different levels of organization by
building up, maintaining and degrading energetic
and materialistic gradients in cylcic process
sequences.
These processes are interrelated within a broad
hierarchy of interactions. We can find gradients
on all relevant scales — from organells to the
whole biosphere—and the degrading flows take
place within a huge continuum of correlated temporal and spatial constants and rates. The integrity of the systems and their long term ability
for further development is strongly influenced by
the adjustment of the gradient hierarchies and
their dynamics. Resuming the respective, hier-
Fig. 8. Unfolding the carbon cycle.
17
archy oriented hypotheses mentioned in section
1.4, we can state the following three points.
“ It is possible to define a hierarchical continuum
of gradients, in which the basic requirements
for the definition of holons are fulfilled. We
can e.g. derive a hierarchy of soil characteristics from the gradient patterns which have been
found in the beech forest (Reiche et al., unpublished). A similar arrangement of structures
and processes has been carried out in reference
to the data of the functional example on the
different carbon cycles in two ecosystems
(Kappen et al., submitted). Both examples
show that gradient systems can be described as
partly ordered sets which are interrelated by
assymetric interactions.
“ Gradients are no constants. They develop in a
typically self-organized way, following the dynamics Holling has proposed (Holling, 1986)
for ecological entities. There is a phase of slow
gradient formation (e.g. the emergence of soil
concentration gradients as a consequence of
tree growth), afterwards the gradients are
maintained in a steady state for long times, and
finally they can be degraded rapidly (e.g. soil
concentration homogenization after tree fall),
opening the stage for a replacement by a new,
maybe better adapted gradient generation.
“ Those gradients which have a large extension,
coupled with a high constancy and resistance,
(e.g. texture, relief in the first case study) define
the degree of freedom of the gradients which
have a restricted extent, and which are changing quicker. As an additional quality, which is
not considered in hierarchy theory, the height
and strength of the gradient has to be mentioned. e.g. the Ca concentrations in the forest
do not only have a high extent but also a high
effect because their absolute values are extraordinary high. Nevertheless the potential consequences of the Ca concentrations are limited by
the system’s environmental envelope, consisting
of the constraints, mentioned above.
Taking into account the above proposed
ecosystem definition and the examples from section 2, we can now also find some answers to the
other general initial questions in section 1.4.
18
F. Müller / Ecological Modelling 108 (1998) 3–21
Fig. 9. Comparison of the gradient patterns in the unfolded carbon budget of two related ecosystems with different landuse
structures (alder forest vs. maize field). The carbon flows and storages are presented as t C/ha/a, the figures have been scaled by the
cubic root of the original data. Abbreviations: GPP, gross primary production, NPP, net primary production, SOC, soil organic
carbon. The measurements have been executed by many colleagues from the Bornhöved-Lakes-Project; they were documented
integratively in Kappen et al. (submitted).
F. Müller / Ecological Modelling 108 (1998) 3–21
“
19
Gradients are good means to integrate structural and functional aspects into a holistic
ecosystem aspect, because both poles are
unified in the dynamics of gradient
development.
“ The gradient approach can be used as a compatible level of translation for the variety of
ecosystem theoretical approaches. Although
this step has not been taken up to now, the
hypotheses of some hardly provable thermodynamic approaches can be transformed into gradient oriented statements rather easily. These
can be used to test the abstract hypotheses.
Some case studies refering to this necessity will
be an interesting next step of concept
development.
“ The general consequence of a further developed gradient concept for ecosystem theory
may be a better compatibility and comparability of different approaches. It may be helpful to
improve the aspired unification of ecosystem
theories into an integrative pattern (Jörgensen,
1992).
Another conclusion arises from the comparisons of differently used ecosystems in reference to
all discussed gradient types. They have shown
that the integrity of ecosystems also can be indicated by their gradient features. In all three cases
extreme changes of gradient structures and dynamics can be observed after human landuse has
become effective. Therefore, the gradient approach may also be helpful to investigate ecological goal functions and to derive environmental
targets from these theoretical considerations.
after internal degradation processes, optimize the
capacity to build up storage gradients, optimize
the entropy exports after the degradation of energetic gradients, and optimize the transformation
of exergy gradients into structural and informational gradients in a system specific way. Summarizing these objectives, landscape management
should enable the ecosystems to realize a long
term creative, self-organized pattern of gradient
dynamics.
3.2. Gradients and en6ironmental management
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Acknowledgements
Many colleagues have indirectly contributed to
this paper by measuring their data as components
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