On the application of stability concepts in ecology

On the application of stability concepts in ecology

Ecological Modelling, 63 (1992) 143-161 Elsevier Science Publishers B.V., Amsterdam 143 On the application of stability concepts in ecology Volker G...

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Ecological Modelling, 63 (1992) 143-161 Elsevier Science Publishers B.V., Amsterdam

143

On the application of stability concepts in ecology Volker Grimm, Eric Schmidt and Christian Wissel Philipps-Universitiit Marburg, Fachbereich Physik und Biologie, Renthof 6, D-3550 Marburg, Germany

ABSTRACT Grimm, V., Schmidt, E. and Wissei, C., 1992. On the application of stability concepts in •ecology. Ecol. Modelling, 63:143-161. Using the example of stability properties, this paper demonstrates the problem of defining and characterizing 'emergent properties' in ecology. The debate about stability in ecological theory is marked by a frightful confusion of terms and concepts. Judgements about stability properties are often far too general. In fact, stability concepts can only be applied in clearly defined ecological situations. The features of an ecological situation determine the domain of validity of statements about stability. We have compiled these features into an "ecological checklist" aimed at making statements on stability complete and more useful as a result. The checklist is also a tool for identifying gaps in previous ecological research on stability. A model is provided to demonstrate that approaches other than local stability analysis (here the "Linear Response Theory") can help close these gaps.

INTRODUCTION W h a t is t h e appropriate level in trying to u n d e r s t a n d ecological processes? Should o n e c o n c e n t r a t e on individuals, populations, communities, or ecosystems? This question would be pointless, if t h e h i e r a r c h y o f ecological levels was only the result o f the way ecological p h e n o m e n a are perceived by m a n (cf. Miiller, 1992). T h e point that turns the h i e r a r c h y f r o m an arbitrary into a logically compelling o n e is the existence o f properties that are specific to certain levels a n d c a n n o t be observed o n others. " T h e whole is m o r e t h a n the s u m o f the parts" most pregnantly describes the belief in the existence o f these " e m e r g e n t properties". Ecosystem theory has b e e n strongly influenced by this point o f view, but in

Correspondence to: V. Grimm, Philipps-Universit~it Marburg, Faehbereich Physik und Biologie, Renthof 6, D-3550 Marburg, Germany. 0304-3800/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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recent research it really was more a creed than a testable scientific hypothesis. One of the main problems ecosystem theory is facing is the question of how to define and characterize "emergent properties". This paper deals with a special group of "emergent properties": the stability properties. This field of research is a good example of the problem just mentioned, as we will see later on. Stability properties in fact are emergent properties. The persistence of a population, for example, cannot be observed at the level of individuals. The constancy of the number of species in a community cannot be studied by examing a single population. The resistance of energy flows or nutrient cycling towards disturbance has to be investigated at the ecosystem level. In ecology, stability properties are not only of theoretical interest. Facing the dramatic decline in global biodiversity, the question of stability must be given high practical relevance. But despite this relevance, there is confusion in this field that is nearly Babylonian: the number of stability definitions to be found in the literature is limited only by the time spent on reading it. The probability of two authors naming one property with the same term is very low. But, unfortunately, the confusion is far from being just a problem of definitions. Many statements about stability have a simplistic or vague character and therefore are of little use. This unsatisfactory situation arose because of the enormous variety of ecological situations. Stability statements always relate to a specific ecological situation and there are a number of features that must be used to characterize this situation. These characteristics have an essential influence on the resulting stability statement and thus are an important part of the statement itself. This paper pursues two aims. First, we want to point out the importance of influential characteristics for stability statements. Second, we want to present a tool - - the "ecological checklist" - - which helps provide precise stability statements and analyze imprecise ones. STABILITY DEFINITIONS The term "stability" has no practical meaning in ecology. Rather, it is a generic term for a whole series of properties of ecological systems. We denote a property as a "stability property" if it can be assigned to one of the following blocks of properties: (1) staying essentially unchanged ( = constancy); (2) returning to the referential state (or dynamics) after a temporal external influence (disturbance) has been applied ( = resilience); (3) persisting through time ( = persistence). A "stability concept" denotes a stability property, a "stability measure" is a related measure used to quantify the stability property in question. It is possible to have several stability measures for one property. It is pointless singling out one property

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TABLE 1 List of stability concepts and measures found in the literature Adjustment st. Amplitude Anthropogenetie st. Asymptotic st. Attractor block Basin of attraction Biomass st. Boundedness C-, t- and o-stability Conservatism Constancy Cyclical st. Domain of attraction Elasticity

Endurance Fragility Global st. Hysteresis Inertia lnvasability Labilit~it Lagrange st. Lyapunov st. Malleability Maturity Natural st. Neighborhood st. Perceived st.

Persistence Qualititative st. Recurrence Relative st. Resilience Resistance Robustness Sensitivity Structural st. Terminally st. Total st. Trajectory st. Variability Vulnerability

to call "stability", because the term covers a whole range of different properties. Much of the confusion in the discussion about ecological stability could be avoided if people stopped talking about " t h e " stability. There is not one stability, there are only many different aspects of stability. This is the reason why the question of stability can be divided into several subquestions (el. in the case of the stability-complexity discussion, see Sutherland, 1981; Pimm, 1984). Table 1 shows stability concepts and measures found in the ecological literature. Altogether we have found more than 140 different definitions of "stability", stability properties and stability measures. Faced with this impressive "diversity" it should be clear that it is an "inexcusable crime" (Yodzis, 1989) to say "stable" in ecology without defining it previously or unless it is absolutely clear from the context what kind of stability is meant. In Table 2 we have tried to put the essential characteristics common to all the different definitions into a small number of short and precise terms. These definitions show that the basic feature of stability terms is the characterization of dynamic properties. Ecology does not appear in these definitions - - except for the case of persistence. AN ECOLOGICAL CHECKLIST Looking at Table 2 we see that the concept of stability consists of only a few clearly definable aspects. This fact rais~,s the question of why there is such confusion in this field of ecology, The answer is that dynamic stability concepts such as resistance or resilience are "inherited" (Holling, 1973)

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TABLE 2 The most important stability concepts, stability properties and measures in ecology Stability concept Constancy Resistance

Resilience

Persistence

Definition of the related stability property Staying essentially unchanged Staying essentially unchanged despite the presence of potentially disturbing external influences (disturbances) Returning to the referential state (or dynamics) after a temporal external influence (disturbance) has been applied Persistence through time of populations (or other ecological units based on populations)

Related measures Standard deviation, annual variability Sensitivity, buffer capacity

Return time, size of the domain of attraction

Mean time to extinction, value of a lower nonzero limit of the state variable

from physics and mathematics. Originally, they were created in order to characterize relatively simple dynamic systems, for example a ball moving over mountains and through valleys. But ecological systems are far from being simple dynamic systems. It is important to be aware of the fact that the application of stability concepts in ecology requires an abstraction, which is several orders of magnitude larger than, for example, that in physics (Slobodkin, 1981). It is impossible to talk about a stability property of " t h e " forest. One has to name the variable in question, the time horizon and the area for which the stability statement is intended to hold. If there is a disturbance, it is nessecary to say which quantity has been disturbed and what the temporal and spatial characteristics of the disturbance look like. All of these characterizations of the specific ecological situation determine the domain of validity for the stability statement. If only one of the characteristics changes, the judgement of stability properties might possibly yield totally different results. The stability discussion in ecology is so confused and unsatisfactory because very often the features of the ecological situation investigated are not specified. The flood of stability definitions can also be explained this way" different authors are often looking at totally different situations when making "their" definitions. To attain clarity in this respect it is necessary that stability statements always consist of two parts: first, the judgement of

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a clearly defined stability property, and second, a complete characterization of the ecological situation. Often it is not easy to keep all those details in mind that are necessary to characterize the ecological situation. The solution we suggest to this problem should be applied in situations where one runs the risk of forgetting essential points because of the complexity of a problem: a checklist. Our "ecological checklist" is a structured compilation of the characteristics of ecological situations. Of course there is no canonical way to compile such a list (compare Orians, 1975; Connell and Slayter, 1977; Gigon, 1983, 1984; Pimm, 1984). But more important than the specific form of our list is the idea of the list: it is necessary to solve the problem of judging stability properties in ecology in a systematic way. The checklist serves the following purpose: only if all points on the list have been checked for a specific stability statement, will the statement be a complete one. Below, we will comment briefly on the six main points of our checklist. All these points have in common the fact that they have also played an important role in other contributions to the workshop "Ecosystem Theory".

1. Level of description Most ecologists are only working on one - - "their" - - level of description, the level of genes, individuals, populations, communities, ecosystems, or on the level of the whole biosphere. This inevitably leads to the fact that they are often trying, more or less consciously, to understand all ecological phenomena by looking at mechanisms at their specific level. There is always a great danger of transferring level specific hypotheses or results to other levels without reflecting on the fact tha't results can change drastically just by moving to anc;ther level of description. For example, it is possible to investigate the conditions of coexistence of two species with a simple two-species model of the Lotka-Volterra type. To transfer these conditions to higher levels of description (e.g., communities or ecosystems) is dubious, to say the least, because the indirect effects can outweigh the direct ones described by Letka-Volterra-type models (Lawlor, 1979; Yodzis, 1988; Patten, 1991).

2. Variable of interest (or state variable) Stabi!~ty statements should always refer to precise and measurable quantiti~.s. It is impossible to talk about the stability properties of an ecological system as a whole, but only about the stability properties of the state variables used to describe the system, e.g. number of species, biomass,

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biomass size distribution (Gaedke, 1992), number of individuals, age distribution, gene frequencies, spatial distribution, etc. It is quite possible that different variables of a system will have quite different stability properties (King and Pimm, 1983; Rahel, 1990).

3. Referential dynamics of the variable of interest The reaction of ecological systems to disturbance is the key topic in the research field of ecological stability. This requires us to define and characterize what is to be regarded as the referential dynamics. The definition and the knowledge of normal systems' behaviour is essential to the question of what has to be judged as a system's reaction towards disturbance. In the literature the term "referential state" is often used instead of "referential dynamics". We prefer the latter, because it emphasizes the possibly very dynamic nature of an undisturbed ecological system. Referential dynamics can mean an equilibrium, more or less regular oscillations, or perhaps even chaotic fluctuations. Furthermore, referential dynamics can display certain tendencies which can easily be misinterpreted as a system's response to the disturbance.

4. Disturbance With reference to the previous section it should now be obvious how to define disturbances. In accordance with a complete systems description, everything not contained in the referential system and its dynamics is to be defined as a disturbance. Delineating a system is a very sensitive point because there is no obvious way to distinguish between normal system dynamics and disturbances. Often, the spatial and temporal scales under investigation decide if something is to be considered as a disturbance or as an inherent part of the system. A forest fSre, for example, can - - locally and on a short time scale - - be considered as a disturbance. Observed over longer times and on greater spatial scales, this "disturbance" can be a crucial factor facilitating the persistence of many species in the system (Sousa, 1984; Pickett and White, 1984; Remmert, 1988). Therefore, it may be convenient to include the fire in the referential system. There are no general rules for the "right" distinction between disturbance and normal system dynamics. This distinction has to be related very closely to the aim and the specific characteristics (spatial and temporal scales, stability property, etc.) of the investigation. Anyhow, this distinction has to be made explicitly and the disturbance has to be characterized in its different aspects: What is the temporal structure of the disturbance? What is the amplitude? Which quantity is disturbed? What is the spatial struc-

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ture of the disturbance? The importance of disturbance characteristics will be demonstrated later in this article. 5. Spatial scales In every ecological investigation, empirical or theoretical, there are certain spatial scales characteristic of that investigation. These are, for example, the size of the research area or the spatial resolution of different quantities (number of individuals, disturbances, number of species, environmental parameters, etc.). The spatial scales underlying an investigation have an eminent effect on the judgement of stability properties. Consider, for example, the constancy of a number of species in a certain area. Just by moving from a low resolution (large area) to a higher one (smaller area) population dynamics can frequently be changed from constant to fluctuating. The question of what spatial scales are the most appropriate for an investigation depends strongly on the situation considered: Which system or species is being considered? Which variable is used for the description? Which disturbance is being considered? What is the problem to be investigated? What stability property is to be judged? All these questions (and others too) determine which spatial scales are "right" or "important" for the investigation (el. Zauke et al., 1992). But in the case of persistence there is a lower limit to the investigation: the question of persistence only makes sense if the research area has a certain minimum size, defined by the prerequisite that all spatial processes necessary for the persistence of the species in question can take place (Connell and Slayter, 1977; Connell and Sousa, 1983). Miiller-Dombois (1987), for example, estimated the minimum size of different types of natural forests that just about guarantees the persistence of these original forests. The minimum size varies betwe~.n 8 and 100 km 2 (see also Miihlenberg et al., 1991). In empirical studies research areas are nearly always much smaller than the minimum size (Kareiva and Anderson, 1988). 6. Time scales In principle, the arguments and problems relating to time scales are equivalent to those for spatial scales. There are always time scales specific to an investigation and the time scales can also have a strong effect on the resulting stability statement. Following Pimm (1984), the minimum time for persistence investigations is determined by the average generation time. For communities, the longest average generation time appearing in the community has to be chosen.

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Stability statements depend on the characteristics of the ecological situation in question. These characteristics have to be delineated explicitly because of their crucial effect on the resulting stability judgement. A lack of specification results in imprecise statements that cannot be used by others and that cannot be compared with other statements. The ecological checklist is a tool for a specification that is as complete as possible. It lists the points that we consider to be the most important in the context of stability statements. With this checklist in mind, it should be easier to make and to interpret stability statements. STABILITY CONCEPTS AND MEASURES The aim of stability research is a critical judgement of the stability properties in question. To serve this purpose it is necessary to quantify these properties or aspects of them. In this context it is useful to ask two questions. The first question is" "What does 'more stable' or 'less stable' mean in the sense of the observed prcperty?". This question stresses the fact that, in ecology, only relative stability statements are possible (J0rgensen and Mejer, 1977). For example, we can only say "the biomass of the forest growing on this soil is more stable with respect to a certain property than the biomass of the forest growing on that soil". There is no natural basis for making absolute stability statements. Nevertheless, there are situations, especially in the field of ecological management, where it is necessary to set absolute limiting values. So a second question to be asked is" "What does 'stable' or 'unstable" mean in this context?". This reminds us of the fact that the absolute judgement of stability properties depends crucially on our own subjective criteria (Gigon 1983, 1984). When do we judge the number of species in a cu,lmunity as unstable in the sense of resilience (see Table 2)? Is it necessary that m after a temporal disturbance - - 1, 5, or even 10% of the species have become extinct? The subjective factor of judgement can also be found in mathematically formulated measures, because in many measures there are limits, breakpoints, or environments that have to be specified in a subjective manner. The question of how to define a certain measure in a given ecological situation depends strongly on the situation itself. This is the reason why the following list of stability measures cannot be complete in principle. We simply want to comment on the most important measures. These measures are not new in ecological theory. What is new is our view that these measures have to be regarded against the background of the ecological checklist.

ON THE APPLICATION OF STABILITY CONCEPTS ~ ECOLOGY

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Measures of constancy and resistance Constancy or its inverse - - variability - - is the only stability concept that, at least in principle, can be determined in a relatively simple way. That is the reason why this concept plays the most important role in empirical ecology. It is convenient to take the standard deviation, s, of the logarithm of the state variable as a measure of this concept. In comparing two different populations, the relation of the numbers of individuals at two succesive points in time serves as a m e a s u r e (nt+l/nt=R; R is the reproduction rate). Wolda (1978) uses the so-called annual variability A V as a measure of variability. Annual variability is defined as the variance of log R (for the relationship between s and A V see WiUiamson, 1984). Quantifying resistance - - or sensitivity as its inverse - - raises the problem of judging two quantities at the same time: the dynamics of the state variable and the disturbance. Because of the difficulties in identifying and quantifying disturbances, resistance commonly cannot be judged in purely descriptive studies, but only in controlled experiments. Another useful tool in quantifying resistance is the buffer capacity (Jcrgensen and Mejer, 1977; Ulrich, 1992). There is one possible source of confusion in the usage of the terms "variability" and "sensitivity". We define them as the inverse of constancy and resistance, respectively. But these terms are also often used for measures of constancy and resistance.

Measures of resilience The stability concept of resilience is concerned with those aspects that arise when we wish to establish if a system "returns" into its original state after a temporal disturbance. First, there is the qualitative aspect, i.e. whether a system returns or not. Second, there is the question of how long the return will take. This aspect is often denoted as elasticity. A third aspect, denoted as the domain of attraction, is related to the question of the maximum strength of a disturbance that still permits a return to the original state. The concept of resilience is mainly used by theoreticians. This is due to the existence of tractable mathematical methods to analyze the resilience of model populations. The local stability analysis is by far the most popular method in this context (see for example, Lewontin, 1969; Wissel, 1989). It allows us to answer the question of return for "sufficiently small" disturbances. The main problem of this method, which in fact is the problem of all local approximations, is the translation of the results to real -,vorld

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situations. Generally this transfer can only be achieved with the help of some additional information about system dynamics. Empirical eco!ogy has to deal with two main problems when attempting to investigate aspects of resilience. On the one hand, there is often a lack of knowledge about the referential system dynamics, and on the other hand there is the problem that empirical data are always of a stochastic nature. These problems raise difficulties in defining the undisturbed state or the referential dynamics of the state variables. Futhermore, they make it hard to decide in which cases a return to the original state has actually taken place.

Measures of persistence Persistence is the only stability concept which is characteristic of ecology. The reason for this lies in the fact that the concept of persistence evolved from the ecological question of species extinction. A measure for persistence is, for example, the mean time to extinction or its logarithm (Nisbet and Gurney, 1982). But except for short-lived species, the time to extinction can hardly be determined empirically. Furthermore, the question of persistence is often raised in order to prevent endangered species from becoming extinct. In this case it is necessary to find indirect measures for persistence (Botkin and Sobel, 1975). Normally these indirect measures are based on the idea that there is a lower nonzero limit which will not be crossed by population size under normal circumstances (i.e. in the absence of disturbances). The higher the limit's value, the lower the risk of the population becoming extinct as a result of a disturbance. AN EXAMPLE: SENSITIVITYOF A PREDATOR-PREY MODEL The example below serves the following purposes: (1) We introduce a new possibility for quantifying sensitivity (or resistance as its inverse) in population models. (2) We want to demonstrate the importance of the different points on our chccklist. (3) We want to show a special advantage of the checklist: the advantage of making one aware of empirical or theoretical gaps, i.e. situations that have not yet been considered by previous research. To start with the third point, there are two facts to be noted when looking at disturbances investigated so far: (1) the temporal structure of disturbances is most often taken only very roughly into consideration (single, periodic, or permanent disturbances); (2) in almost all cases it is the state variable that is disturbed. These two statements indicate a gap which is the starting point for our following research. Our topic of investigation is

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the sensitivity of the state variables towards certain disturbances. The disturbances we are considering do not act upon the state variable, but on some system parameter (indirect disturbances). These disturbances can be seen as changes in the environment of the populations due to human influences, for example. The temporal structure of the disturbances is explicitly taken into account and is of central importance. For our investigation we used a simple predator-prey model of the Lotka-Volterra type. This does not mean that we favour this model as a very "realistic" one. We use the model to demonstrate the relevance of time scales in the case of indirect disturbances in population interactions. As soon as possible we will leave the model-specific level in order to formulate general statements. Therefore, the model is mainly to be understood as an object of demonstration. Our model is a simple predator-prey model of the Lotka-Volterra type: dt = f l ( N l ' N 2 ) f r N ! 1 - - - ~ -

-fN1N 2

dN 2 dt =fz(N~, N~) = -gN2 + wfN,.N2

(1)

where N 1 denotes the number of prey and N 2 the number of predators. Looking at the prey dynamics one recognizes the logistic grovdh as having capacity K and intrinsic rate of increase r, followed by a negative interaction term due to predation (predation rate f ) . In the absence of prey, the predators show an exponential decline at a rate g. The presence of prey leads to a growth in the predator population as expressed by the interaction term. Here, w is a measure of the reproductive yield one predator obtains by eating one prey. The model has an equilibrium at

N~" = - ~ , N~' =

i

(wfK)

(2)

Biologically plausible parameter values have to be positive. Furthermore

g < wfK is required to obtain a positive predator equilibrium. For any combination of biologicaly plausible parameter values, equilibrium (2) shows local stability (in the mathematical sense). Our main question is how the sensitivity of the state variables depends on the temporal structure of the indirect disturbance. Consider, for example, that the capacity K is not a constant, but varies in time owing to environmental influences: K(t) = Ko + d(t), where d(t) is the disturbance. The system will react with a deviation at(t) from the equilibrium:

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Ni(t) = N j * + aj(t) ( j = 1, 2). For the calculation of the relationship between disturbance and the system's response a(t) we used a method well known in physics: the Linear Response Theory (LRT) (see Appendix for a general description). By considering aj(t) and d(t) to be small quantities this method yields a direct relationship between disturbance and system's response in frequency space by making a linear approximation followed by a Fourier transformation:

Aj(,o) = b(o,)Dj(o,) Here, D(to) and Aj(to) are the Fourier transforms of d(t) and a~(t). Vj(to) is called the transfer function. It determines how the frequency components of the disturbance are transferred to the frequency components of the response. The absolute value of the (complex) transfer function creates a relationship between the absolute values of disturbance and response. It can be interpreted as a direct measure of the frequency-dependent sensitivity of the state variable. Similar to the local stability analysis the LRT deals with approximations. Disturbance and response are considered to be "sufficiently small" in a mathematical sense. We can try to overcome this restriction with the reasonable argument that a strong sensitivity towards small disturbances will probably result in a strong sensitivity towards large (real) ones. But this need not be the case in general! To check our results for real-size disturbances we simulated the system explicitly. RESULTS

Before presenting the results, a short technical comment is necessary: calculations can be simplified by scaling the variables in a suitable way. Notice that the scaling procedure fixes the time scales of our investigation (point 5 of the ecological checklist). There are two parameter combinations that totally determine the system's behaviour and the results:

g fl~m,

(wfk - g ) or

r

r

/3 can be interpreted as the degree of monophagous behaviour of the predator (monophagous predators have a high /3 value, polyphagous predators have a small one). or can be interpreted as the relationship between the typical time scales of predator and prey growth (a high or means that the predator population grows significantly faster than the prey population). By using the method described in the Appendix we were able to compute the transfer function of predator and prey populations. The

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Fig. 1. (a) Predators' sensitivitytowards disturbance of f[~ = 100, o- = 100; see equations (1) and (3)]. Disturbance at non-characteristic frequency (o =2.0 [r] (b) yields almost no response of the predators (c), whereas disturbance at characteristic frequency ~o-- 7.071 [r] (d) yields a very strong response of the predator population (e).

absolute value of the transfer function serves as a direct measure for frequency-dependent sensitivity. The statements obtained in this way were tested by numerical simulations of the system's response towards parameter disturbances at special frequencies. H e r e we want to present only the most important results.

Characteristic time scales The first point to mention is that the system's sensitivity towards disturbances is explicitly frequency dependent. Figure 1 shows this for disturbances of parameter f . / 3 and o- have large values - - typical of host-parasitoid systems. Figure l(a) shows the transfer function of the predator population. It peaks very sharply around a characteristic frequency (oc. This characteristic frequency corresponds to a characte~stic time scale on which the predator population is extremely sensitive towards disturbance of f. To demonstrate this there are two simulations of the predators' response, one for a disturbance at the characteristic frequency [Fig. l(d), (e)] and one for a disturbance at the non-characteristic frequency [Fig. l(b), (c)]. Although

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I0.6 "mm~llvlthmjv,(c~)lendlw(w)lI OA v~ 0.2 0 -" 1_ o o.s 1.o I I~1 frequen0yc~lrl I ml.di:tueaancer(O-ro[ro]

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(a) S e n s i t i v i t i e s o f l~rey a n d p r e d a t o r t o d i s t u r b a n c e o f r [/3 = 0.1, ~r = 0.1; s e e equations (1) and (3)]. The predators show a strong response (e, lower curve) towards disturbance at low frequency to = 0.005 [r] (b), whereas they show nearly no response (e, lower curve) towards disturbance at a high frequency to = 1.0 Jr] (d). For the prey population [upper curves in (c) and (e)] the opposite is true. Fig. 2.

the amplitudes of the disturbances are of equal size, the predators' response shows an impressive difference.

Separation of time scales In the case of small/3 and o- values, typical of herbivore-plant systems, another effect can be recognized: for disturbance of parameter r, a separation of characteristic time scales appears, while the predator population reacts most sensitively to disturbances at small frequencies the prey population shows minimum sensitivity at these frequencies. The judgement of sensitivity therefore can essentially depend on which state variable is considered.

Disturbance of different parameters By comparing Fig. l(a) and Fig. 2(a), the importance of the disturbed parameter can be seen. Differences in sensitivity towards different parameter disturbances can be ~xtreme, both i n qualitative and in quantitative terms.

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Generalization of results Up to now the results have explicitly referred to the model under consideration. Still, these results can be generalized. The common forms of the transfer function are mainly determined by the dimension of the model, i.e. the number of state variables. This is due to the fact that, despite the specific model equations, LRT always starts with a linear approximation. In this respect the results above (characteristic time scales, separation of time scales, differences for different disturbance parameters) are of a general nature. Even so, the explicit (quantitative) aspects and results are of course determined by specific model characteristics. CONCLUSIONS One of the central questions of the workshop "Ecosystem Theory" was how the "emergent properties" of ecosystems can be defined and characterized. In this paper we only deal with a special group of emergent properties: the stability properties. Our proposals on how to define and characterize stability properties can be transferred to other emergent properties as well. (1) There has to be a clear distinction between the definition of a property and its characterization (its measure). The definition of a property is valid generally, whereas its characterization (measure) may depend crucially on the special ecological situation considered. (2) As the characterization and judgement of a property can be strongly situation-dependent, it is often not possible to transfer the results and judgements to other situations. If a transfer takes place, it has to be justified in very great detail. (3) An essential aspect of scientific research is communication within the scientific community. Therefore, statements about properties of ecological systems have to include all the information necessary for an unambigous interpretation of the results (see also Jax et al., 1992). Without complete specification of the important characteristics, results are worthless or even harmful. Worthless, if they cannot be interpreted at all, and harmful, if they lead to misinterpretations. With respect to ecological stability the points we wanted to make in this paper are: (1) "The" entity stability does not exist, but only many different aspects of stability. A complete stability judgement of an ecological system requires the investigation of several stability properties, which may possibly show quite different qualitative characteristics.

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(2) Stability statements are very closely related to the specific ecological situation. In this respect it is necessary to include situation characteristics in the statement itself. Without doing so a generalization or a transfer to other situations is dangerous and can easily yield wrong results. (3) In view of points (1) and (2), it is obvious that across-the-board stability statements in ecology make no sense. (4) A complete description of the ecological situation, as required in point (2), can be achieved by listing all characteristics relevant to the stability statement (level of description, variable of interest, disturbance, etc.). The "ecological checklist", introduced in this paper, is a tool for achieving this complete characterization as well as for a systematic way of looking at questions of stability. (5) The model introduced in this paper demonstrates the crucial importance of different aspects of the point 'disturbance' on our checklist. The main result is the frequency-dependence of sensitivity when applying indirect disturbances. This paper may leave the impression that, ultimately, the set of all possible stability statements about ecological systems will be as confusing as the ecological systems themselves. One should not forget, however, that dynamic stability concepts such as resistance and resilience are basically only tools for understanding the most important stability property of ecological systems: persistence. The dynamic concepts have no intrinsic meaning for organisms and what they are doing: trying to produce as many copies as possible of themselves (or their genes). But the persistence of populations is closely related to the fitness of individuals (although this relationship is not yet well understood; see Henle, 1991). A good strategy in any future stability discussion in ecology will be to attempt to relate all particular questions about stability properties to the central property of persistence. Perhaps this will avoid further confusion and enhance the level of the stability discussion in ecology. ACKNOWLEDGEMENT This work was supported by a grant from the Bundesministerium fiir Forschung und Technik (BMFT). APPENDIX: THE METHOD OF THE "LINEAR RESPONSE THEORY" (LRT) The starting point of the method is the model of an ecological system in the form of a set of differential equations: dN d--T -- F(N, P) (A.1)

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where N is the vector of the state variables, F is the vector of the growth functions, and P is the system parameter to be disturbed. For the undisturbed parameter value Po the system rests in a locally stable equilibrium N*: N*: F(N*, P o ) = 0 (A.2) When disturbing the parameter P: P = P o ~ P = P 0 + d(t) (A.3) the system will respond with a deviation from equilibrium: N = N o --* N = N o + a(t ) (A.4) The basic assumption of LRT is a linear relationship between the system's response a(t) and the disturbance d(t) when the disturbance is small: a(t) = kd(t) (k.5) With this assumption the original equations (A.1) can be linearized: da(t) dt = Ca(t) + bd(t) (A.6) where C is the community matrix and b the vector which describes how the growth functions Fj in equation (A.1) depend on the disturbed parameter: with c 0 =

dF~(N*, Po) dNj

b = (b~) r with bj=

dFi(N*, eo) dP

C = (cij)

(A.7)

By a Fourier transformation of (A.6) we change to a description of the system in frequency space: itoA(to) = CA(to) + D(to) (A.8) where A(to) and D(to) are the Fourier transforms of a(t) and d(t), respectively. The solution of (A.8) can be written as: A(to) = V(to)O(to) (A.9) where V(to) is the vector of the transfer function: V(to) = (/toE - C ) - l b (A.IO) E is the identy matrix. A more convenient representation of V(to) can be achieved by the usage of the eigenvalues and eigenvectors of the community matrix C: V(to) = ~ [ (ito f~ - [l~,) ]r (~)

(A.1I)

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v. GRIMMEl"AL

w h e r e I ~ is the tt-th eigenvalue o f t2 and r t " ) the v,-th right eigenvector o f 12. fl~ are the coefficients o f the expansion of b by the right eigenvectors b - - ~". fl~, r ~0 /z

(A.12)

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