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Pubblicazione N°9


Pubblicazione N° 9



Atti delle II Giornate di Studio su 'Metodi Numerici, Statistici e Informatici nella Difesa delle Colture Agrarie e delle Foreste: Ricerca e Applicazioni' - Pisa, Scuola Superiore Sant'Anna, Maggio 2002.


1 International Centre of Insect Physiology and Ecology (ICIPE), Nairobi, Kenya
2 Dipartimento di Agrochimica e Agrobiologia, Universitā di Reggio di Calabria,
Gallina di Reggio di Calabria, Italy
3 Memorial University of Newfoundland, St. John's, Newfoundland, Canada
4 Istituto di Scienze dell'Atmosfera e del Clima (ISAC-CNR). Sezione di Roma. Area di Ricerca 'Tor Vergata', Via del Fosso del Cavaliere, 100. I-00133 Roma.


Ecological research on populations, communities and ecosystems is guided, among others, by principles originating from social sciences including economics, informatics, biochemistry and biophysics. In our work, however, we primarily rely on biophysical approaches and use mathematics to formalize a coherent and concise framework for the analysis of structure and function of ecological systems as well as for the design of management schemes. 
Hierarchy refers to discrete levels in organized systems, whereas scale denotes the spatial or temporal dimension of an object or process characterized by both grain and extent. Examples of scale-dependencies in time, space and biomass dimensions are presented. In general, scaling dependencies are not considered in population model development. Population ecologists should also be aware that measurements are taken in observational windows of time, space and biomass, and conclusions are specific to these windows.



La ricerca ecologica su popolazioni, comunitā ed ecosistemi č guidata, tra gli altri, da principi tratti dalle scienze sociali, dall'economia, dall'informatica, dalla biochimica e dalla biofisica. Nel nostro lavoro noi ci basiamo soprattutto su approcci biofisici ed usiamo la matematica per formalizzare una base coerente e concisa per l'analisi della struttura e della funzione dei sistemi ecologici cosė come per il progetto di schemi di gestione. 
La gerarchia si riferisce a livelli discreti in sistemi organizzati, mentre la scala denota la dimensione spaziale o temporale di un oggetto o processo ed č caratterizzata sia dalla risoluzione che dall'estensione. Vengono presentati esempi di dipendenza dalla scala nelle dimensioni di tempo, spazio e biomassa. Generalmente, la dipendenza dalla scala non viene considerata nello sviluppo dei modelli di popolazione. Gli ecologi delle popolazioni debbono anche avere la coscienza che le misure sono prese all'interno di 'finestre' di tempo, spazio e biomassa e che le conclusioni sono specifiche per queste finestre. 


Ecology is a branch of science concerned with the interrelationships of organisms and their environment and relies on principles derived from many different disciplines. In our work, biophysical elements related to matter and energy applied to biological systems are very important (CURRY & FELDMAN, 1987; GUTIERREZ, 1996; JØRGENSEN, 2001). Mathematics provides a coherent and concise framework for the analysis of structure and function of ecological systems as well as for the design of management schemes. Biochemistry deals with the composition, properties and transformation of substances relevant to biological systems. Of interest to us are not the substances but their effects on organism behaviour and physiology including nutritional influences on population development . For example, the demand for energy depends on the ratio of nitrogen supply to demand (WERMELINGER et al., 1991). This is a substantial aspect in resource-based functions used for representing population interactions (GUTIERREZ, 1996). The combination of biophysical and biochemical elements has been referred to as physiological based approach to populations system study (GUTIERREZ & BAUMGÄRTNER, 2002). Informatics deals with the collection, storage, retrieval and dissemination of recorded knowledge and has found wide applications in information dissemination (XIA & BAUMGÄRTNER, 1999), had a profound influence on ecological concepts and methods of ecological investigations. Moreover, it is an important tool for the analyses of biological systems at the cellular level (KITANO, 2002), However, in our work, we are primarily interested in information dissemination technologies (XIA & BAUMGÄRTNER, 1999; BAUMGÄRTNER et al., 2002). 

Human ecology deals with human communities and populations as concerned with the preservation of environmental qualities. Of interest to us is the valuation of ecological system performance in the framework of ecosystem services that are conditions and processes through which natural ecosystems sustain and fulfill human life (DAILY, 1997). Sociology is the systematic study of the development, structure, interaction, and collective behavior of organized groups of human beings. It is the basis for hierarchy theory with a profound influence on ecology (AHL & ALLEN, 1996) and the provision of a framework for our research activities. In a hierarchical organization, discrete levels emerge as a result of the interaction between the observer and the part of the universe observed (ALLAN & HOEKSTRA, 1992). When striving for management schemes of sustainable ecological systems, we agree with GOODLAND (1995) who linked environmental and social sustainability to economic sustainability but considered environmental sustainability as a basis for social sustainability (BAUMGÄRTNER et al., 2002). Environmental sustainability is achieved by holding the scale of the human economic system within the biophysical limits of the ecosystem.

This is a brief overview on hierarchy and scale theory elements with particular reference to population study as well as management with references to the relevant modern literature (e.g. GARDNER et al., 2001). LEVIN (1992) stated that problems of patterns and scale are the central problem in ecology, unifying population biology and ecosystem science, and marrying basic and applied ecology. When presenting examples, we primarily rely on our work because of familiarity with both objectives and explicit as well as implicit assumptions of relevance to methodology selection. The selection of case studies, however, does not imply comparisons and evaluations of population models constructed and implemented by different researchers or research teams. The work focus on populations, but their place within the biological system of individuals, cohorts, communities and ecosystems (Fig. 1) as well as the implications for study and management are recognized. Populations studies are based on methods outlined by GILBERT et al. (1976), CURRY & FELDMAN (1987), SEVERINI et al. (1990, 1996), GUTIERREZ (1996), DI COLA et al. (1998, 1999) and GUTIERREZ & BAUMGÄRTNER (2002), while population control is sought by means of Integrated Pest Management (IPM) systems (FLINT & VAN DEN BOSCH, 1981; KOGAN, 1998; KOGAN et al., 1999, BAUMGÄRTNER et al., 2002).


Brief hierarchy theory review

The hierarchical organization of nature has been referred to as a basic presumption in modern ecology (NAVEH & LIEBERMAN, 1984). Hierarchically organized systems have levels that are distinguished by rates that differ by one or more orders of magnitude (O'NEILL & KING, 1989). Hence, in order to describe adequately a complex system several levels should be addressed simultaneously (AHL & ALLAN, 1996). Levels are a system characteristic that depends on the system's internal structure and complexity. This feature emerges from observation (AHL & ALLAN, 1996), or in ALLAN & HOEKSTRA's (1992) words, results from the interaction between decisions of the observer and the part of the universe observed. Thus, hierarchy theory is also a theory on the observer's role in any formal study of complex systems (ALLAN & STARR, 1982, ULANOWICS, 1997, O'NEILL & KING, 1998, ULANOWICS, 2000) and takes into account the problem of aggregation errors by identifying levels of nested systems within larger systems (SCHNEIDER, 1994). Hence, the passage from a level to a succeeding one is not a simple aggregation of subsystems (AGENO, 1994). Hierarchical organization means that higher level processes can steer and constrain lower level processes, while, at the same time, high level features might emerge from low level dynamics (ANONYM, 2001). At each level of organization phenomena exist that requires new laws and principles, that cannot be predicted just from those at more fundamental levels (WILSON, 1998).

There are two principal ways to characterize a hierarchical system. At any pre-defined levels of organization, elements that bear a certain set of common structural and/or process features may be treated as functional groups (KÖRNER, 1993). Alternatively, levels of organizations are occupied by entities, and those entities are responsible for the characteristics of the level in question (ALLAN & HOEKSTRA, 1992). Accordingly, populations can be considered as entities or functional groups that operate within hierarchically organized biological, spatial, temporal and organizational systems (Fig. 1). 

Fig. 1 The hierarchical organization of ecological and decision-making systems of relevance to population study and integrated management. 

As an aside, several levels of explanation may similarly be required for an abstract system (AHL & ALLEN, 1996). In population ecology, we may seek a deep level of understanding as obtained by thermodynamics principles (CURRY & FELDMAN, 1987; JØRGENSEN, 2001) or statistical relationship may provide an adequate understanding (BAUMGÄRTNER & GUTIERREZ, 1989). 


Hierarchically organized systems can easily be recognized in Integrated Pest Management (IPM) research and implementation work (CONWAY, 1984; KOGAN, 1998; KOGAN et al., 1999; BAUMGÄRTNER et al. 2002; Fig. 2). The organizational structure of the systems, resulting from human activities (BAUMGÄRTNER & SCHNEIDER, 2001), may be represented by fields, farms, communities and regions well as decision making levels are represented by farmers, extensionists, administrators and policy makers (Fig. 2). The subject of this work is population study and management. Nevertheless, IPM should not be restricted to a population but consider communities and ecosystem levels as well (Fig. 2). 

Fig. 2 The hierarchical organization of systems relevant to Integrated Pest Management (IPM) as identified by Conway (1984), Kogan (1999), Kogan et al., 1999) and Baumgärtner et al. (2002). The roman numbers refer to different integration categories.

In the next sections, we disregard IPM scheme implementation and focus on demographic models as the appropriate instrument to study population dynamics at predefined hierarchical levels. In general, a population is pragmatically defined as a collection of individuals of the same species occupying a spatial interval. When seeking a mechanistic representation of population development, the VON FÖRSTER (1959) model and its discretized and stochastic variants (SEVERINI et al., 1990, 1996; DI COLA et al., 1998, 1999), accounting for time varying population age-structures, are often appropriate


with the boundary condition


where N(t,a) = population density, t = time, a = age, m(t,a) = specificmortality rate, l(t,a) = specific fecundity rate.

For poikilothermic organisms, chronological age should be replaced by physiological age, and aging is expressed by a temperature-dependent rate function r(.) 


At the organizational levels of fields and farms, eqs. 1a, 1b have been used to model the dynamics of thrips and ticks (TAMŌ & BAUMGÄRTNER, 1993; TAMŌ et al. 1993; MWAMBI et al., 2000a,b). These models represent the temporal dynamics of arthropod populations and have been developed for strategic and policy purposes (CONWAY, 1984).

In many cases, the consideration of population changes relative to biomass is required. This is done by the modified SINKO-STREIFER (1967) model and its discrete and stochastic variants (GUTIERREZ, 1996; DI COLA et al. 1998, 1999) 

[3a]with boundary conditions


where = rate of development, wo = initial biomass of the newly born individuals (note that in this model the initial weight is the same for all individuals), g = growth rate with reference to the biomass and w = biomass and other variables defined above. In the case of poikilothermic organisms, the rate of development is expressed as a function of temperature GILBERT et al., 1976).

At the organizational level of a field, eqs. 3a,b have been used to model the dynamics of maize crop yield formation (BONATO et al., 1999), of cassava including the crop and an arthropod community (GUTIERREZ et al., 1988), and of other systems (GUTIERREZ & BAUMGÄRTNER, 2002). The models have been developed for research and population management purposes including for demonstrating the impact of classical biological control to policy makers and administrators.

Some limitations

The population models discussed so far operate at the predefined hierarchical level of the field, and little attempt has apparently been made so far to develop multi-level approaches according to hierarchy theory (AHL & ALLAN, 1996). Moreover, the consideration of ecological processes, such as reproduction, in the definition of populations (ALLAN & HOEKSTRA, 1992) might improve the understanding of population dynamics and lead to new management strategies.

Hierarchy theory was adopted by ecologists because of familiarity with phenomena segregated into discrete levels and the belief they could be named a priori, and interactions would occur at specific levels only (O'NEILL & KING, 1998). However, we agree with O'NEILL & KING (1998) who state that in general, levels of explanation must be extracted from data, not pre-imposed. They added that to date, empirical evidence shows that the levels extracted from data do correspond in any simple way to traditional levels of biological organization. For example, the population is assigned to a lower level than the ecosystem (Fig. 1). However, the foraging strategy of large herbivores depends on moving across spatial scales usually identified with ecosystems. If levels in hierarchy are distinguished by rates that differ by one or more orders of magnitude, O'NEILL & KING (1998) can easily cite additional examples of ambiguity. 

Probably, the use of pre-imposed levels is valid and useful for man-made ecological and socio-economic systems such as the levels of fields, farms, social communities and decision-making bodies (BAUMGÄRTNER & SCHNEIDER, 2001). Nevertheless, the identity of at least some levels might require verification by data analysis, while other levels should be obtained from data (O'NEILL & KING, 1998) or might be derived from thermodynamics theory (JØRGENSEN , 2001). WIENS (1989, 2001) for example, expects that there are discontinuities in trajectories from small to large spatial scales (see below). These discontinuities and plateaus (Fig. 3) may reveal hierarchical levels and the boundaries separating them (WIENS, 1989, 2001; KEMP et al. (2001). This provides a link between hierarchy and scale theories.

Fig. 3 Conceptual relationship between scale of observation and ecological variance, including scale grain (a) and scale extent (b) influences. Dashed lines indicate discontinuities in the trajectory from small to large scales. With kind permissions from editors and authors, this figure has been reproduced from Kemp et al. (2001), who adapted it from Wiens (1989).


Brief scale theory review

The spatio-temporal framework for the explanation of natural phenomena can be traced back to the philosopher Kant (HONDRICH, 1995). In population ecology, numbers and biomass have long been treated as state variables in models on spatio-temporal dynamics. Disregarding the spatial dimension, this is exemplified by the aforementioned von FÖRSTER (1959) model and its variants. To our knowledge, the biomass dimension has been added to the time and space dimension in the recent past only (SCHNEIDER, 1994). The widely used SINKO & STREIFER (1967) model and its variants disregard spatial processes but consider changes in population densities in both time and biomass dimensions (DI COLA et al., 1998, 1999). 

BARENBLATT (1996) states that physical laws do not depend on an arbitrarily chosen basic unit of measurement, such as the ones taken here in the time, space and biomass dimension. He draws a simple conclusion from this observation: the functions that express physical laws must possess a certain fundamental property, which in mathematics is called generalized homogeneity or symmetry. 

TURNER & GARDNER (1991) define scale as the spatial or temporal dimension of an object or process characterized by both grain and extent. Grain is the spatial and temporal resolution chosen to analyze a given data set, whereas extent is the size of the study and the total duration over which measurements are made. To the commonly used space and time dimensions SCHNEIDER (1994) adds the biomass dimension. Accordingly, the concept of scale is different from the aforementioned concept of hierarchy; scale is recorded as a quantity and involves (or at least implies) measurements and measurement units (O'NEILL & KING, 1998).
Scale dependent patterns can be defined as a change in some measure of patterns with a change in either the resolution (grain) or the range (extent) of measurement (WIENS, 1989, 2001). For both observation and experimentation, a particular property is said to be scale dependent if the magnitude (or variability) of that property changes with a change in either the grain or the extent of the measurement (SCHNEIDER 1994).
Specifically, a quantity Q may scale with another measured quantity Y (e.g. area, duration, mass) as follows 


Hence         [4b]

The exponent of these power laws can sometimes be obtained by dimensional analysis. In ecology, power laws are typically obtained empirically, by regressing the logarithm of Q against the logarithm of Y. Examples of power law scaling in ecology include body size allometry (Y= mass), species area curves (Y = area), and Taylor's Power Law (TAYLOR, 1961)(Y = mean density) and are presented below.
Scale theory applications

Rather than hierarchical levels, we consider spatial as measurable quantities in the spatial dimension and explore the dynamics of populations in the spatio-temporal dimension. It is often assumed that natural or man-made pest population control occurs at a particular local scale such as a mosquito breeding site (FOUQUE & BAUMGÄRTNER, 1996). This is the case in weed, but not necessarily in arthropod control (FIRBANK, 1993). For example, AEBISCHER (1991) found that many arthropods showed few differences between fields and farms and seemed to respond to large scale factors. Nevertheless, many population models such as the one for cowpea infesting thrips (TAMŌ & BAUMGÄRTNER, 1993; TAMŌ et al. 1993) and cassava infesting mealybugs (GUTIERREZ et al., 1988) have been developed for a micro or local scale. Ticks are moving at a micro-scale, but their occurrence depends on the movement of their mammal hosts and hence, control operations should be undertaken at a larger scale (MWAMBI et al., 2000a, 2000b). Highly mobile tsetse fly populations are operating over large geographical areas and herd- or farm-specific control operations are less efficient than measures undertaken at a meso-scale (SAINI et al., 1999; ODULAJA et al., 2001). The African Bollworm Helicoverpa armigera (HÜBNER) is a pest in fields of vegetables and cotton. Field-specific control systems such as the inundative releases of virus and egg parasitoids are combined with naturally occurring control. However, population movement occurs over waste geographical areas requiring wide scale extents to be considered in population study and management. Pest control methods have been developed for farms (placement of crops in space and time), for communities (autodissemination of pathogens) as well as for regional (or meso) scales (pest prediction and early warning systems). For this purpose, a multi-scale control system has been proposed (SITHANANTHAM et al., 2001). Such a system may also be required for study and management of Plum Moth Cydia funebrana TR. populations (SCIARRETTA et al., 2001) as well as for many other pests. In these examples, reference has been implicitly or explicitly made to scale, but little attempt has been made to consider scale dependencies in model development. 
In population ecology, scale dependent patterns can be identified in the spatial, temporal and, according to Schneider (1994), biomass dimension. Here, we examine some scale dependencies and draw some conclusions regarding the relationship between measurements, population dynamics and management.
Spatial dimension. In a homogeneous habitat, the sample variance d2 can be related to the mean u according to Taylor's power law (TAYLOR, 1961) that can be derived empirically from sample data or analytically from models on population movements (SAWYER, 1989) 

. [5]

A single b describes a consistent relationship between sample variance and sample means over a range of densities, on a spatial scale related to the size of the sample unit (SAWYER, 1989). KENDAL (1995) derived eq. 5 from diffusion-limited aggregation and discovered therein an inherent symmetry 

. [6]

That is, for obtaining the variance d2 of quadrat size q, the variance for a unit size quadrate is scaled up by a facto fb when the quadrate size is increased by the factor f. KENDAL (1995) also observed that the exponent b can be interpreted as a fractal dimension and hence, showed the link between scaling and fractal geometry (see GARDNER, 1998). In conclusion, different sample unit sizes detect different spatial distributions. The grain effect is important in demographic studies and has implications in population management strategies. The effect of extent on the other hand, was studied by SAWYER AND HAYNES (1978). They observed that the parameter a of eq. 5 was extent-dependent, but recognized the need to validate the inherent assumption of density invariant distribution within the strata. KEMP et al. (2001) refer to WIENS (1989, 2001) when discussing the effect of both grain and extent to spatial distributions (Fig. 3). 

Temporal dimension. Scale grain appears as time step length in simulation models, but detailed considerations go beyond the scope of this paper. For example, FIRBANK (1993) cites a study on the temporal coincidence of weed-crop interactions, where a scale of more than 12 hours affects yield predictions. Moreover, spectral analyses the measure of association between measurements taken through time depends on the grain of scale used (SCHNEIDER, 1994). WIENS (1989) describes scale extent influence as represented in Fig. 3 (KEMP et al., 2001).

Mass dimension. Respiration rates are related to body weight m according to


that is, to obtain the respiration rate z of body weight m, the respiration rate for a unit mass z(1) organism is scaled up by a factor mc. The scaling relationship between the breathing rate of animals and their mass reflects the fractality of respiratory organs (BARENBLATT, 1996). In a narrow range of arthropod body mass, biomass consumption rates may be approximately proportional to biomass (GUTIERREZ, 1996). For a wider range, however, scaling functions may be required for both respiration and consumption rates. This is important in the development of resource-based functional and numerical response models for representing population interactions (GUTIERREZ, 1996; GUTIERREZ & BAUMGÄRTNER, 2002).

Spatio-temporal dimensions. In ecological processes there is a connection between temporal and spatial scales (O'NEILL & KING, 1998; SCHNEIDER, 1998). SCHNEIDER et al. (1999) and SCHNEIDER (2001) analysed scaling effects in

. [8]
where a = birth rate, d = death rate and F = net movement rate

They relied on dimensional analyses and studied scaling effects on movement and death processes. For the case of a bivalve under study, they concluded that the changes in density are controlled by movement at low spatio-temporal scales, while mortality becomes the predominant factor at high spatio-temporal scales (Fig. 4). 


According to hierarchy theory, populations as subjects of study and management are placed in a system comprising individual, cohort, community and ecosystem levels (Fig. 1). Moreover, organizational systems provide additional levels for study and management (Fig. 1). When relying on hierarchy theory, population studies and IPM schemes might benefit from approaches in that the levels are simultaneously addressed and receive more attention than until now.

The population can be defined as a collection of individuals belonging to the same species. However, the definition says nothing about the time frame over which populations might exist, and nothing about the spatial coherence of the population on the landscape (ALLAN & HOEKSTRA, 1992). In the aforementioned examples, the appropriate levels were identified according to pragmatic criteria. For example, the populations were studied as collections of individuals inhabiting specified habitats, fields, farms or regions. However, if populations are delimited in space and time by coherent internal processes (AGENO, 1994),

Fig. 4 Biological effects of spatial and temporal scales. Scales where mortality prevails over lateral movement and where lateral movement prevails over mortality for adult stages of the bivalve Macoma liliana, indicated by dashed lines. Solid lines have been obtained by Monte Carlo simulation methods (Schneider et al., 1999; Schneider, 2001). With kind permissions from editors and authors, this figure has been reproduced from Schneider (2001) in the book edited by Gardner et al. (2001).

or in ALLAN & HOEKSTRA's (1992) words, if spatially aggregated and historically defined populations are seen as arising from ecological processes such as reproduction, than spatially aggregated and historically defined populations may arise . The study of so defined populations may provide new insight into the dynamics and open new possibilities for management.
From a theoretical standpoint, in most of the population models under consideration, scale dependencies remain to be investigated. From a practical and theoretical standpoint, population ecologists should be aware that measurements are taken in observational windows of time, space and biomass and that conclusions are specific to these windows.

Scaling theory suggest that hierarchical levels occur at distinct breaks in the scale continuum (Fig. 3), i.e. that levels can be recognized by discontinuities (O'NEILL & KING, 1989; KEMP et al., 2001; WIENS, 1989, 2001). This provides a link between hierarchy and scale theories and might open a way to derive hierarchical levels from field data. If possible, the application of hierarchy theory in ecology would be strengthened. Very likely, the combination of hierarchy theory and scale theory elements will profoundly influence the methodology in population ecology, provide new insights in population systems and result to new management solutions.

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