Ecosystems are often made up of interactions between large numbers of species. They are considered complex systems because the behaviour of the system as a whole is not always obvious from the properties of the individual parts. A complex system can be represented by a network: a set of interconnected objects. In the case of ecological networks, the objects are species and the connections are interactions between species. Work on the structure of complex systems and networks (Newman 2003) has drawn on methods developed in condensed matter physics, such as statistical mechanics, quantum mechanics and field theory (Albert & Barab´asi 2002). Many complex systems are dynamic and exhibit intricate time series (Boccaletti et al. 2006; Barrat et al. 2008). This thesis deals with the structure, dynamics, and robustness of ecological systems (Chapters Two and Three), the spatial dynamics of fluctuations in a social system (Chapter Four), and the analysis of cardiac time series (Chapter Five).
In a network, objects are called nodes or vertices, and the connections between objects are called links or edges. Properties of these abstract systems have been studied by mathematicians since the 1950s in the field of graph theory (Erd¨os & R´enyi 1959; Bollob´as 2001). An example of a real network is a social one: humans are represented by nodes and are connected to acquaintances by links. In 1967, Milgram published the findings of his famous “six degrees of separation” letter forwarding experiment (Miligram 1967). He showed that the average (shortest) number of links between individuals was smaller than previously expected. This became known as the “small-world” effect (Watts & Strogatz 1998).
Advances in the theory of networks have been driven by the availability of data (Newman et al. 2006). Theory is developed to understand a wide range of empirical systems. Examples of biological systems include metabolic networks, protein-protein interaction networks, and neural networks (ibid). Examples of technological systems include the internet, the world wide web, and product supply-chains (ibid). Examples of social systems include friendship networks, disease spread on human-contact networks, and networks of inter-bank loans (ibid).
The structure of empirical networks often change thorough time (Dorogovtsev & Mendes 2002). Work to date has focused on growing networks: where the number of nodes and links increases through time. This is largely due to lack of data on networks that are decreasing in size. Only recently have contracting networks been considered (Saavedra et al. 2008).
Ecological networks comprise many complex interactions between species (Pascual & Dunne 2006). Biodiversity on Earth is decreasing and ecological networks are losing species, largely due to human-induced causes (Millennium Ecosystem Assessment 2005). My work seeks to understand the effects of anthropogenic change on the structure and dynamics of ecological networks. In Chapter Two, I investigate predator adaptation on food-web robustness following species extinction. In Chapter Three, I study changes in parasitoidhost (consumer-resource) interaction frequencies between complex and simple environments. Ecological networks are embedded in spatially-heterogeneous landscapes. In Chapter Four, I assess the role of geography on population fluctuations in an analogous social system: the number of venture capital firms registered in cities in the United States of America (US) between 1981 and 2003.
Time series analysis has been developed to investigate a wide range of natural phenomena (Kantz & Schreiber 2003). Understanding how physiological signals vary through time is of interest to medical practitioners (Richman & Moorman 2000). In particular, the study of electrocardiograms has lead to significant improvements in patient care (Braunwald 1997).
In Chapter Five, I present a technique for quickly quantifying disorder in high frequency event series (published, Staniczenko et al. 2009). The method uses changes in frequency-domain entropy to identify periods of irregular rhythm. I use this method to distinguish two forms of cardiac arrhythmia— atrial fibrillation and atrial flutter—from normal sinus rhythm. Applying the algorithm to patient data provides a rapid way to detect arrhythmia, demonstrating usable response times as low as six seconds (with correct assessment of 85.7% of professional beat-classifications).
Statistical approaches have required around two minutes to detect changes in rhythm (Tateno & Glass 2000; Sarkar et al. 2008). By contrast, the entropy-based method is applicable to short sections of data, enabling quicker response times. Combination of these approaches is desirable in an automatic detector of arrhythmia. Such a detector would be clinically useful in monitoring for relapse of fibrillation in patients and in assessing the efficacy of anti-arrhythmic drugs (Israel 2004).
Theoretical ecology, unlike other natural sciences, has no widely accepted first-principle laws such as gravity, conservation of mass, or inheritance. Different theories and models must be invoked to answer the questions posed in population, community and conservation ecology. Nevertheless, ecological theory has a unifying intention: as Ilkka Hanski (1999) writes, “Mathematical models [. . . ] are constructed in the hope that they will clarify our thinking, reveal unexpected and significant consequences of particular assumptions, and lead to interesting new predictions that could be tested with observational and experimental studies.”
A central goal of ecological research is to understand the mechanisms influencing the persistence of ecosystems. Studies of the complex interactions between species (Darwin 1859; Hutchinson 1957) have played a significant role in the development of ecology as a scientific discipline (Hardy 1924; Elton 1927). The approach of population and community ecology (following Elton 1927; MacArthur 1955) considers individual species as the fundamental unit of study. Interactions between species can be formulated in terms of ecological networks (Montoya et al. 2006). Networks may comprise species and interaction presence-absence data (binary), or contain information on species abundances and interaction strengths (weighted).
Natural ecosystems comprise a range of interactions. But research to date typically distinguishes between three types of network (Ings et al. 2009): (i) predator-prey food-webs; (ii) parasitoid-host webs; and (iii) mutualistic webs. Predator-prey and parasitoid-host webs describe antagonistic relationships between species, while species in the mutualistic web benefit from interacting. The impact of anthropogenic environmental change (e.g., Sala 2000) has motivated studies of the stability and robustness of ecological networks. This is because the communities of species described by ecological networks often provide ecosystem services that are of great practical benefit to humankind (Costanza et al. 1997).
Seminal work by Robert May (1972) used random matrix theory to assess the stability of random assemblages of interacting species to perturbation. He found that increasing interaction complexity led to reduced system stability. This relationship questioned the observation that empirical data repeatedly demonstrated the prevalence of complexity in nature (Polis 1991; Williams & Martinez 2000). One possible explanation for this difference is May’s assumption of random interactions: the structure of empirical networks was subsequently shown to follow non-random distributions (Dunne et al. 2002a).
Structural food-web research has received renewed interest following a series of highly critical reviews in the late 1980s and early 1990s (Paine 1988; Hall & Raffaelli 1993). This may be attributable to the collection of improved empirical food-webs as well as an influx of new analytical methods from other disciplines. Studies of biological, technological and social networks have provided new ideas and new perspectives from which to study ecological networks (Proulx et al. 2005).
Of the different types of ecological network, predator-prey food-webs have received the most attention during the early development of the field (Pimm 1982; Cohen et al. 1993). The current “second-generation” empirical food webs (see Allesina & Pascual 2009 for a collection) have been thoroughly studied with respect to their structural properties and theoretical robustness to secondary extinctions (summarised in Pascual & Dunne 2006). One study (Dunne et al. 2002b) found that robustness increases with connectance (a structural measure of food-web complexity), in direct contrast to the finding of May. Modelling secondary extinctions has informed structural traits that may identify keystone species: typically understood as a species that has a disproportionate effect on its environment relative to its biomass (introduced in Paine 1969, review in Mills et al. 1993). The identification and study of keystone species is important in conservation ecology (ibid).
Robustness studies to date have only considered static food-web structures (but see Kaiser-Bunbury et al. 2010). This is despite the widely held view that there are many possible types of compensatory dynamics in ecosystems that may alter food-web structure (e.g., Brown et al. 2001). Indeed, in Jennifer Dunne’s study (Dunne et al. 2002b) of food-web robustness she writes, “. . . our simple algorithm for generating secondary extinctions is limited, and may overestimate secondary extinctions since species can survive by switching to less preferred prey.”
In Chapter Two, I present a model that introduces structural dynamics into the framework of secondary-extinction robustness analysis (published, Staniczenko et al. 2010a). In the model, trophic links may be rewired following the loss of a predator species from the food web. Due to reduced competition, species loosing a predator become more available to other, biologicallyplausible, predators. I compare the increase in robustness conferred through rewiring in 12 empirical food webs. Using the model, I identify a new theo-retical category of species—overlap species—which promote adaptive robustness. These findings underline the importance of compensatory mechanisms that may buffer ecosystems against environmental change, and highlight the likely role of particular species that are expected to facilitate this buffering.
The introduction of structural dynamics represents a significant advance in the theoretical treatment of food-web robustness. The method may be incorporated into other theoretical frameworks (e.g., population dynamics), extending the realism of community-level models. The identification of overlap species raises important practical questions in conservation biology. Which species in an ecosystem enable adaptation and hence additional robustness? What mechanisms underlie this form of adaptation? And what is the relationship, perhaps phylogenetic, between these species? Thus, in complement to keystone species, whose removal causes large cascading effects, we must ask: which species provide ecosystem stability in the first place? In addition to protecting keystone species, conservationists must preserve the diversity of overlap species in order to maintain functional ecosystems.
Models that aim to describe the structure of food webs typically fall into two broad categories (Stouffer 2010): (i) phenomenological models and (ii) population-level models. Phenomenological models rely on heuristic rules to determine how species select their prey and thus generate food-web structure (Cohen & Newman 1985; Williams & Martinez 2000; Cattin et al. 2004). Population-level models prescribe an ecologically-motivated generative mechanism and resulting interactions produce food-web structure (e.g., Loeuille & Loreau 2005). Mechanistic models based on first principles are generally preferred to phenomenological models due to their inherent predictive, rather than pattern-fitting, nature (Ings et al. 2009).
Recent work by Beckerman, Petchey & Warren (2006) used foraging theory (MacArthur & Pianka 1966; Pulliam 1974; Stephens & Krebs 1986) as an ecological basis to determine some emergent properties (e.g., connectance) of food-web structure. This work was then extended to include species allometries (body-size) to predict interactions observed in empirical predator-prey food webs (Petchey et al. 2008; but see Allesina 2010). However, these mechanistic models are currently limited to size-structured, binary, food-webs.
It is well known that not all species and interactions are equally important (Paine 1980; Benke & Wallace 1997). The prevalence of weak interactions in nature (Berlow et al. 1999) has cast new light on the complexity-stability debate (Polis 1998; McCann 2000). Several key studies (Paine 1992; McCann et al. 1998) suggested that weak links tended to stabilise local community dynamics. The advent of increasingly quantified webs (e.g., M¨uller et al. 1999) has enabled more rigorous testing of the role interaction strength plays in determining food-web stability. It has been shown that the configuration of weak and strong links, not just the presence of weak links, has implications for ecosystem functionality (Bascompte et al. 2005, 2006).
Methods summarising the information contained in quantitative webs (Bersier et al. 2002) has facilitated detailed studies of the effects of habitat modification on species interaction patterns (Klein et al. 2006; Tylianakis et al. 2007; Albrecht et al. 2007). Across these distinct studies, structural metrics describing parasitoid-host webs were observed to change with similar pattern along increasing land-use gradients. However, the mechanisms responsible for these structural changes are unknown. One study of host-parasite interactions suggested that observed topological patterns arise from species abundance distributions (Vazquez et al. 2005, 2007).
In Chapter Three, I show that the feeding preferences of consumer species can actively change in response to habitat modification (in preparation, Staniczenko et al. 2010b). Parasitoid species focused on particular trophic interactions within their existing set. Their distribution of interactions differed significantly from what would be expected if density-dependent reallocation is assumed. I present a model of consumer feeding reallocation that generates quantitative food webs in simplified environments and test the model against empirical data. I show that consumer preference for resource species can alter between environments, resulting in corresponding changes to the structural properties of their community food webs. My findings suggest that in environments where communities are more impacted by habitat modifi- cation, interaction patterns will increasingly depart from density-dependent resource selection.
The active reallocation model is able to generate quantitative interaction frequencies in non-size-structured food webs. This represents a large step forward in modelling realistic consumer feeding behaviour. Since parasitoids are natural enemies of many crop pests (Hawkins 1994), knowledge of altered interaction pattern in modified environments could be exploited to control outbreaks of previously less abundant pests. Understanding the mechanisms underlying species interactions subject to environmental change will help with the planning of habitat restoration and with assessing its efficacy. Active reallocation is a significant and functionally important process that needs to be taken into account when developing forecasts of the effects of humaninduced disturbances on community structure and composition.
I argue that active reallocation is consistent with differences in parasitoid foraging behaviour between forested and unforested habitats (Lalibert´e & Tylianakis 2010). This suggests that foraging behaviour is a strong candidate for the ecological mechanism causing structural differences between quantitative webs. Not unsurprisingly, the environment in which a species is located has direct influence on its foraging behaviour. This explains the variable trophic breadths of parasitoid species observed in empirical data.
The traditional approach to population ecology assumes that individuals in a (species) population share the same environment (Kingland 1985; McIntosh 1985). However, populations are often non-homogenously distributed throughout the spatial landscape (Turner 1989; Wiens 1997). Metapopulation ecology provides an explicit treatment of space within its conceptual framework (Hanski 1999). Hanski (ibid) describes metapopulation studies as “typically assum[ing] an environment consisting of discrete patches of suitable habitat surrounded by uniformly unsuitable habitat.”
Metapopulation ecology is primarily concerned with the density of populations within patches and the emigration and immigration of populations between patches (Levin et al. 1993). Metapopulation theory, along with population ecology in the wider sense, is often restricted by the large scale of the study phenomena required to test model predictions (Hassell et al. 1989). Lack of extensive field studies has hampered progress in refining models dealing with fragmented landscapes. This is despite growing evidence linking species extinction to habitat fragmentation (e.g., Pimm 1998). Although large-scale population data is sparse, field studies have provided some evidence in support of metapopulation dynamics. Notably: population density is significantly affected by patch area and isolation (Turner 1989; Wiens 1997) and migration and immigration (Krebs 1994).
Spatial synchrony refers to coincident changes in the time-varying characteristics of geographically separated populations (see Liebhold et al. 2004 for a review). The concept of population synchrony is particularly relevant to metapopulation systems because synchrony is directly related to the likelihood of global extinction (Heino et al. 1997). Synchrony is typically measured by correlation in abundance and many studies found that synchrony declines as the distance separating populations increases (ibid). Due to dif- ficulty collecting extensive spatiotemporal ecological data, metapopulation and population dynamics studies have focused on spatial correlations of population density in fragmented landscapes. The dynamics of fluctuations in species populations has received little or no attention.
Within metapopulation ecology (and population ecology more generally), fluctuations in population have two important consequences: (i) positive fluctuations can lead to local population outbreaks and (ii) negative fluctuations can lead to local population extinctions. Studies relating heterogeneous spatial landscapes to synchronous population extinctions have been almost exclusively theoretical (Liebhold et al. 2004). A prototypical example is the study by Earn et al. (2000) in which the authors used a simple spatial population model to assess the influence of “conservation corridors” on population synchronicity. In this and similar studies, the aim was to understand the analytical relationship between mathematical parameters governing synchronicity and fluctuations leading to local, and ultimately global, extinctions. A systemic study of the spatial-temporal patterns of fluctuations has, to my knowledge, yet to be studied.
In Chapter Four, I analyse the spatial and temporal pattern of fluctuations in the number of venture capital firms (VCFs) registered in US cities (in preparation, Staniczenko et al. 2010c). Data comprise the number of registered VCFs in 509 cities (VCF populations) sampled yearly over the period 1981 to 2003. I argue that VCF dynamics, in addition to being of interest to the social sciences, has implications for spatial ecology where suitable data are less available for analysis. In the metapopulation-analogous framework of cities (patches) non-homogenously distributed through US states, I show that fluctuations in VCF populations are consistent with spatial contagion. That is, fluctuations in a city are more likely to occur if neighbouring cities demonstrated fluctuations during the preceding year.
To describe the observed VCF fluctuation dynamics, I propose a model that posits three phenomenological features: (i) cities strongly induce self- fluctuations; (ii) the (fluctuation) influence of cities on proximate cities follows an exponentially-decaying function; and (iii) the influence of proximate cities on the fluctuation behaviour of a city is cumulative. The model provides a good fit to the empirical data compared to two null models. One null model assumes fluctuations are independent of city identity and geographical location; the other null model incorporates the empirical observation that some cities experience greater numbers of fluctuations than others.
Although the study in Chapter Four involves populations of VCFs, the findings have direct relevance to problems in spatial ecology. Primarily: are fluctuations in local species abundance spatially contagious? A more thorough investigation of population synchronicity, beyond simple density effects, may lead to more effective methods to control or eradicate invasive species. Furthermore, the simple model of spatial contagion can be used to improve our understanding of the effects of habitat fragmentation—patches of land joined by conservation corridors—on metapopulation persistence.
Ecological theory, in turn, may provide candidate mechanisms for the phenomena observed in the VCF data. Four areas of the population synchrony literature motivated my explanation of VCF fluctuation dynamics: (i) observed patterns in the dispersal of species populations; (ii) impact of habitat quality on population density; (iii) synchronicity of population density with exogenous factors; and (iv) focal species’ interactions with other species populations demonstrating synchrony. Indeed, the social sciences have drawn greatly on ecology: not least in organisational ecology. Insights from ecology and biology have been combined with economics and sociology to understand the conditions under which organisations emerge, grow and die (Hannan & Freeman 1977, 1989).
The theme of this Introduction has been the interplay of theory and data. From this interplay emerge phenomenological and mechanistic descriptions of the natural world. The success of ecology, as with all the natural sciences, relies on the work of both experimentalists and theoreticians. As we experience rapid environmental change, the questions asked in ecology are becoming in-creasingly important. The methods used to find answers and solutions may improve, but the fundamental goal remains the same: as Athelstan Spilhaus wrote, “[Ecological] models as they develop will not only provide understanding, but also when we build a highway, dam, city or pipeline—predict the consequences!”
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