1.1 Surprise!

While investigating the effects of bird guano runoff on intertidal ecosystems in southwestern South Africa, a group of scientists observed that two neighbouring islands (4 kilometres apart) had very different benthic communities: one was teeming with lobsters while the other was covered in mussels and whelks. According to the local fishermen, lobsters were present in both locations till the early 1970s, but then mysteriously disappeared from the second island. After a series of horror-inducing experiments, where lobsters were re-introduced to the second island, the scientists realised that the whelks had turned the tables on their erstwhile predators and now preyed on the lobsters (Reference Barkai and McQuaidBarkai & McQuaid, 1988).Footnote ² Another example of surprise comes from a species of butterfly that lays its eggs in a particular host plant (Reference Singer and ParmesanSinger & Parmesan, 2018). An invasive plant outcompeted the butterfly’s native host, but the butterfly adapted to the invader. Around twenty years later, the invader was eradicated, but the butterfly could not switch back to its original host and went locally extinct. A third example comes from interactions between plants and soil microbes, which have been known to reverse, changing from positive to negative feedback (Reference Casper and CastelliCasper & Castelli, 2007; Reference KlironomosKlironomos, 2002; see also Section 2.2). Finally, Reference Benincà, Huisman and HeerklossBenincà et al. (2008) observed unexpected and significant changes in species abundances and community structure in an experiment on a plankton community isolated from the Baltic Sea, even though the experiment was conducted in a controlled laboratory setting with most conditions kept constant.

The frequency of surprises like these in ecology has been documented. Reference Doak, Estes and HalpernDoak and colleagues (2008) conducted a survey and found that surprises are far from rare. They outlined at least sixteen cases of famous surprises just within the subfields of population and community dynamics and reported that 98 per cent of established field ecologists affirmed that they had encountered surprise events akin to the sixteen cases. Moreover, many of the respondents revealed that the majority of surprising results had not been subsequently sent for publication, ‘the implication being that these observations were uninteresting, bothersome, embarrassing, or not sufficiently well chronicled and understood through proper application of the scientific method, and thus were underreported in the scientific literature’ (p. 956, my emphasis).

But what does the existence and frequency of surprises mean for ecological research? In the philosophical literature, a few surprises are viewed as a positive and integral aspect of scientific practice. Scientists learn from surprises, as understanding why they occur leads to scientific progress (Reference MorganMorgan, 2005; Reference ParkeParke, 2014). However, too many surprises are problematic. In the above examples, the scientists had identified patterns in nature (or the laboratory) and formulated expectations based on those patterns. Yet these patterns were ephemeral: they existed for a while, but at some point, they ceased, resulting in surprise. This explains why surprises are problematic: scientists rely on identifying patterns to generate generalisations, on which they base explanations, predictions and interventions. A surprise indicates that the explanation, prediction or intervention has failed or is likely to fail.

Ecological complexity (which includes the notion of causal heterogeneity) explains both the frequency and magnitude of surprises in ecology. Ecological phenomena have numerous, diverse and variable causes, so the behaviour of ecological systems does not always go as expected. This diversity and variability is the reason why the patterns that scientists detect are likely to be ephemeral, resulting in surprise. Nonetheless, as we shall see in the next section, there is a view which states that frequent surprises only occur in disciplines that are immature or whose theories or methods are somehow flawed (Reference Doak, Estes and HalpernDoak et al., 2008; Reference Hitchcock and SoberHitchcock & Sober, 2004).

1.2 The Scientific Status of Ecology

Some ecologists believe that the frequency of surprises in their discipline should be taken as an indication that there are gaps in our knowledge of ecological systems. In some sense, worrying about the quality of a discipline’s research is something that all scientists (ought to) do, as it helps to maintain standards and improve methods in scientific practice (Reference Hitchcock and SoberHitchcock & Sober 2004).Footnote ³ However, there are ecologists who go much further, questioning whether ecology should be considered a science at all, or lamenting that it is at best a ‘soft’ science. Ecologists are often said to suffer from ‘physics envy’, wishing that their theories, methods and results would more closely emulate those in physics (Reference EglerEgler, 1986; Reference KingslandKingsland, 1995 p. 234; Reference McIntoshMcIntosh, 1987; Reference Shrader-Frechette and McCoyShrader-Frechette & McCoy, 1993 p. 34). Another phrase sometimes invoked is that of ‘stamp collecting’, which is the lot of scientific endeavours that are merely descriptive, lacking general theories, predictive power and the ability to be expressed mathematically (Reference JohnsonJohnson, 2007; Reference KingslandKingsland, 1995 p. 200). As historian Sharon Kingsland points out, the introduction of mathematical models into ecology was viewed by ecologists themselves as an important step towards the discipline becoming a real science, and that this trend is ongoing, as ‘ecologists continue to look towards mathematics and the physical sciences for ideas, techniques and models of what science should be’ (1995, p. 234).

Despite these ‘advances’, there is a small but persistent and vocal group of ecologists who continue to worry. Every few years publications appear, often in monographs or the opinion section of major journals, expressing misgivings about the scientific status of ecology or one of its sub-disciplines. Perhaps the most famous of these critiques is Peters’s aptly titled Critique for Ecology (Reference Peters1991), which criticised ecologists for not providing testable hypotheses in the form of precise predictions. Moreover, Peters argued that theory did not play a significant enough role in ecological research as it did little more than provide the conceptual inspiration for a scientific investigation. It seems that many ecologists took this criticism to heart, as ‘across the western world there were professors who removed the book from library shelves to prevent their students from reading it, lest they became demotivated’ (Reference GraceGrace, 2019). A more recent version of this view appears in Reference Marquet, Allen and BrownMarquet et al. (2014), who argue that ecology does not have enough ‘efficient theories’, by which they mean theories that ‘are grounded in first principles, are usually expressed in the language of mathematics, make few assumptions and generate a large number of predictions’ (p. 701). On a similar note, Reference Houlahan, McKinney, Anderson and McGillHoulahan et al. (2017), argue that ecology has ‘abandoned prediction [and] therefore the ability to demonstrate understanding’ (p. 1). Moreover, it is still an ‘immature discipline … [that] must move beyond such qualitative coarse predictions to riskier, more quantitative, precise predictions, sensu Popper’(p. 5).

There are also critiques of sub-disciplines or types of research, which are in some sense more alarming, as they could be used by universities or funding bodies to limit the amounts allocated to those disciplines or methods. For example, Reference Valéry, Fritz and LefeuvreValéry et al. (2013) argued that their inability to find a process or mechanism specific to invasion biology ‘eliminates any justification for the autonomy of invasion biology’ (p. 1145). Reference Courchamp, Dunne and Le MahoCourchamp et al. (2015) state that ‘one of the central objectives and achievements of fundamental ecology is to develop and test general theory in ecology’ (p. 9). Here, fundamental, or ‘pure’ ecology is contrasted to ‘applied’ ecology, which is aimed at solving particular problems and/or intervening on the world. The authors worry that applied ecology has seen an increase in support (economic and otherwise) in recent years, at the expense of fundamental ecology, which should be reversed (Reference Courchamp, Dunne and Le MahoCourchamp et al., 2015).

Though it is primarily ecologists who worry about the scientific status of their field, these ideas are rooted in philosophy of science. As stated above, the ability to generate laws and the ability to make precise and accurate predictions used to be seen as the hallmarks of true scientific disciplines (Reference Hempel and OppenheimHempel & Oppenheim, 1948; Reference SalmonW. C. Salmon, 2006). A discipline that could not provide either, would traditionally be considered at best ‘immature’ and at worst ‘soft’ or ‘unscientific’ (Reference RosenbergRosenberg, 1989; see discussion in Reference WintherWinther, 2011). The more extreme versions of the positions are nowadays viewed as outdated in philosophical circles, yet aspects of them are still deemed important. Many philosophers arguing for a revised notion of laws, do so partly in order to show that sciences like biology are on a par with other sciences. For example, Reference Linquist, Gregory and ElliottLinquist et al. (2016) argue that as ecology has resilient generalisations which ought to count as laws, this ‘should help to establish community ecology as a generality-seeking science as opposed to a science of case studies’ (p. 119).

Thus, there seems to be a general worry that ecology is far from an ideal science. The suggestions for how to improve the quality of ecological research vary: some argue that the answer is to find more or better laws, others argue for more focus on explanations or predictions, others still argue for more integration between sub-disciplines, and so on. My view is different, as I do not believe that there is anything, in principle, wrong with ecological research, merely that ecological research is particularly difficult in certain ways. These difficulties stem from the particular way in which ecological systems are complex.

1.3 Outline

My aims, in this Element, are to show that (i) our current views about complexity do not capture how complexity works in ecological systems (ii) we should reconceptualise complexity to include causal heterogeneity (iii) this reconceptualisation explains some of the important difficulties that ecologists face and (iv) this reconceptualisation can point to some ways of mitigating these difficulties. The argument will proceed as follows. In Section 2, I examine the concept of ‘complexity’ in ecology. I start by providing a brief sketch of the main characteristics associated with complexity and then move to an in-depth account of some of these characteristics, that is, those that affect the study of ecological systems. I then turn to Reference LevinsLevins’s (1966) account of complexity and trade-offs between model desiderata, which I subsequently extend and refine. In Section 3, I examine some of these trade-offs in more detail, showing how causal heterogeneity creates difficulties for generalisations, predictions and interventions. In Section 4, I argue that this explains but does not justify the worry that ecology is not a true or sufficiently mature science. I show that even if we give up on extensive generalisation in ecology, ecologists are capable of making successful predictions and interventions. Rather than being embarrassed by the modesty of ecological generalisations, ecologists and philosophers should recognise the scientific and practical value of ecology’s methodological toolkit. In Section 5, I outline some concluding remarks on generalisation and prediction in science more broadly.

This Element is not just meant for a philosophical audience. I hope that any ecologists looking for an alternative philosophical view of science, that accounts for the peculiarities and idiosyncrasies of their discipline, will find the arguments I present helpful. Moreover, I hope that this philosophical approach can be used by practicing scientists to support the alternative, undervalued research strategies examined in Section 4. Finally, this discussion of ecological complexity could also be helpful for scientists in other disciplines whose systems are also causally heterogeneous, such as Economics or Climate Science.

2.1 Ecological Complexity in Context

Though aspects of complexity have been studied in various disciplines for more than 150 years, interest in complex systems began in earnest in the 1960s and 1970s, and gained momentum in the 1980s (Reference HookerHooker, 2011; Reference Miller and PageMiller & Page, 2009; Reference SimonSimon, 1962; Reference WimsattWimsatt, 1972). The subsequent explosion of research on complexity and its effects on a variety of phenomena, had important and long-lasting implications for scientific practice, as it contributed to the establishment of a framework for anti-reductionist philosophy of science (Reference HookerHooker, 2011; Reference MitchellMitchell, 2009), along with the recognition that the emergence and manifestation of complexity, especially in biological systems, is an worthwhile and fruitful research topic (Reference McShea and BrandonMcShea & Brandon, 2010; Reference MitchellMitchell, 2009; Reference WimsattWimsatt, 1972).

Despite – some might say because of – the level of interest and research in complex systems, a single, unified definition of complexity has yet to be agreed on (Reference HookerHooker, 2011; Reference Ladyman, Lambert and WiesnerLadyman et al. 2013; Reference Miller and PageMiller & Page, 2009). In lieu of a precise or formal definition of complexity, scientists and philosophers usually list some characteristics that tend to appear in complex systems. It is worth noting that there is quite a bit of overlap, as some characteristics appear frequently. One such characteristic is having ‘multiple interacting parts’, which seems to be a basic requirement of complexity. The idea is that a high number of parts and (dynamic) interactions between them increase the likelihood of complex behaviours. Table 1 contains a representative selection of characterisations of complexity, from complex systems science and (philosophy of) biologyFootnote ⁵. As we can see, multiple interacting parts (highlighted in bold), features prominently.

Within the discipline of ecology, the context of complexity has its own history. Here, discussions of complexity were originally related to a question that was considered fundamental, namely, ‘how are ecological systems possible?’ Early influential ecologists, such as Odum, Elton and MacArthur, found the ability of populations, communities and ecosystems to persist, in spite of internal and external disturbances, quite remarkable, hypothesising that this apparent stability was caused by the diversity and connectivity of ecological communities (Reference KingslandKingsland, 2005; Reference McCannMcCann, 2000 Reference Odenbaugh and HookerOdenbaugh, 2011). The general idea is that complex communities are more able to adapt to changes, such as disturbances or perturbations (fires, new competitors/predators, sudden climatic changes), without falling apart. This view was famously disputed by Reference MayRobert May (1973), who used mathematical models to show that we should expect complex communities to be less stable. Subsequent generations of ecologists have refined the concepts of diversity, complexity and stability in order to bolster their favoured side of the debate, while philosophers of ecology have provided their own clarifications and categorisations of the various views (Reference McCannMcCann, 2000; Reference Odenbaugh and HookerOdenbaugh, 2011). A current consensus seems to be that complexity is indeed an inherent feature of healthy and mature ecological systems, even though such systems may be susceptible to particular disturbances (Reference Hooper, Chapin and EwelHooper et al., 2005; Reference Loreau, Naeem and InchaustiLoreau et al., 2001; Reference McCannMcCann, 2000; Reference ParrottParrott, 2010).

The brief outline of the context of ecological complexity highlights two important points for our discussion. First, it is uncontroversial, indeed quite common to consider multiple interacting parts as key features of complex systems, including biological systems. Thus, there is also no difficulty in recognising that it is also a key feature of complex ecological systems. Second, whether or not ecologists agree that complexity leads to stability, they seem to agree that complexity is an inherent feature of (at least healthy) ecosystems. This is an important theme that will appear throughout the Element: the complexity of ecological systems is an inherent feature of the systems themselves. In other words, it is inescapable.

2.2 Key Features of Ecological Complexity

Now that we have a rough idea of how scientists and philosophers characterise complexity, we can take a closer look at the most important characteristics or features that are usually associated with complexity. This is not meant to be an exhaustive list of all the features ever to be associated with complexity, but a summary of the characteristics that are relevant for the context of ecological research, namely, multiple parts, diversity, interaction, emergence, non-linearity and historicity.

Multiple Parts. Complex systems tend to be composed of many parts. Increasing the number of parts allows for more and varied interactions between them, which facilitates complexity. Even though this characteristic is frequently included in characterisations of complexity, it is usually not discussed in great detail (especially outside biology), perhaps because it seems conceptually straightforward. An example of this characteristic ‘in action’ can be seen in Reference McShea and BrandonMcShea and Brandon’s (2010) account of complexity in evolutionary biology. Here, a high or increasing number of parts is taken as a marker of increase in complexity, which in turn is associated with evolutionary progress. The basic idea is that, in the absence of other constraints, evolution creates organisms of increasing complexity. Even if some species are less complex than their ancestors, the claim is that complexity increases over evolutionary time.

Diversity (aka heterogeneity). There are various notions of diversity in ecology, the most famous of these being biodiversity (Reference JustusJustus, 2021; Reference LevinLevin, 2002; Reference Maclaurin and SterelnyMaclaurin & Sterelny, 2008; Reference ParrottParrott, 2010; Reference SantanaSantana, 2014). Biodiversity is a notoriously difficult concept to define and measure (Reference Maclaurin and SterelnyMaclaurin & Sterelny 2008) so much so that some have questioned the usefulness of the concept for ecological research (Reference SantanaSantana 2014, see also discussion in Reference JustusJustus 2021, section 5). I will not delve into the debates surrounding the definition, measurement and value of biodiversity. To avoid entering the territory of these debates, I will follow Reference LevinsLevins (1966) and Reference MatthewsonMatthewson (2011) in focusing on heterogeneity: the diversity between parts of a system or between systems. For example, a population can be heterogeneous because its individuals are diverse in terms of genetic and behavioural traits, a community can be heterogeneous when it is composed of populations of different species, while an ecosystem can be heterogeneous when it is composed of multiple communities. How exactly heterogeneity should be understood and how it relates to other characteristics of complexity is a central theme in this Element. I worry that when heterogeneity is subsumed under the notion of multiple parts (see for example Reference LevinLevin, 2002; Reference McShea and BrandonMcShea & Brandon, 2010; Reference Odenbaugh and HookerOdenbaugh, 2011; Reference ParrottParrott, 2010; Reference PotochnikPotochnik, 2017; Reference WimsattWimsatt, 1972), it is too easily overlooked. In Sections 2.2 and 2.3 I examine my view of heterogeneity and its relationship to the characteristic of ‘multiple parts’.

Interaction. Another important characteristic of complex systems is that the parts of the system interact with each other. That is, the parts of a system stand in cause-and-effect relationships. These effects are often (but need not be) non-linear or non-additive. Outside ecology interaction is not always treated as an independent characteristic, but often subsumed under one of the others, such as multiple parts, feedback, non-linearity or emergence (Reference HookerHooker, 2011 but see Reference Ladyman, Lambert and WiesnerLadyman et al., 2013). Within ecology, interaction of parts features more prominently as a characteristic in its own right. For example, Stuart Pimm’s influential account of complexity included interaction as part of its definition, which he further categorised into connectance, that is, the number of interspecific interactions out of those possible, and interaction strength, that is the average value of the interspecific interactions in the community (Reference Odenbaugh and HookerOdenbaugh, 2011).

Emergence. This refers to the phenomenon of lower-level parts within a system giving rise to higher-level behaviours. For example, density dependence is the effect that the size of a population has on its members. Individual members of the population cannot display density; it is a property of the population as a whole. Discussions of emergence often have ideological connotations, in the sense that they imply a deeply anti-reductionist basis for complex systems research (Reference MitchellMitchell, 2009). The idea is that even though systems are made up of parts, it is (sometimes, often or always – depending on the strength of the view) better to investigate the behaviour of the system at the higher level – the level of the emergent property, rather than at the level of the constituent parts. What exactly is meant by ‘better’ here also depends on the view of emergence, but usually refers to some aspect of explicability: it is easier, more understandable, or more comprehensive to provide an explanation that includes the higher-level property, rather than one that occurs just at the lower level. Some emergent properties are considered to be representative characteristics of complex systems. For example, complex systems are organised hierarchically, that is, their components are grouped at different levels, and higher-level groups constrain the behaviour of lower groups. This contributes to self-organisation and causal autonomy that is, the ability of the system to regulate its own states and/or behaviour, creating and maintaining the processes that enable it to function (Reference LevinLevin, 2002, Reference Levin2005). An example of a biological hierarchical, self-organised and autonomous system is an organism’s metabolism, which creates and maintains the processes enabling the organism to live (Reference HookerHooker, 2011).

Non-linearity. A system is linear if ‘one can add any two solutions to the equations that describe it and obtain another, and multiply any solution by any factor and obtain another’ (Reference Ladyman, Lambert and WiesnerLadyman et al., 2013, p. 36). Complex systems do not possess this quality hence they are non-linear. In linear systems, a change in one factor will result in a similar or at least proportional change in the behaviour of the system. In non-linear systems, a change in one factor can result in a disproportional change in the behaviour of the system. For example, the growth rate of populations is positive when these populations are small and resources are abundant, yet as the population grows it does not do so proportionally. If resources continue to be abundant, the growth rate itself increases, yet if they become scarce, the growth rate becomes negative (i.e. the population decreases).

Non-linearity is sometimes linked with other characteristics of complexity, most notably chaos or sensitivity to initial conditions (Reference Anand and OrlóciAnand & Orlóci, 1996; Reference Benincà, Huisman and HeerklossBenincà et al., 2008; Reference Ladyman, Lambert and WiesnerLadyman et al., 2013). It is often difficult to distinguish between these concepts and their definitions: sometimes they are used interchangeably, sometimes one is considered to be a characteristic of another. For example, Reference Bishop and HookerBishop (2011) defines sensitivity to initial conditions as ‘the property of a dynamical system to show possibly extremely different behavior with only the slightest of changes in initial conditions’ (p. 108), and characterises the property as a feature of both chaos and complexity.

Another characteristic that is sometimes linked to non-linearity is feedback. Feedback can greatly magnify an initial (small) effect, contributing to the non-linearity of the system. For example, removing a keystone species from an ecosystem can result fundamental changes in the ecosystem, including a cascade of local extinctions and the eventual collapse of the entire ecosystem (Reference LevinLevin, 1998).

Path dependence (aka historicity). Put simply, systems display this characteristic when their later states depend on their previous states. Sometimes, development along a certain path becomes increasingly entrenched. For example, an initial mutation that provides a competitive advantage then spreads through the population (Reference HookerHooker 2011, p. 33). An important ecological version of path dependence is that of Simon Levin. Here, path dependence is an effect of non-linearity, as it occurs when ‘the local rules of interaction change as the system evolves and develops’ (Reference LevinLevin 1998, p. 433). A typically ecological example of path dependence is the colonisation of new areas, such as islands or forest patches. The final composition of the system (i.e. which species persist and in what proportion) will depend on which species are the original colonisers and how they interacted with each other.

2.3 The Effects of Complexity

So far, I have given a brief outline of what complexity is, in terms of some key characteristics. The aim of this Element is to examine the characteristics that are relevant for a particular context: what difficulties complexity creates for ecological research. The starting point for this discussion is the work of Richard Levins, specifically his (Reference Levins1966) paper ‘The structure of model building in population biology’, which, as we shall see, explicitly focuses on the effects of complexity for biological research. Levins’s argument was subsequently refined (Reference LevinsLevins 1993), and his ideas have received attention in philosophy of science, especially in the literature on modelling (Reference Godfrey-SmithGodfrey-Smith, 2006; Reference JustusJustus, 2005, Reference Justus2006; Reference MatthewsonMatthewson, 2011; Reference OdenbaughOdenbaugh, 2003; Reference OrzackOrzack, 2005; Reference Orzack and SoberOrzack & Sober, 1993; Reference WeisbergWeisberg, 2006). This body of philosophical literature forms the conceptual background for many of the ideas in this Element and I will be drawing from it heavily, especially for the remainder of this section.

Levins starts by pointing out that population biologists must deal with systems that are incredibly complex in the sense that: (i) they are made up of many different species, (ii) there is genetic, physiological and age diversity within each population, (iii) there are demographic interactions within and between populations and (iv) all this is happening in heterogeneous environments (p. 421). He recognises that dealing with such a complex system is problematic, as capturing all this complexity in a mathematical model is highly impractical. Even if it were feasible, it would yield results that made little sense to us (Reference LevinsLevins, 1966). Thus, scientists must decide how much complexity to include in each model. The issue is that there is no universal optimal level of model complexity, as scientists use models for multiple distinct purposes (i.e. understanding, predicting and modifying nature) (p. 422). The optimal level of model complexity can thus differ, depending on the model’s intended purpose and the system it is applied to.

Much of the complexity of ecological systems is represented within models by what Levins calls realism: how accurately a model captures the causal structure of the world. In practice, this is achieved by models containing many variables. That is, models are realistic when they are not overly simplified or idealised, for example, when they include many variables that correspond to real world factors, represent (dynamical) links between these variables and relax simplifying assumptions such as symmetry (Reference LevinsLevins 1993, pp. 548–52). In contrast, examples of highly unrealistic models in biology include those that ‘omit time-lags, physiological states, and the effect of a species’ population density on its own rate of increase’ and contain assumptions analogous to ‘frictionless systems or perfect gasses’ (Reference Levins1966, p. 422). I should note that Levins’s notion of realism is relevant in current ecological literature, as the term ‘model complexity’ is still used to refer to the number of variables represented in a model (Reference Clark, Ann Turnbull and TredennickClark et al., 2019; Reference Ward, Holmes, Thorson and CollenWard et al., 2014).

Capturing system complexity is not the only requirement of a model. Levins identifies two other ‘desiderata’ that scientists aim for in their models. The second desideratum is generality. Put simply, a model is general when it applies to many systems in the world, that is, when a modeller can use the model to gain information about many systems (I explain model applicability in the next two paragraphs). The third desideratum is precision, that is, how finely specified a model’s predictions are (see footnote 5, Box 1 C and Sections 3.2.1 and 4.3 for a full explanation of precision). On Levins’s view (Reference Levins1966, Reference Levins1993), modellers aim to increase all three desiderata as much as possible within each model. However, after a certain point, only two out of the three desiderata can be increased further. This gives rise to three strategies for building models and three corresponding types of models (Figure 1).

Figure 1 Levins’s 3-way trade-off

Before I move on to the three types of models, a note on ‘model applicability’. In the previous paragraph, I pointed out that models are general when they apply to many systems, which means that they provide information about those systems. This is an intentionally weak constraint, as Levins wants to distinguish between a model merely applying to a system and applying well to a system. In principle, anything can be a model for any phenomenon or system. For example, my coffee cup and water bottle could be a model of the earth and the sun. This view may seem controversial, at first glance, but it is espoused by many philosophers of science, who believe that a model’s use is partly determined by the intentions of the modeller (see for example Reference GiereGiere 2004; Reference Knuuttila and LoettgersKnuuttila & Loettgers, 2016a; Reference WeisbergWeisberg, 2013). The weakness of the constraint has an additional benefit, namely that it gives scientists the freedom to use models in novel and creative ways. For example, it allows scientists to use models developed for quite different systems, such as using physical or chemical models in biology or biological models in economics (Reference Knuuttila and LoettgersKnuuttila & Loettgers, 2016b; Reference WeisbergWeisberg, 2007).

Of course, just because a model could be applied to a system, does not mean that it should be applied to that system. This depends on the model, the system and the phenomenon we want to investigate. For example, if I want to use my beverage holder model to explain rudimentary planetary motion to my five-year-old nephew, then the model will probably apply quite well to the system. If, on the other hand, I want to use the beverage holder model to predict the next lunar eclipse, then it will be woefully inadequate. There are many accounts in the philosophical literature detailing what it means for a model to apply well to a system (Reference FriggFrigg, 2009; Reference Morgan and MorissonMorgan & Morisson, 1999; Reference Van Fraassenvan Fraassen, 2008; Reference WeisbergWeisberg, 2013) but adjudicating between these views is not relevant for this discussion. For now, it is more important to be able to identify when a model has been applied well to a system. A common view in the scientific literature, shared by Levins, is that we can test that a model applies well to a system, by whether it yields accurate predictions about the system’s behaviour (Reference Beckage, Gross and KauffmanBeckage et al., 2011; Reference Dambacher, Li and RossignolDambacher et al., 2003; Reference Kulmatiski, Heavilin and BeardKulmatiski et al., 2011; Reference Stillman, Railsback, Giske, Berger and GrimmStillman et al., 2015; Reference StockwellStockwell, 1999; Reference Tompkins and VeltmanTompkins & Veltman, 2006).Footnote ⁶

Back to the three types of models. Type I models (perhaps the most common models in population ecology) maximise generality and precision at the expense of realism. The main advantage of these models is their ability to apply widely. By omitting all the causal factors specific to particular systems, these models can identify factors that are common across many systems. For example, the logistic model of population growth shows how populations grow when they are limited by the carrying capacity of the environment (see Box 1A). This dynamic is thought to be the core factor of population growth, present in all populations, even when other factors are also present. Thus, the model that describes the process common to all populations is general. Moreover, these models allow scientists to make precise predictions about the future size of a population given a set of initial conditions. The main disadvantage of type I models is that they cannot capture many of the relevant factors and dynamics that are idiosyncratic (i.e. unique to particular systems), so their predictions are often inaccurate (Reference JustusJustus, 2005; Reference Novak, Wootton and DoakNovak et al., 2011). In other words, even though they apply to many systems, they frequently do not apply well to these systems.

Box 1. Type I, II, III models

Type II models (e.g. ecosystem network models) maximise precision and realism at the expense of generality (Box 1B). They are constructed with a particular system in mind, and contain many factors present in the real-world system. The main advantage of these models is that they can capture many more factors and dynamics and thus have the potential to provide much more accurate predictions (Reference Fischer, Rodig and HuthFischer et al., 2018; Reference Phillips, Ibanez and D’OrangevillePhillips et al., 2016; Reference Travis, Coleman and AusterTravis et al., 2014). The flip side of the coin is that by including many system-specific factors and dynamics, these models only apply well to the systems they are created for; they do not allow scientists to generalize their results beyond the system under investigation (Reference Wenger and OldenWenger & Olden, 2012). Another worry for type II models is that they do not always fulfil their potential in terms of predictive accuracy (Reference Schindler and HilbornSchindler & Hilborn, 2015; Reference Ward, Holmes, Thorson and CollenWard et al., 2014). This is a version of the problem of ‘overfitting’, where a complex model includes or accommodates noisy data resulting in inaccurate predictions (Reference Hitchcock and SoberHitchcock & Sober, 2004). The quality of available data is often low in ecology, as it can be partial, gappy or even false – this is referred to as noise (see Section 3.3 paucity of data). Type II models, which can incorporate many variables and data points can be ‘led astray’ by low quality data. They can end up being more susceptible to low quality data than type I models, simply because they can incorporate more of it. Thus, type II models can end up yielding predictions that are highly inaccurate (but see discussion in 4.2.1).

Type III models (e.g. loop analysis) maximise realism and generality but sacrifice precision (Box 1 C). The outputs of these models (their predictions) are not finely specified. They can be imprecise probabilities (e.g. the population will rise by more than 2 per cent), ranges of values (e.g. the time to extinction is 47–60 years) or qualitative trends (e.g. the effect of population A on population B is negative). Their main advantage is that they can maximise both realism and generality simultaneously. They can capture all the relevant variables and dynamics in a system and can also apply to many diverse systems. A second advantage is that they are less susceptible to errors (Reference JustusJustus, 2006). Unlike type I models, they can include many more parameters, thus capturing all relevant dynamics. At the same time, unlike type II models, they are better at avoiding or at least minimising the effect of errors in parameter estimation (because their parameters and outputs are imprecise). However, this way of minimising errors (by widening what can be accepted as accurate) is often seen as a major disadvantage of type III models (Reference Orzack and SoberOrzack & Sober 1993). The worry is that the imprecise predictions could be masking systematic errors in the models, so that even if the predictions are accurate, they cannot be relied on as tests of how well the model applies to a system, nor can the models be relied on to determine how best to intervene on the systems under investigation (Reference Orzack and SoberOrzack & Sober, 1993; but see Section 4.3, Reference Elliott-GravesElliott-Graves 2020a and Reference JustusJustus 2005, Reference Justus2006).Footnote ⁷

Levins’s trade-off framework can be understood both as a diagnosis of the effects of complexity for scientific practice and as a suggestion for how best to deal with these effects. The diagnosis is that complexity creates difficulties for scientific practice: it imposes limits on how scientists can investigate natural systems. An important assumption of this diagnosis is that complexity is not an artefact of our scientific methods, but a feature of biological systems. This means that scientists have no control over complexity itself (e.g. they cannot reduce the complexity of the systems they study, no matter how much science progresses). They can only get better at dealing with its effects. But this leads to another question: Are the trade-offs also caused by complexity or are they artefacts of our methods and/or our epistemic limitations (Reference MatthewsonMatthewson, 2011)? If the trade-offs are mere artefacts, then there is no reason to adopt Levins’s suggestions for how to deal with complexity.

Levins stated (and subsequent commentators have explicated) that this is a false dichotomy (Reference LevinsLevins, 1966; Reference MatthewsonMatthewson, 2011; Reference WeisbergWeisberg, 2006). Trade-offs exist because of the combination of biological complexity and our epistemic limitations. Complexity ensures that each system has hundreds of parameters, and multiple dynamic interactions between them. Our epistemic and technological limitations mean that the collection of data is difficult and prone to introduction of error. Even if all this data was collected and put in a model, the model would probably be analytically insoluble. Even if we managed to create a model that provided analytical solutions, these solutions would be meaningless to us (Reference LevinsLevins 1966, p. 421).Footnote ⁸ In other words, even if part of the problem is that we are limited beings and cannot adequately capture complexity in our models, these limitations are not temporary (Reference MatthewsonMatthewson 2011; Reference WeisbergWeisberg 2006). Technological advances are likely to make data collection easier or less biased or even result in models that are better at capturing complexity. Yet there are limits to these advances so we will never be able to fully capture the complexity of natural systems. It therefore behoves us to adopt scientific strategies that best deal with the effects of complexity. I address this in Section 4. In the meantime, I return to the discussion of how we should understand ecological complexity.

2.4 Which Characteristics of Complexity Give Rise to Trade-offs?

In the previous section, I outlined the contextual backdrop which frames our discussion of complexity, namely the effects of complexity on model construction. We can now fine-tune our characterisation of ecological complexity in light of this contextual backdrop. From Section 2.1, we have the following characterisation of complexity (à la Levins): ‘Population biology must deal simultaneously with genetic, physiological and age heterogeneity within species of multispecies systems changing demographically and evolving under the fluctuating influences of other species in a heterogeneous environment. The problem is how to deal with such a complex system’ (Reference Levins1966, p. 421). I identified the following characteristics from this description: complex systems in biology are (i) made up of many different species, (ii) there is genetic, physiological and age diversity within each population, (iii) there are demographic interactions within and between populations and (iv) all this is happening in heterogeneous environments (p. 421). If we cross-reference this description with our list from Section 2.1, we can detect three features that together give rise to complexity: multiple parts, interaction and heterogeneity.

Levins, like many other scientists and philosophers, believes that complex systems are made up of many parts. But what counts as a system and what as a part? Levins identifies individuals, populations and species and bits or aspects of environments as possible ‘parts’. This is intentional: a scientist can categorise the world into different sorts of systems, depending on the type of phenomenon they are investigating (Reference Elliott-GravesElliott-Graves, 2020b). So, for example, ecological systems can be populations, meta-populations, communities, ecosystems and so on. Parts of populations typically are individuals, parts of ecosystems can be populations or species, abiotic factors (e.g. nitrogen) and so on. Notice that for each system the parts need not occur at the same hierarchical level but can cut across levels (so one part of an ecosystem can be a population, another part can be an element and another can be a species). The second characteristic is interaction: the multiple parts of the system interact with each other. Here, Levins mentions demographic interactions, which occur within and between populations, but given his inclusion of environments as a characteristic, I think we can safely assume that he would also count interactions between individuals (or populations) with parts of the environment as relevant. The third characteristic is heterogeneity, which Levins mentions in relation to species (i.e. many different species), individuals within a population (genetic, physiological and age diversity) and types of environment.

The next question to ask is what is the relationship between these three characteristics? This is where things really start getting interesting. On the face of it, it seems that for Levins, all these characteristics are aspects of complexity. So, all these three characteristics together make up the notion of complexity. All these aspects together cause the trade-offs. However, John Matthewson argues that complexity and heterogeneity are distinct concepts. In his 2011 paper, which critiques but also builds on Levins’s trade-off account, Matthewson points out that ‘complexity’ can be defined in many different ways, including the ‘technical definition’: ‘it has many interacting parts and perhaps exhibits some kind of emergent behaviour’ (Reference Matthewson2011, p. 331). He then seems to adopt this technical definition, which centres on the ‘many interacting parts’ characteristic (Reference Matthewson2011). Moreover, he argues that complexity alone cannot account for the trade-offs that biologists routinely face. After all, scientists in other fields that study complex systems, such as physics and chemistry, do not seem to face such extensive trade-offs (see also Reference JustusJustus 2005, Reference Justus2021, section 2). Matthewson asks us to consider a group of Airbus A330s. They are clearly complex systems, in the sense of being made up of many interacting parts. Yet we can construct successful models that simultaneously capture all the relevant variables, apply to all Airbus A330s, and provide precise predictions about their flight trajectories. In other words, they are maximally general, realistic and precise. There must be some characteristic other than complexity, that causes the trade-offs between desiderata in biological systems.

This is where heterogeneity comes in. Biological systems (e.g. ecosystems), like aeroplanes, are made up of many interacting parts, but unlike aeroplanes, each ecosystem is unique. A marine ecosystem and a forest ecosystem might have similar trophic levels, but the species in each level are very different (in the aeroplane analogy, this would be akin to a situation where the engines of various A330s were actually different). Thus, the knowledge we gain from examining one ecosystem does not transfer to others, even if they seem similar. For example, a model of the marine ecosystem might not generalise to a forest ecosystem and vice versa. In short, heterogeneity magnifies the trade-offs between desiderata, by restricting a model’s ability to generalise (a similar view of heterogeneity as magnifying certain trade-offs can be found in (Reference WeisbergWeisberg, 2004)).Footnote ⁹

I propose two refinements to Matthewson’s account of heterogeneity. The first is a shift in focus from ontological to causal heterogeneity. Matthewson’s conception of heterogeneity is ontological, in the sense that it refers to differences in the nature of systems’ parts. Thus, Airbus A330 parts (e.g. the engine, the wings) are the same type of thing across different individual Airbus A330s, whereas the components of ecosystems (e.g. plants and animals) are different types of things: each ecosystem has a unique collection of species. The problem is that ontological differences between systems are neither necessary nor sufficient for heterogeneity. For example, the phenomenon of niche overlap occurs when different species such as sardines and anchovies fulfil very similar functions within an ecosystem, so the populations can be substituted without affecting the functioning of the marine ecosystem (Reference Ricklefs and MillerRicklefs and Miller 2000). Here, ontological differences are not sufficient for heterogeneity, as ontologically diverse species are causally homogeneous.

Even more interesting, for our purposes, are the cases where ontological heterogeneity is not necessary for causal heterogeneity, that is, when ontologically similar or even identical species are causally heterogeneous. Consider the case of plant-soil feedback (PSF) (Figure 2). Plants interact with microbes in the soil, for example, arbuscular mycorrhizal fungi and nitrogen fixing bacteria. These interactions can be beneficial to the plant’s growth (positive feedback), for example, displacing plant pathogens or detrimental (negative feedback), for example, less access to nitrogen (Figure 2A). PSF can affect the outcome of competition between species, as positive feedback can confer a competitive advantage to a plant, while negative feedback can significantly reduce a plant’s ability to compete (Figure 2B). The interesting thing about PSF is that the type of feedback (positive, neutral or negative) can vary even when these systems are made up of the same species of plants and soil microbes (Reference KlironomosKlironomos, 2002; Reference van der Putten, Bardgett and Bevervan der Putten et al., 2013). For example, in field experiments, the sign of the PSF has been known to reverse in a single community, changing from positive to negative feedback (Reference Casper and CastelliCasper and Castelli 2007; Reference KlironomosKlironomos 2002). That is, the same soil biota that were boosting plant growth begin to hinder the growth of those same plants. The opposite switch has been observed in plant invasions, where plant species experience negative feedback with soil microbes when they first move to a new area, but then begin to experience positive feedback in the same system (Reference van der Putten, Bardgett and Bevervan der Putten et al., 2013).

(A) Plants interact with microbes in the soil, leading to positive, neutral or negative feedback;

(B) PSF can affect the outcome of competition between species, as positive feedback can confer a competitive advantage to a plant, while negative feedback can significantly reduce a plant’s ability to compete

Figure 2 Plant-soil feedback.

In short, it is the behaviour of systems and their parts that is relevant, rather than the makeup or nature of these systems. Of course, there will be many cases where the ontological differences between systems or their parts is the reason why they behave differently. In other words, ontological heterogeneity often contributes to or even entails causal heterogeneity. Still, even in these cases, it is the causal heterogeneity that is most relevant for studying the behaviour of ecological systems. Moreover, causal heterogeneity is the more useful of the two concepts: it encompasses all the instances where ontological heterogeneity leads to causal heterogeneity but excludes all those where ontological heterogeneity leads to causal homogeneity.

The second refinement concerns the units of heterogeneity. Matthewson focuses on ‘inter-system heterogeneity’, that is, heterogeneity between different systems. Each Airbus A330 or ecosystem is a complex system, and heterogeneity occurs between these systems. Thus, two Airbus A330s are homogeneous with respect to each other whereas two ecosystems are heterogeneous. However, causal heterogeneity can also manifest within each system, as differences between the parts of a system can have different effects. I call this intra-system heterogeneity (Reference Elliott-GravesElliott-Graves, 2018). Both types of heterogeneity are important because each causes a different type of problem for ecological research. Inter-system heterogeneity hinders generalisation across different systems and intra-system heterogeneity hinders generalisation within a system. This usually occurs when the future behaviour of a system is different to its past behaviour, which predominantly affects our ability to make predictions. I will examine the effects of inter- and intra-system heterogeneity in the next section (3.1 and 3.2).

So much for multiple parts, interaction and heterogeneity. What about the other characteristics? I believe that they are not as important for this context, that is, identifying the epistemic effects of ecological complexity, such as trade-offs. Let me clarify. The trio of multiple parts, interaction and (causal) heterogeneity can capture the main aspects of the other characteristics, in this context. In other words, the other characteristics can be seen as special cases of one, two or all three of the trio. For example, aspects of non-linearity, such as sensitivity to initial conditions and feedback can be seen as instances or aspects of causal heterogeneity. Changes in initial conditions cause systems to behave differently, while feedback does the same because of magnification of an effect. Both are captured by causal heterogeneity (see discussion on PSF above and example 2 – Emerald Ash Borer). I should note that my claim is not that the other characteristics are superfluous or irrelevant generally, merely that we do not need to discuss them separately in the context of their epistemic effects – we can account for these effects by focusing on the trio.

2.5 Ecological Complexity

To recap the main points of the discussion so far, Levins believed that complexity should be understood as a combination of three characteristics, multiple parts, interaction between the parts and heterogeneity of the parts themselves. Matthewson argued that despite Levins’s inclusion of heterogeneity in the notion of complexity, what people usually mean when they use the term ‘complexity’ is multiple interacting parts. This is a problem, because leaving heterogeneity out of the picture means that our notion of complexity does not account for the differences between biology and other sciences in terms of the extent and magnitude of trade-offs. However, Matthewson goes a bit too far in the other direction, as he claims that heterogeneity alone explains the trade-offs. My view is that heterogeneity generates new trade-offs and magnifies existing ones, so it should be viewed as an important characteristic of complexity, but not as a concept entirely independent of complexity. I further refined Matthewson’s notion, arguing that causal rather than ontological heterogeneity accounts for the magnification of trade-offs.

Thus, finally, we have our answer: ecological systems are complex in the sense that they are made up of multiple causally heterogeneous interacting parts. I should quickly point out, once again, that I am restricting my analysis to this particular context: understanding the epistemic difficulties in ecological research. It is within this context that this definition is meant to apply. There may well be other contexts in which other notions of ecological complexity turn out to be more useful. But if we are trying to understand why ecological systems are difficult to investigate, it is because they are made up of many causally heterogeneous interacting parts.

3.1 Generalisation

3.2 Prediction

3.2.2 Causal Heterogeneity and Prediction

How does causal heterogeneity affect predictions? The main effect of causal heterogeneity is on their accuracy. This is because predictions are based on generalisations, which, as we saw in the previous sections, are based on patterns. Scientists make predictions of the behaviour of a system based on the past behaviour of that system or the current behaviour of a similar system. If the system behaves differently with respect to its past behaviour or in comparison to similar systems, then the pattern breaks and the prediction fails. The more causal heterogeneity a system exhibits, the less likely it is that predictions will turn out to be true. To illustrate, I will examine the use of Species Distribution Models (SDMs) for predicting biological invasions.

• Example 2. Distribution of the Emerald Ash Borer

The Emerald Ash Borer (EAB) is a beautiful beetle (see Figure 3D) native to East Asia, that lays its eggs under the bark of ash trees (Reference Cuddington, Sobek-Swant, Crosthwaite, Lyons and SinclairCuddington et al., 2018). The larvae hatch under the bark and feed on the ash tree until they become adults. Asian ash trees have a certain level of resistance to the EAB, so EAB populations in Asia are relatively small. However, EABs are highly destructive to ashes in Europe and North America. The USDA Forest Service estimates that EAB has killed hundreds of millions of ash trees in North America and cost hundreds of millions of dollars (USDA Forest Service, 2020).

Figure 3 A, B & C GARP for Emerald Ash Borer

(adapted from Sobek-Swant et al. (2012)). D Photo by U.S. Department of Agriculture, on Wikimedia Commons

Scientists can use SDMs, such as GARP and MAXENT, to predict the distribution of a potential invasive species in a new area (Reference Sobek-Swant, Kluza, Cuddington and LyonsSobek-Swant et al., 2012; Reference Wang and JacksonWang & Jackson, 2014). The first step is to describe the species’ niche, which (in this context) refers to the environmental conditions under which the species can maintain a population. Based on the values of each parameter within the species’ native distribution, the model calculates the ranges where the species can survive. The model then predicts where the species will be able to maintain populations in the new area. Figure 3 shows the native and projected distribution of the EAB using an SDM (GARP).

On the left (A) we see the model projection of the native range of the EAB. This model projection is used as a test of the accuracy of the model. If the modelled distribution corresponds to the known actual distribution of the EAB in the native range, then the scientists move on to the next step, projecting the EAB distribution in the invaded range. The top right projection (B) is the prediction of the EAB’s range given the data from the native range. The bottom right projection (C) is the projection given known occurrences of EAB in North America. The overall prediction is that the EAB will likely increase its range from C to B, that is, northwards and westwards. Importantly, the scientists predicted that there would be a limit to the northern spread of EAB because of harsh winter conditions.

The underlying assumption in using SDMs for prediction is that there is a certain level of causal homogeneity across different systems. At first glance, this assumption makes a lot of sense as there are niche parameters that are shared between different areas. For instance, there are ash trees in East Asia and North America, while both areas have similar precipitation and temperature patterns.

Unfortunately, however, recent research has revealed that this prediction (the northern limit of EAB distribution) was inaccurate (Reference Cuddington, Sobek-Swant, Crosthwaite, Lyons and SinclairCuddington et al., 2018). The primary reason for this is climate change. Both the native and predicted ranges are based on actual temperature data (in this case, going back 50 years). These data sets reflected ‘extreme cold events’ that happened with some regularity, that is, about every six years. These events help to keep the EAB populations below certain levels. The problem is that these extreme cold events are happening less and less frequently, so the EAB populations are not being kept in check, thus allowing them to spread beyond the originally predicted range. This is an example of causal heterogeneity, as some of the causal factors that are relevant to the behaviour of the system (the frequency of extremely low temperatures) have changed.

The lesson to learn from this example is that causal heterogeneity is not just rampant, but it can be unexpected. Despite their best intentions, scientists are not always able to predict all the factors that might change so as to accommodate this change in their predictions. Surprises lead to more surprises. This points to an interesting implication of causal heterogeneity, namely that scientists are often unable to make accurate predictions, even though they are able to provide explanations of the same phenomena.

3.3 Effective Interventions

A substantial part of ecological research is ‘applied’, in the sense that it is aimed at solving real-world problems in real time, by intervening on ecological systems. Typical examples include pest control, saving a species from extinction and preventing or halting biological invasions. These interventions are based on what I call ‘applied predictions’: predictions of where/when an intervention will be needed and predictions of an intervention’s effect on the system. Inaccurate applied predictions result in less effective or even failed interventions. A rather tragic example is the introduction of a predatory snail Euglandina rosea to the Pacific islands, based on the prediction that it would prey on and thus control the population of the invasive Achatina fulica. Unfortunately, this prediction turned out to be false, as the introduced predatory snail (E. rosea) preferred the native snail species to the one it was brought in to control (A. fulica). In fact, E. rosea subsequently became an invader in its own right, probably causing much more devastation on the native ecosystem than A. fulica (Reference Thiengo, Faraco, Salgado, Cowie and FernandezThiengo et al., 2007). Overall, the intervention was a resounding and dramatic failure.

Ecological complexity also affects the effectiveness of interventions indirectly. The main indirect effect is paucity of data. Gathering data in ecology is often quite difficult. Even state-of-the-art methods are prone to patchiness and bias. Consider the task of estimating the size of a population. This is an essential step in a lot of ecological research, including most conservation efforts, yet it is exceedingly difficult and costly (Reference Akamatsu, Wang, Wang and WeiAkamatsu et al., 2001; Reference Kaschner, Quick, Jewell, Williams and HarrisKaschner et al., 2012; Reference Tyne, Loneragan and JohnstonTyne et al., 2016). The most widely used sampling method is the capture-release-recapture method, which does not work equally well for all species/populations and suffers from a number of biases (Reference Boakes, Fuller, McGowan and MaceBoakes et al., 2016). For example, it is not very useful in cases with sparse data, because each capture is difficult. For species with large distribution ranges, such as marine mammals or migratory birds, the available data is often patchy, so there are gaps in the information scientists have about the population’s seasonal distribution, behaviour and population dynamics (Reference Tang, Wu and LiuTang et al., 2019). It can also be biased in terms of the internal structure of the population. For example, in some fish populations, the method is biased towards larger individuals, which can skew the demographic information on age-structured populations, essential in most models of population growth (Reference Pine, Pollock, Hightower, Kwak and RicePine et al., 2003).

Sometimes ecologists rely on or supplement their own data with data collected by others. This can include historical data sets (whose quality cannot be checked) and data collected by non-experts. In some cases, such as various citizen science initiatives, these data sets prove to be extremely valuable, and of very high quality. However, in other cases, the people providing the data have incentives to distort it. This problem is especially pertinent for species that are part of commercial fisheries (e.g. cod) or species whose conservation is at odds with the interests of the fishery (e.g. dolphins), as ecologists rely on data provided by those very people with the opposing interests.

The difficulties created by ecological complexity for interventions are very important, because they have real-world consequences. In other words, they usually have high stakes. For example, failing to halt an invasion can incur huge economic costs in addition to the negative effects on the native ecosystem. A failure of conservation targets can lead to the extinction of a species, which is mostly irreversible. In addition, interventions are usually time-sensitive. Finding a solution to a problem is often not sufficient; the solution must be discovered and implemented within a particular timeframe. The scientists tasked with solving the problem must determine the extent of the danger, identify the causes of the danger, find ways to mitigate the threat and determine the best ways to implement the mitigation. All this must be achieved before the population drops below a certain level.

Why are these characteristics worse for applied rather than confirmatory predictions? After all, scientists making predictions for confirmation purposes don’t always have optimal data sets, unlimited time or low stakes. Still, these effects tend to be much larger in the case of applied predictions. There is quite a big difference between failing to confirm a theory and failing to save a species from extinction. Similarly, time-constraints may exist in confirmatory contexts, but they are usually not as small. Poor data sets are problematic because they limit the accuracy of our best available models. For example, it is very difficult to predict the future population size after an intervention, when you cannot even determine the current population size with the accuracy required by the best available models. Moreover, in applied contexts, these three characteristics tend to exist simultaneously and even magnify each other. For instance, when paucity of data causes an inaccurate prediction, there is usually insufficient time to adopt an alternative method, even if such a method exists.

In order to fully appreciate how ecological complexity directly and indirectly affects predictions aimed at interventions, let us examine the case of the Yangtze River porpoise.

• Example 3. The Yangtze River finless porpoise

The finless porpoise (Neophocaena asiaeorientalis asiaeorientalis), endemic to the Yangtze river, is the world’s only surviving freshwater porpoise (Reference Huang, Mei and HaoS.-L. Huang et al., 2017) (Figure 4). Its population has been steadily declining, and despite the existing conservation measures, in 2016 it became critically endangered. In light of this, additional, more extensive conservation measures were adopted to protect it. An essential piece of information for determining a conservation intervention is the ‘time to extinction’ (TE), because it affects which species are prioritised and which policies are adopted. An important paper was published in 2016, which predicted that the TE was significantly lower than previously thought (37–43 years, instead of 63 years from 2012) (Reference Huang, Mei and HaoHuang et al., 2017).Footnote ²¹

(A) Yangtze finless porpoise in the Institute of Hydrobiology, Chinese Academy of Sciences on Wikimedia Commons;

(B) Yangtze finless porpoise in Poyang Lake, Jiangxi, China, on Wikimedia Commons

Figure 4 Yangtze finless porpoise.

This case bears all the hallmarks of the difficulties associated with applied predictions. The prediction here is the TE, which is used to determine the type and extent of the intervention. Typically, this prediction involved high stakes. An inaccurate prediction would incur huge costs, as the porpoise population was already critically endangered, and there was little room for error. Local extinction, in this case, would mean global extinction, as the porpoise is endemic, and individuals tend not to survive in captivity. Moreover, the whole situation was extremely time-sensitive; the scientists were literally racing against the clock to determine what policies needed to be implemented or adapted before the porpoises dropped below a certain threshold.

Paucity of data was also an issue. The porpoises are extremely difficult to locate and the scientists had important gaps in their information concerning the porpoises’ behavioural patterns, population dynamics and seasonal distribution (Reference Tang, Wu and LiuTang et al., 2019). The scientists have to rely on sightings of dead porpoises by fishermen, though as the porpoises are most likely to die from illegal fishing methods, this data is usually biased. This patchiness or bias in the data introduces uncertainty into the models used to estimate population size, which reduces the likely accuracy of their predictions. After all, only four years before the study, the same group of scientists had predicted a significantly higher TE.Footnote ²²

Luckily, the scientists’ warnings seem to have been heeded and additional conservation measures were put in place (including increased protection level of the species, increase in the size and number of protected areas, fishing bans and closer monitoring of fisheries) (Reference Huang, Mei and ChenJ. Huang et al., 2020). These measures are considered successful, as even though the population is still decreasing, it is doing so at a much slower rate. There is now hope that the new decreased rate of decline gives us enough time to save the species from extinction, provided that the existing conservation measures are extended to increase migration between the three sub-populations (Reference Huang, Mei and ChenJ. Huang et al., 2020; Reference Tang, Wu and LiuTang et al., 2019).

This has been an admittedly pessimistic discussion of the difficulties caused by causal heterogeneity. Luckily, I believe that there are a number of options open to ecologists for conducting research that is undeniably useful and of high quality. In the next section I will examine some of these research methods and argue that they have more value for ecology than they are currently given credit for.

4.1 Reducing Complexity: The Traditional Approach

The traditional approach to dealing with complexity is to reduce it, by simplifying aspects of the phenomenon or system (Reference LevinsLevins, 1966; Reference MitchellMitchell, 2003; Reference WeisbergWeisberg, 2007). In experiments, this can be achieved by omitting or controlling causal factors, whereas in models it can be achieved through abstraction (the omission of factors) and idealisation (the distortion of factors) (Reference Elliott-Graves and WeisbergElliott-Graves & Weisberg, 2014). This traditional approach works well in causally homogeneous systems, because it allows scientists to distinguish between real causal factors and noise – the real causal factors are present in many systems, whereas the noise is idiosyncratic. Thus, reducing complexity provides the additional benefit of identifying generalisations across systems. As we saw earlier, this approach is much less likely to be useful in causally heterogeneous systems, as the idiosyncratic factors of each system are often real (not mere noise) and relevant, so omitting or distorting them leads to inaccurate explanations and predictions.

Nonetheless, there are some contexts where reducing complexity can be beneficial even in causally heterogeneous systems. The first is when scientists want to identify all the instances where a particular factor is present, even when this factor is not relevant to the functioning of a particular phenomenon. This can enhance the scientists’ understanding of the natural world, as it shows them which factors are common across different phenomena. This type of generalisation is often useful for pedagogical purposes as it shows how disparate phenomena might have some factors in common. For example, most populations seem to have the potential to grow exponentially, if there are no other factors curbing their growth (limited resources, competitors, predators, etc.). Of course, almost no populations grow exponentially in real-life because there are almost always multiple other factors curbing their growth. Thus, it would not be particularly useful to use an exponential growth model to predict the growth of any particular real-world population. The existence of these other factors that affect real-world populations will render the predictions of an exponential growth model inaccurate. Still, it is useful to know that most populations would grow exponentially if they could, that is, in the absence of other factors. In other words, it can be useful to know that a particular causal factor is general, even though other factors overpower it, and it does not actually manifest in real-world phenomena.

The second type of situation, where reducing complexity and applying large-scale generalisations is useful, occurs when the common causal factors are relevant for the phenomenon and its idiosyncratic factors are (i) either not relevant, (ii) are overshadowed by other factors or (iii) give rise to factors that are shared across other systems. It just so happens that in some cases scientists are lucky, so that simple, general (type I) models do capture the relevant causal factors of the phenomenon and actually yield accurate explanations, predictions and interventions. An example of an exceedingly simple model being used successfully can be found in the case of the Vancouver Island Marmots.

• Example 4. Allee effect in the Vancouver Island Marmots

The marmots of Vancouver Island (Marmota vancouverensis) (Figure 5) are classified as critically endangered. It was estimated that their population had dropped 80 per cent –90 per cent since the 1980s and by the mid-2000s consisted of roughly 200 individuals (Reference Brashares, Werner and SinclairBrashares et al., 2010). The cause of this rapid decline was a mystery. The marmots were not hunted, their sources of food were unaltered, there were no new predators or competitors and the small disturbances to their habitat (small-scale logging) seemed to have a positive effect on the population, as the absence of thick tree roots in clearings made the building of burrows much easier. Nonetheless, the marmot population began to decline in the 1980s and kept on declining despite some early conservation efforts (such as the expansion of Strathcona Provincial Park in 1995). Something had to be done quickly to preserve the species from extinction.

Figure 5 Vancouver Island Marmot.

Photo by Alina Fisher on Wikimedia Commons

Reference Brashares, Werner and SinclairBrashares et al. (2010), who took on the marmot case, decided to use a simple, general (type I) model to try and determine the cause of the decline. They reasoned that, as there was no recent change in predators, competitors, disturbance to their habitat, food sources or hunting, a simple demographic model of population growth could shed light on the situation. Thus, they applied the logistic model of population growth to the Vancouver Island (V.I.) marmots (see Box 1 A and Figure 6A solid line). This model describes how a population grows, given (i) its intrinsic growth rate r (which is the maximum possible growth rate of the population and approximately corresponds to the number of births minus the number of deaths), (ii) the density of the population and (iii) K the maximum size of the population that the environment can support. At first glance, the choice of model seems wrong as the growth of the V.I. marmots diverges from the model predictions (Figure 6A red line vs. Figure 6B). That is, the V.I. marmot population growth rate seems to be dropping even at low densities, in contrast to standard logistic growth.

(A) Per capita logistic growth (red line) and per capita logistic growth with Allee effect (dotted line). K is the carrying capacity (the maximum population that the environment can support). In standard logistic growth (red line) the growth rate drops as the population size increases. With an Allee effect, the population falls at low densities, despite an abundance in resources;

(B) Allee effect in the V.I. marmots. The graph shows annual counts of free-living V. I. marmots 1970 to 2007. There is a clear Allee effect, a strong inverse density dependence in per capita growth rate (y axis) with respect to population size (x axis). Trend line represents least-squares quadratic fit (R² = 0.55).

Data exclude animals introduced from captivity (Figure 6B and its explanation is adapted from Reference Brashares, Werner and SinclairBrashares et al., 2010).

Figure 6 Logistic growth and Allee effect.

Nonetheless, the V.I. growth rate conforms to a recognized variation of logistic growth, termed the ‘Allee Effect’ (Figure 6A dotted line), which describes populations that behave normally (i.e. according to logistic growth) at high densities, but have a positive correlation between the density of the population and its growth rate at low population densities (Reference Courchamp, Clutton-Brock and GrenfellCourchamp et al., 1999). In other words, the Allee effect describes cases where once the population drops below a certain level, it cannot bounce back, even though there are plenty of resources to go around. This is exactly what was happening with the V.I. marmot population: the data (from 1970 to 2000) show that at low population sizes the V.I. marmots cannot bounce back (Figure 6B).

The next question to ask is why the V.I. marmots exhibit an Allee effect. As it turns out, these rodents are quite special, because they are highly social. They live in groups of 5–15 individuals, have an intricate pattern of social interactions and a variety of alarm calls. In fact, they normally spend most of their time socialising, as they forage communally, taking it in turns to look out for predators and share in the upkeep of their burrows. When a marmot out foraging encounters another marmot, they greet each other by touching their noses together, while the youngsters, who stay on with the family until they are around two years old, spend most of their time play-fighting. The problem is that when the population falls below a certain density, these marmots find it hard to locate potential mates. The smaller populations mean that there is less division of labour (caring for young, guarding against predators) so each marmot spends more time and effort foraging and can afford to devote less time to searching for mates. In addition, as the population decreases, the potential mates are spread more thinly across the territory, so they become even harder to locate. All this results in lower instances of mating, which leads to further decrease in the population, despite the abundance of resources.

The case of the V.I. marmots is an example of an extremely simple and general model being used to successfully explain a particular phenomenon, predict the future of behaviour of the system and form the basis of a successful intervention.Footnote ²³ It is a case where the complexity of the system was significantly reduced, as the study focused only on a few demographic properties of the V.I. marmots, excluding many other factors (competitors, predators, pathogens, habitat loss, disturbance, etc.). Moreover, this model worked despite the fact that the V.I. marmots had a particular idiosyncratic characteristic that is not shared by other species of marmot, that is, sociality. Therefore, one may ask why I have included this example which seemingly undermines my claims that causal heterogeneity decreases generalizability.

The answer is that this is a very special case – and one of the few cases where type I models are genuinely useful. It is special for two reasons. First, the complexity could be reduced: the factors normally operating on a population, that is, competitors, predators, pathogens, habitat loss and so on, were not relevant causal factors in this case. In fact, the scientists already knew that these factors were not relevant because they had remained constant before, during and after the drop in the V.I. marmot population. Second, the apparent heterogeneity of the system, that is, the uncommon sociality of the V.I. marmots, actually made this system homogeneous to other systems, such as plankton, plants and sessile invertebrates, because it led to mate limitation, which is a common cause of Allee effects in these other systems (Reference Berec, Angulo and CourchampBerec et al., 2007). In other words, this is a rare example of apparent heterogeneity resulting in causal homogeneity across different systems. As stated at the end of Section 3.1.3, the absence of causal homogeneity explains when and why some generalisations are actually possible.

Nonetheless, it is important to reiterate that this confluence of reducible complexity and effective homogeneity is far from common in ecological systems. Thus, while we should be happy when such situations occur, we should not expect them to occur frequently. More importantly, we should not criticise scientific investigations of systems that do not allow this type of generalisation. What should ecologists do then? I turn to this issue next.

4.2 Retaining Complexity: Modest Generalisations

4.3 Incorporating Uncertainty: Imprecise Predictions

The most controversial strategy for ecological research is to incorporate uncertainty in models by sacrificing the precision of their predictions. In general terms, imprecise predictions are those that refer to the existence of a phenomenon or effect, without specifying its extent or magnitude. For example, a model could predict a trend, such as ‘population x will increase’, without specifying by how much it will increase. Some predictions are imprecise because they predict a range of values for a particular variable. For example, the temperature in 2030 will increase by 1.5–3.5 degrees. Other predictions are imprecise because they include imprecise probabilities; for example, there is 0.6–0.8 probability that organism y will be extinct in twenty years.

It should be clear from these examples and from the discussion in Section 3.2, that precision is not a binary characteristic, but comes in degrees. In fact, imprecision is a way of representing uncertainty (Reference Elliott-GravesElliott-Graves, 2020a; Reference ParkerW. Parker, 2010; Reference Parker and RisbeyW. Parker & Risbey, 2015; Reference Smith and SternSmith & Stern, 2011). That is, scientists can use imprecision to convey that they are not certain about a model’s predictions. The more uncertain the result, the less precise the prediction. Table 2 shows predictions for variable x in decreasing levels of precision (adapted from Reference Parker and RisbeyParker & Risbey 2015).

In ecology, imprecision usually appears as the output (predictions) of models. The imprecision in these models usually corresponds to levels (c)–(e) in Figure 6. The models themselves correspond to Levins’s type III strategy (Section 2.2.1). The main attribute of these models is that they can achieve generality without sacrificing realism. By representing each factor imprecisely, they can include many more factors than a type I model. They are thus said to contain idealisations of specificity rather than idealisations of veracity (Reference JustusJustus 2006, p. 659). That is, imprecise models represent systems veridically (without omitting or distorting most of their variables). In contrast, maximally precise models employ different methods of simplification, that is, making unrealistic assumptions and/or omitting causal factors altogether, which decreases the ‘veracity’ of the model. This type of idealisation often mischaracterizes salient features of systems, resulting in inaccurate explanations and/or predictions (Reference JustusJustus, 2005, Reference Justus2006).

Despite these advantages, imprecise predictions have traditionally been viewed with suspicion in philosophy of science, for two main reasons.Footnote ²⁴ First, they are thought to be insufficiently risky to provide strong enough tests for theory confirmation (Reference Orzack and SoberOrzack & Sober, 1993; Reference RosenbergRosenberg, 1989, see also Section 3.2). If predictions are the primary method by which we test and choose between theories, then they must not be too easy to obtain. The worry is that as precision trades off with accuracy, the less precise the prediction, the more likely it is to be accurate. Thus, if we sacrifice too much precision, the predictions will not be able to discriminate between higher and lower quality theories. The second criticism of imprecise predictions is that they could lead to ineffective, insufficient or inappropriate interventions (Reference Houlahan, McKinney, Anderson and McGillHoulahan et al., 2017; Reference Orzack and SoberOrzack & Sober, 1993; Reference RosenbergRosenberg, 1989). For example, if scientists are trying to save a particular population x from extinction, they might use a model that predicts that population’s qualitative trend after an intervention: a drop in the predator population y will cause the population of the prey x to rise. The scientists will then intervene to reduce population y. The worry is that if the scientists do not know by how much the predator population is expected to fall with the intervention, their intervention could be ineffective. They could reduce the predator population y, but not reduce it enough, which would not save species x from extinction. Alternatively, they might reduce the predator population too much, spending scarce resources that could be put to better use. Thus, the story goes, it is better to have specific information about how much the population of x will rise given a specific intervention on population y.

These criticisms might seem plausible, but they are not warranted. First, it should be noted that imprecise model predictions are not exceedingly imprecise; that is, they do not correspond to levels (f) and (g) in Table 2. Rather, they correspond to levels (c) and (d), and occasionally level (e). That is, there are many values that they could take which would render them false, so they are not altogether without risk. Second, in causally heterogeneous systems, predictive accuracy is much more difficult to achieve than the critics presuppose. It is not the case that most predictions turn out to be true, so that we need stronger tests for our theories. Rather, as explained above and in the previous section, predictions in ecology tend to come out false quite often. In these cases, even an imprecise but accurate prediction is better than a number of precise but wildly inaccurate predictions.

Third, imprecise (type III) models are often better than both type I and type II models at dealing with biased, patchy or low-quality data (see Section 3.3). This is because poor data sets lead to errors in the estimation of parameters, which result in errors in the model’s output. Often (especially when the models have large numbers of parameters), these errors can persist despite statistical methods used to reduce or correct them (Reference Dambacher, Li and RossignolDambacher et al., 2003; Reference Novak, Wootton and DoakNovak et al., 2011). In contrast, type III models are built for imprecise parameter estimates, so even though their predictions are imprecise, the entire range that the prediction spans, tends to be more accurate. In other words, the predictions of precise models are often further away from the actual value than the entire range of the imprecise model prediction. For example, Reference Novak, Wootton and DoakNovak et al. (2011) found that even if there is observational and experimental data on each particular species within an ecosystem but insufficient data on the indirect effects of each population on other populations within the ecosystem, the predictive accuracy of highly precise models dwindles rapidly (e.g. to about the level of flipping a coin).

In order to fully appreciate the value and potential of imprecise models, one only has to look into the case of the kōkako. The scientists here used two imprecise models to bring back the kōkako from the brink of extinction. Importantly, these models yielded predictions that were sufficient in terms of both riskiness and for formulating and implementing a successful intervention.

• Example 6 – Saving the kōkako from extinction

The North Island kōkako is a bird endemic to New Zealand (see Figure 7). In 1999 the species was reduced to 400 pairs, because of predation from the ‘unholy trinity of pestilence’: possums, stoats and rats (Reference HansfordHansford, 2016) (Figure 7). The scientists tasked with saving the kōkako from extinction faced three difficulties (Reference Ramsey and VeltmanRamsey & Veltman, 2005):

1. (i) Limited resources. Ideally, the scientists would have simply eradicated all three predator populations, but this was not feasible. Resources only allowed for a reduction of predator populations.

2. (ii) Limited time. As the kōkako population was already so low, the conservation strategy needed to be conceptualised and implemented quickly – before the kōkako population fell to a level from which it could not be restored.

3. (iii) Poor/patchy data. It was practically impossible to determine the exact size of each population and consequently the precise rates of competition and predation within the community.

(A) Photo by Doug Mak on Wikimedia Commons;

(B) Photo from Reference HansfordHansford (2016) Radio New Zealand

Figure 7 North Island kōkako and predators.

The conservation strategy was based on a combination of two imprecise models (Reference Ramsey and VeltmanRamsey & Veltman 2005). The first, loop analysis (Box 1 C I), was used to identify all the dynamic relationships between populations in the community, along with their quality (positive (1), neutral (0), negative (−1)). Scientists can use the conjunction of qualitative effects to determine which predator population should be the focus of an intervention so that the prey population is saved from extinction. In this case, loop analysis predicted that an increase in all predator populations would have a negative effect on the kōkako population, but that an increase in the rat population would have a stronger effect than the other two predators because it would result in a net total of two negative feedback cycles (one from predation and one from competition) (Box 1 C II).

The second model was a ‘fuzzy interaction web’ (FIW). FIWs take imprecise information on the size of populations within an ecological community, such as population abundance data that is incomplete, and create ‘fuzzy sets’. Recall that precise and accurate data on population sizes is impossible to attain. With FIWs, however, scientists have a way to get around this problem. They can conduct the entire investigation without knowing the exact size of any population in the community. They do this by measuring ‘tracking rates’, that is, the percentage of traps that are full on any given night. These are used to determine what counts as low, medium or high abundance for this community. Thus, for example, high abundance could occur when 20 per cent or above of the traps each night are full and low abundance could occur when less than 10 per cent of traps are full. Values between 10 per cent and 20 per cent could mean ‘quite high’, ‘moderately high’, ‘quite low’ and so on (Figure 8).

Figure 8 Fuzzy set membership for kōkako community. (A) Fuzzy set membership functions for linguistic descriptions of species abundances (‘low’, ‘(mod)erate’, ‘high’) for each of the animal species in the kōkako Fuzzy Interaction Web

(reprinted from Reference Ramsey and VeltmanRamsey & Veltman 2005, supplementary materials)

The ‘fuzziness’ is a way of representing this uncertainty about population sizes. The concept comes from ‘fuzzy logic’, an approach where a variable can take multiple values. In classical logic, conclusions are either true or false, and each variable either is or is not a member of a set. For example, a monochromatic marble is either blue or it is not blue; it cannot belong and not belong to the set of blue things. However, in fuzzy logic, the boundaries between sets are not ‘crisp’, so membership of a set is partial. This does not mean that the marble is and is not blue at the same time, but that we are not certain about whether the marble is or is not blue. In the case of the kōkako, the fuzziness represents the fact that the scientists are not 100 per cent certain about where the boundaries between the categories of low, medium and high are. Figure 8 shows the fuzzy set membership for each of the species in the community.

The scientists can use these fuzzy sets to make predictions about the effect of each population on the other populations in the community. Figure 9 summarises the predictions yielded by the FIW. It shows that controlling all three populations would have a high effect on the kōkako population, whereas controlling rats and possums would have a moderate effect on the kōkako population. As a moderate effect is sufficient for bringing back the kōkako population to acceptable levels, the most efficient intervention should focus on the rats and possums, and not worry about the stoats. More specifically, the FIW predicted that the rats need to be kept at a tracking rate of ‘about 11 per cent or lower’ and possums at a tracking rate ‘below about 10 per cent’, so that the kōkako population is maintained at ‘moderate levels’ (Reference Ramsey and VeltmanRamsey & Veltman 2005, p. 914).

Figure 9 Kōkako fuzzy interaction web predictions. Imprecise predictions of the magnitude of the effect on the equilibrium kōkako fledgling rate resulting from sustained single and multispecies control of nest predators from the FIW ‘trained’ model

(reprinted from Ramsey &Veltman 2005)

How does this example hold up to the criticisms of imprecise models? Recall that the first criticism was that imprecise models are not sufficiently risky. A non-risky prediction, in this context, would be that all predator populations should be controlled. The prediction is not risky because it would be satisfied if the desired effect was achieved by culling any one, two or all three predator populations. However, neither loop analysis nor the FIWs made that sole prediction. While both models yielded the prediction that culling all three predators would have a positive effect on the kōkako population, they also made the much riskier prediction that this extensive intervention was unnecessary, and that the intervention could focus on just the rat population (loop analysis) or the rat and possum population (FIW).

The second criticism was that interventions based on imprecise models are ineffective. Yet the kōkako case is a perfect example of an effective intervention. The populations have indeed increased, and additional populations have been established. Their range has expanded to twenty-two different sites. In fact, the kōkako is now the poster child of successful conservation in New Zealand. But how do we know that these models were influential in the actual intervention? As it turns out, one of the paper’s authors (Veltman), was based at the Science and Research Unit of the New Zealand Department of Conservation, which was in charge of the kōkako conservation project. In addition, other papers she (co)authored include research on direct and indirect effects of pest (rat, stoat and possum) control on ecological communities (Reference Tompkins and VeltmanTompkins & Veltman, 2006). This work was part of a larger project within the Department of Conservation, contributing to a paper on imprecise models for deer control (also a pest in New Zealand) (Reference Ramsey, Forsyth and VeltmanRamsey et al., 2012). Finally, subsequent publications of the Department of Conservation incorporate the research and recommendations from papers by these scientists (see for example, Reference Brown, Elliott and KempBrown et al., 2015). So, it is extremely likely that these model predictions were actually used in the intervention on the kōkako and were instrumental in saving the population from extinction.

I hope that the discussion in this section has dispelled any pessimism regarding the ability of scientists to deal with ecological complexity in the systems they investigate. I have outlined examples from three distinct types of investigation, all of which allow ecologists to conduct high quality research and can form the basis of successful interventions. These types of research are (i) type II models (that yield modest generalisations), (ii) statistical tools such as meta-analysis (that give scientists the means to test and probe the scope of existing generalisations), and (iii) imprecise (type III) models that do not suffer from the limitations of the other models in cases involving poor and patchy data. I should note that my claim is not that these are the only tools that ecologists should use. I do not think that ecologists should drop all other types of inquiry and focus exclusively on these three types of research. I am also not arguing for the complete elimination of the traditional approach (exemplified by simple, general, type I models). There will undoubtedly be cases, such as the case of the V.I. marmots, where type I models are likely to be successful and should therefore be preferred. The point is that these cases are not the norm, but quite special. Their frequency is much lower than the advocates of the traditional approach seem to expect. In contrast, the situations where the other approaches outlined in this section will be useful are going to be much more frequent in complex, heterogeneous systems. Therefore, we should not be surprised, much less embarrassed, when an ecological investigation calls for type II or type III models, or when the only generalisations that can be made are quite modest.