Structural Loop Analysis of Complex Ecological Systems☆
Introduction
Socio-ecological system models represent the interconnected nature of society and the environment. These systems are complex, able to exhibit emergence and self-organisation, with behaviours arising endogenously through non-linear dynamics (Güneralp, 2006). A defining characteristic of socio-ecological systems is multiple feedback loops which collectively form the internal structure of the system (Meadows, 2009), but disentangling and prioritising those feedbacks in order to understand system behaviour and develop effective policy is no simple task. Indeed, a necessary condition for labelling a system a socio-ecological system is that of a feedback loop operating between social and ecological elements. It is such feedback loops that are often the primary drivers of emergent system behaviour (Sterman, 2001). Here, we use the term ‘driver’, in the context of structural feedback loops, to mean the main endogenous cause of a system's behaviour. Systems may exhibit strong non-linear dynamics which are explored with the concepts of critical transitions between alternative stable states, regime shifts, and tipping points with potentially hard or effectively impossible to reverse changes in state due to properties of hysteresis (Scheffer, 2009; Carpenter, 2005). Consequently socio-ecological systems are hard to understand, hard to predict and difficult to manage (Meadows, 2009). Maintaining socio-ecological systems in desirable states and understanding why their behaviours change through time is fundamental for economic growth, poverty alleviation and general wellbeing (United Nations, 2015; Scheffer, 2009).
Process based, mechanistic, bottom up modelling has been used to understand socio-ecological systems (Verburg et al., 2016). System dynamics is one methodology that can be used to increase our understanding of such systems. System dynamics models are structural representations of dynamic real world systems. They take a resource based view of the world, characterising a system through a set of stocks and flows in order to represent its structure. Stocks are often, but not only, material goods, and flows are pathways of material between stocks (Ford, 2010).
System dynamics has an established track record of being applied to ecological and socio-ecological modelling (e.g. Ford, 2010; Meadows, 2009; Dyson and Chang, 2005; Saysel et al., 2002; Vezjak et al., 1998). Powerful and intuitive software packages such as Vensim (Ventana Systems, Inc., 2006) and STELLA (isee systems, 2016), the exchange of established models and modelling libraries allows potentially very complex systems to be represented with models that produce output via computationally efficient numerical integration schemes. While this has allowed a wide range of system dynamic models to be developed, it has, at times, produced models that are difficult to assess with regards to their overall utility in increasing our understanding of real world systems. Such models can be challenging to parameterise, validate and interpret (Voinov and Shugart, 2013). The risk is that some system dynamics models are essentially black box representations of the target system making them effectively as hard to interpret as their real world counterparts (Voinov and Shugart, 2013).
In this study, we investigate a methodology that could increase our understanding, and potentially prediction, of large changes in system structure and functioning through a quantitative analysis of feedback loops as endogenous determinants of system behaviour. Rather than searching system-level properties and variables for statistical properties of impending critical transitions (Scheffer, 2009; Scheffer and Carpenter, 2003), we instead focus on the structural properties of the system which drives such behaviour.
We are motivated to understand how these sub-processes function collectively in producing system behaviour. One analogy is that if the system dynamics model is an organism that we can observe via its output, then we seek to understand the processes that drive such behaviour by peering within the model in order to identify ‘organs’ and ‘physiological processes’. This analysis can be used in conjunction with evaluation of the system output, the system's stability, and identification of the most important components with respect to specific behaviours. In this study we investigate the mechanisms responsible for generating stability and instability within a system, how these change through time, whether stability or instability is dominated by an individual driver or generated by several, and how these drivers change in dominance as a system undergoes a transition between alternative stable states.
The technique explored within this study is known as Loop Eigenvalue Elasticity Analysis (LEEA). LEEA expands on the knowledge gained from linear stability analysis and graph theory, identifying a set of feedback loops within a system's structure known as the Shortest Independent Loop Set or SILS (Oliva, 2015; Oliva, 2004), which are collectively responsible for generating stability and instability within the system. A description of SILS, what it does and, and how it is applied can be found within the Supplementary information Section 1 of this paper. LEEA then structurally analyses the loop set, identifying which feedback loops are dominating the system's behaviour at any point in time, generating a hierarchy of the influential feedback loops of the system.
Exploring ecosystem dynamics through the study of feedback loops has already shown potential to improve our mechanistic understanding of critical transitions and stability within lake systems (Kuiper et al., 2015). While the methodology of Kuiper et al. (2015) focusses primarily on food webs, their motivations of finding feedback loops within a lake ecosystem in order to determine stability and critical transitions between two regimes is similar to this study.
Previous research has demonstrated that LEEA can increase understanding of system behaviour and causal drivers across a range of model systems (Oliva, 2015; Kampmann, 2012; Kampmann and Oliva, 2008; Güneralp, 2006; Güneralp, 2005). Thus far the method has only seen limited use in the field of socio-ecology in the context of agriculture (Bueno, 2013; Bueno, 2012) and the Baltic cod fishery as a potential practice to be undertaken after conducting generalized modelling (Lade et al., 2015). Here we extend this work and evaluate LEEA in the context of critical transitions and regime shifts, implementing loop analysis of a small lake model which can undergo critical transitions between clear and turbid states as a consequence of human drivers.
A full explanation of the limitations of the LEEA technique, along with many solutions to these limitations has been addressed by Güneralp (2006). Efforts to make the technique more automated have been conducted by Sergey Naumov and Rogelio Oliva and can be found online (Naumov and Oliva, n.d.).
The model chosen to demonstrate the application of LEEA has been developed from Carpenter (2005) which formulated a simple model of a shallow lake, Lake Mendota in Wisconsin, USA, using empirical data for soil, lake and sediment phosphorus levels. The model was bistable as increasing phosphorus input in the lake produced a critical transition with a sudden shift from a clear to a turbid state. Shallow lakes are classic examples of bistable systems, capable of discrete transitions from clear to eutrophic conditions (Wang et al., 2012) and their properties are relatively well known (Carpenter et al., 2011; Carpenter, 2005; Ludwig et al., 2003; Scheffer and Carpenter, 2003; Scheffer, 1998) with current theories attributing many eutrophic regime shifts to large influxes of phosphorus through anthropogenic activity such as fertiliser runoff from farms in the lake catchment area. The model has been chosen for two principle reasons: 1) the main focus of the model's dynamic behaviour is a critical transition, allowing for an investigation of feedback loop behaviour around the point of a critical transition. 2) The model is relatively simple, allowing for a quantitative account of LEEA to be presented, and assessment of LEEA's utility for the analysis of such systems.
Section snippets
Lake Eutrophication
Lake Mendota is a shallow freshwater lake surrounded by agricultural fields which receive ample supplies of phosphorus fertiliser. Soil erosion leads to excess phosphorus from the fertiliser, not taken up by vegetation, to be washed into surrounding streams and rivers, eventually leading to the lake. This process concentrates phosphorus runoff from the lake's catchment area into lake water where the excess of nutrients causes algal blooms to form. The formation of these blooms leads to plant
Methodology
The following section gives an overview of the steps required to undertake LEEA. The formalism and expansions on the following steps can be found in the Supplementary information Section 1.
Calculation of loop eigenvalues and loop influence values is conducted via the following process:
- 1)
The Jacobian Matrix of the linearized dynamical system model
- 2)
Eigenvalues of the Jacobian Matrix
- 3)
Loop Gain
- 4)
Loop Elasticity & Loop Influence
- 1)
The Jacobian Matrix is an n × n square matrix where each element of the matrix
The PLUM Model
Carpenter's study of Lake Mendota (Carpenter, 2005) is represented using system dynamic terms in the form of the PLUM model ((P)Phosphorus (L)Loops in (U)Soil and (M)Sediment) (Fig. 1).
System dynamic modelling is carried out using Vensim, which numerically integrates Carpenter's series of ordinary differential equations. The model allows the user to visualise the number, polarity, position and interaction of the system's feedback loops alongside providing the ability to easily implement and
Discussion
Overall the results of LEEA between the forward and reverse critical transition present very similar results. The instability that is seen building up in the system years prior to the critical transition events can largely be attributed to Loop 5, a positive feedback loop (Fig. 4a) containing phosphorus recycling, while stability in the system is maintained by Loop 2, a negative feedback loop (Fig. 4b) responsible for the outflow of phosphorus from the lake.
In this study, LEEA has been shown to
Conclusion
This study has shown how a structural loop analysis technique known as Loop Eigenvalue Elasticity Analysis (LEEA) can be applied to analyse the dynamics of a simple model of a shallow lake system that undergoes critical transitions between clear and turbid states. Analysing a system which contains feedback loops can help reveal significant loop structures operating within a system's behaviour: a non-trivial problem even with apparently simple systems. LEEA is able to show how feedback loops can
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This work was supported by an EPSRC Doctoral Training Centre grant (EP/G03690X/1).