Robustness and resilience are concepts in systems thinking that have grown in importance and popularity. For many complex social-ecological systems, however, robustness and resilience are difficult to quantify and the connections and trade-offs between them difficult to study. Most studies have either focused on qualitative approaches to discuss their connections or considered only one of them under particular classes of disturbances. In this study, we present an analytical framework to address the linkage between robustness and resilience more systematically. Our analysis is based on a stylized dynamical model that operationalizes a widely used conceptual framework for social-ecological systems. The model enables us to rigorously delineate the boundaries of conditions under which the coupled system can be sustained in a long run, define robustness and resilience related to these boundaries, and consequently investigate their connections. The results reveal the trade-offs between robustness and resilience. They also show how the nature of such trade-offs varies with the choice of certain policies (e.g., taxation and investment in public infrastructure), internal stresses, and uncertainty in social-ecological settings.
The concepts of “resilience” and “robustness” have grown considerably in popularity as desirable properties for a wide range of systems. Terms like “resilient communities” and “robust cities” have been used more frequently in public discourse (e.g., Chang and Shinozuka, 2004; Longstaff et al., 2010; Chang et al., 2014). The UK's Water Act 2014 even included “primary duty to secure resilience” as one of the general duties of its Water Services Regulation Authority (Water Act, 2014). Growing with that popularity is some confusion and potential misuse of the terms “robustness” and “resilience” due to imprecision, vagueness, and multiplicity of their definitions. Such a lack of consistency and rigor hinders advances in our understanding of the interplay between these two important system properties.
Relatively speaking, robustness has been defined more consistently and rigorously – as it can be linked to a more familiar concept of sensitivity. For example, according to Carlson and Doyle (2002), robustness in engineering systems refers to the maintenance of system performance either when subjected to external disturbances or internal uncertain parameters. In other words, in robust systems, performance is less sensitive to disturbances or uncertainty.
Robustness may very well be a desirable property of a system but it seems to come with a price. Recent research shows that tuning a system to be robust against certain disturbance regimes almost always reduces system performance and likely increases its vulnerability to other disturbance regimes (Ostrom et al., 2007; Anderies et al., 2007; Bode, 1945; Csete and Doyle, 2002; Wolpert and Macready, 1997). Now, if resilience is also a desirable property of the same system, does it also come at the expense of performance and robustness? Put it another way, is there a trade-off among performance, robustness, and resilience? Such a trade-off, if it exists, is a crucial consideration for governing and/or managing social-ecological systems (SESs).
But resilience, as alluded to above, is trickier to define. According to
Holling (1973), resilience refers to the amount of change or disruption
required to shift the maintenance of a system along different sets of
mutually reinforcing processes and structures. In other words, resilience can
be thought of as how far the system is from certain thresholds or boundaries
beyond which the system will undergo a regime shift or a quantitative change
in system structure or identity. Holling (1996) categorized resilience into
two types: engineering resilience, which refers to the ability of a system to
return to a steady state following a perturbation and ecological resilience,
which refers to the capacity of system to remain in a particular stability
domain in the face of perturbations. The latter category is used by many
researchers to discuss resilience of SESs, or more generally, coupled
infrastructure systems (CISs; Carpenter et al., 2001; Anderies et al., 2006;
Folke, 2006; Biggs et al., 2012; Barrett and Constas, 2014; Redman, 2014;
Walker et al., 2002, 2004; Gunderson and Holling, 1995; Berkes and Folke,
1998; Carpenter et al., 1999a, b; Scheffer et al., 2000; Berkes et al., 2003;
Carpenter and Brock, 2004; Janssen et al., 2004; Folke et al., 2002, 2010, 2016; Cote and
Nightingale, 2012; Mitra et al., 2015; Cumming and Peterson, 2017). The CISs term has been introduced to generalize the
notions of coupled natural human systems (CNHSs) and SESs; in this context,
But these knowledge gaps need to be filled if one wishes to make advances in understanding the interplay between social dynamics and planetary boundaries. Given the magnitude of impacts that human activities have on pushing Earth systems toward their planetary boundaries, we need clearer understanding of how social and biophysical factors come together to define the nature of these boundaries. This paper is a step in that direction. In particular, we will build on recent work that mathematically operationalizes the robustness of a SES framework (Anderies et al., 2004) into a formal stylized dynamical model (Muneepeerakul and Anderies, 2017). We will exploit the relative simplicity of the model to rigorously define robustness and resilience of the coupled system. The modeled system will be subject to uncertainty in social and ecological factors, which will affect the well-defined robustness and resilience, thereby enabling us to investigate the interplay and trade-offs between these important properties, as well as investigate how the nature of the interplay and trade-offs are affected by policies implemented by social agents.
Here we analyze a mathematical model developed by Muneepeerakul and Anderies (2017) by subjecting the coupled system to uncertainty in ecological and social factors. The model captures the essential features of a system in which a group of agents shares infrastructure to produce valued flows. Such a system is the archetype of most, if not all, of human sociality: groups produce infrastructure that they cannot produce individually (security, defense, irrigation canals, roads, markets, financial systems, coordination mechanisms, etc.) that significantly increases productivity. The challenge is maintaining this shared infrastructure (e.g., decaying infrastructure is a major problem in the US at the time of writing; ASCE RCIA Advisory Council, 2013). The model allows for mathematical definitions of the boundaries of policy domain that result in a sustainable system in which both human-made and natural infrastructure can be maintained over the long run. Based on these boundaries and uncertainty in the exogenous factors, we define metrics of resilience and robustness associated with each policy choice and investigate the trade-off between them. The basic model presented by Muneepeerakul and Anderies (2017) is described in the Appendix.
Here a policy is defined as a combination of taxation level
Direct measurement of the above-mentioned resilience, as a specified form of resilience (Walker et al., 2004), in SESs is difficult because boundaries and thresholds that separate domains of dynamics for SESs are difficult to identify (Carpenter et al., 2005; Scheffer et al., 2009, 2012). In this stylized model, however, such boundaries can be clearly identified by the stability condition (SC) and the PPC. Here, we are interested in the resilience of a system's ability to provide sufficient livelihoods for the PIPs and RUs. The basin of attraction for system resilience is defined by those system states (i.e., infrastructure state) in which this is possible, and these system states are directly mapped to the SC and PPC. We will thus define resilience metrics based on the SC and PPC boundaries. Here our goal is to develop resilience metrics that can be meaningfully compared to one another. As such, we identify some desired properties that guide the definitions of these resilience metrics. First, they should be zero at their respective boundaries. Second, positive values indicate greater resilience of the system in a desirable state. These first two properties align with how resilience has been measured, i.e., the distance from the boundary of a basin of attraction (e.g., Anderies et al., 2002; Carpenter et al., 1999a). Third, to facilitate the consideration of relative risks associated with different types of regime shifts that the system may be facing, the metrics should be comparable in magnitude. These properties guide us toward the following definitions of the resilience metrics:
Resilience metrics for a specific setting (a
We define the resilience of the system against abandonment by PIPs as
follows:
Numerical analysis of the model indicates that the equilibrium becomes
unstable when the following Routh–Hurwitz condition (e.g., May, 2001; Kot,
2001) is violated:
Following the guideline provided by the three desirable properties above, we
rearrange terms in Eq. (2) and define the resilience of the system against
instability (increased probability of collapse of infrastructure) as follows:
This allows us to meaningfully define the overall system resilience as the
minimum between the two resilience metrics, namely
As discussed earlier, robustness can be thought of as the opposite of
sensitivity. A commonly used measure of sensitivity is variance. Thus,
variance of a given function under specific disturbance or uncertainty
regimes may be used to indicate robustness of that function against those
disturbances or uncertainty regimes (robustness of what to
what). In this case, the system function of interest is the system
resilience
Variation of
Following this logic, we propose to use a “below-mean mean” as a new robustness metric: the mean of all resilience values lower than the mean. This new definition of the robustness metric has several desirable features. First, it can now be appropriately thought of as a robustness metric in the sense that the higher the value, the more robust the system (unlike the variance for which low variance means high robustness). Second, by using the mean as the threshold value for bad deviations, we remove some arbitrariness associated with prescribing a certain quantile (e.g., 5th or 10th quantile) in calculating the conditional value at risk. Third, it still carries some information about the sensitivity of the resilience metric to outside factors – the information that variance conveys; that is, the higher the below-mean mean (i.e., the bad deviations from the mean are small and the below-mean mean is close to the mean), the less sensitive – and thus more robust – the resilience metric.
In this study, we subject the modeled system to uncertainty in one natural
factor and one social factor, namely the natural replenishment rate of the
resource
The surfaces and contours of the system resilience metric,
The surfaces and contours of the robustness of
The mean,
We explore the interplay between
Resilience–robustness trade-off. Each point represents,
Pareto-optimal policies, represented by black dots, in the policy
space (
In this paper, we exploit the simplicity of a stylized model to quantitatively link resilience and robustness by computing how the CIS resilience to shocks in state variables changes with parameters. In this way, we compute the robustness of CIS resilience to uncertainty in the underlying CIS setting. The resilience metric developed here is a measure of how far the CIS is from the boundaries beyond which it will collapse. The model affords us with expressions of these boundaries, which clearly show how social and biophysical factors interplay to define these boundaries. With a concrete definition of resilience, resilience itself can be considered as the “of what” in the “robustness of what to what” notion. In particular, we use the below-mean mean of the quantitatively defined resilience metric as the metric of robustness. Consequently, this enables us to rigorously investigate the interplay between the two important, but not always well-defined, system properties. A key finding is the fundamental trade-off between resilience and robustness: there are no perfect policies in governing a CIS, only Pareto-optimal ones. Specifically, policies designed to maximize the resilience of a CIS to shocks on timescales at which the state variables play out may be very sensitive to being wrong about our understanding of the underlying dynamics of the CIS in question.
Importantly, we hope this work will stimulate further advances in rigorous studies of CISs that address such subtle, policy-relevant questions, a few of which we briefly discussed here. More dimensions can be considered in defining Pareto-optimality. Figure 5 may give an impression that the set of Pareto-optimal policies is confined to a small region in the policy space, which would imply that PIPs do not have that many choices – even in a simple CIS like the one studied. But that would be a wrong impression. In addition to resilience and robustness (as defined here), a policy maker or a social planner may be interested in other types of robustness with different “of what” and “to what” components. They may also be concerned about other system properties, e.g., productivity, user participation, etc. As more dimensions are considered, the set of Pareto-optimal policies grow. In the same spirit as that of the work done here, these other dimensions should be defined rigorously.
This work also lends itself to more rigorous studies of “adaptive
governance.” In the present study, the governance structure, represented by
a policy (a combination of
Additionally, agents in a CIS may have the capacity to change their behavior in response to changes in policy, environmental conditions, technological changes, and so on. In this study, strategic behavior and the decision-making process are assumed unchanged in the analysis. Adaptation in strategic behavior of agents will subsequently alter the nature of resilience, its robustness, and their trade-offs. Capturing such effects of adaptation requires structural changes to the model, e.g., in terms of specification of payoffs or even the formulation of the dynamical equations. With such adaptive social agents, how should one devise adaptive governance to enhance resilience and robustness of a CIS? Addressing such a question is a theoretically intriguing future research direction with great practical implications.
In keeping with the theme of “social dynamics and planetary boundaries in Earth system modelling,” our results shed light on how social and biophysical factors may interplay to define boundaries of a sustainable coupled infrastructure system. While the modeled system here is admittedly simple, our methodology and results constitute a step toward quantitatively and meaningfully combining social and biophysical factors into indicators of boundaries of more complex systems. Just as in this work, once those boundaries are clearly defined, calculation and discussion of resilience and robustness can become concrete.
No data sets were used in this article.
In this context,
Schematic diagram of the dynamical system model. Taken from Muneepeerakul and Anderies (2017).
The second variable is the resource level,
The strategic behavior of the resource users (RUs) is captured by employing
a replicator equation. Indeed, replicator dynamics provide modelers with
simple, realistic social mechanism where agents follow and replicate
better-off strategies. The two possible strategies considered for RUs are
staying inside the system with the associated payoff of
Based on the system of three differential equations (Eqs. A1, A4, and A5), the
sustainable equilibria, i.e., long-term system outcomes that satisfy the
stability condition and PPC, can be expressed
as follows:
MH, RM, JMA, and CPM designed the study. MH carried out the analysis. MH and RM prepared the manuscript with contributions from all co-authors.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Social dynamics and planetary boundaries in Earth system modelling”. It is not associated with a conference.
John M. Anderies and Rachata Muneepeerakul acknowledge the support from the grant NSF GEO-1115054. Rachata Muneepeerakul and Mehran Homayounfar acknowledge support from the Army Research Office/Army Research Laboratory. The research reported in this grant was supported in whole or in part by the Army Research Office/Army Research Laboratory under award no. W911NF1810267 (Multi-University Research Initiative). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies either expressed or implied of the Army Research Office or the US Government. The authors also thank the editor and three anonymous referees for their constructive and useful comments. Edited by: Jonathan Donges Reviewed by: three anonymous referees