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**Earth System Dynamics**
An interactive open-access journal of the European Geosciences Union

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- Articles & preprints
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ESD | Articles | Volume 9, issue 4

Earth Syst. Dynam., 9, 1159–1168, 2018

https://doi.org/10.5194/esd-9-1159-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/esd-9-1159-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Social dynamics and planetary boundaries in Earth system...

**Research article**
05 Oct 2018

**Research article** | 05 Oct 2018

Linking resilience and robustness and uncovering their trade-offs in coupled infrastructure systems

^{1}Department of Agricultural and Biological Engineering, University of Florida, Gainesville, Florida, USA^{2}School of Sustainability and School of Human Evolution and Social Change, Arizona State University, Tempe, Arizona, USA^{3}independent researcher, Gainesville, Florida, USA

^{1}Department of Agricultural and Biological Engineering, University of Florida, Gainesville, Florida, USA^{2}School of Sustainability and School of Human Evolution and Social Change, Arizona State University, Tempe, Arizona, USA^{3}independent researcher, Gainesville, Florida, USA

**Correspondence**: Mehran Homayounfar (homayounfar@ufl.edu) and Rachata Muneepeerakul (rmuneepe@ufl.edu)

**Correspondence**: Mehran Homayounfar (homayounfar@ufl.edu) and Rachata Muneepeerakul (rmuneepe@ufl.edu)

Abstract

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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.

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Homayounfar, M., Muneepeerakul, R., Anderies, J. M., and Muneepeerakul, C. P.: Linking resilience and robustness and uncovering their trade-offs in coupled infrastructure systems, Earth Syst. Dynam., 9, 1159–1168, https://doi.org/10.5194/esd-9-1159-2018, 2018.

1 Introduction

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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, *infrastructure* is broadly defined
to include human-made, social, and natural infrastructure (e.g., see
Anderies et al., 2016). The problem is that these CISs are complex and thus
identifying thresholds and potential regime shifts associated with those
thresholds is often difficult, if not impossible. In many cases, major
aspects of resilience in CISs may not be directly observable and must be
actualized indirectly via surrogate attributes (Carpenter et al., 2005;
Kerner and Thomas, 2014). Recent significant advances have been made toward
identifying early-warning signals that indicate whether a critical threshold
is being approached for a wide class of systems (Scheffer et al., 2009,
2012). Still, there are gaps in our understanding of how indicators of
resilience and robustness will behave in more complex situations. This lack
of a rigorous metric for resilience makes the investigation into their
connections, interplay, and trade-offs with robustness and performance
impossible.

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.

2 Methods

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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 *C* and the
proportion of tax revenue invested in infrastructure maintenance *y* that the
public infrastructure providers (PIPs) decide to implement in the system. The
infrastructure (e.g., canals) enable resource users (RUs) to produce valued
goods from a natural resource. The two uncertain factors are the
replenishment rate of the natural resource *g* and the wage *w* that RUs
would earn from working outside the system – a combination of *g* and *w*
defines a “social-ecological setting” or simply “setting”. There are two
boundaries that, once crossed, will cause the system to collapse. The first
boundary is called PIP participation constraint (PPC): when the PIPs must
invest too much in maintaining the public infrastructure (exceeding the
opportunity cost of *w*_{P}) and/or cannot retain enough revenue for
themselves, they will abandon the system for another. The second boundary is
the stability condition of the nontrivial equilibrium point (i.e., the
“society” in which both PIPs, for example the state, and RUs, for example
citizens, participate in and public infrastructure is sufficiently maintained
in a long run). Together, these two boundaries delineate a set of policies
(*C*–*y* combinations) that correspond to sustainable outcomes. The
resilience metric, to be developed below, can be thought of as a metric of
how far the system is from these boundaries. As the two exogenous factors
defining settings, namely *g* and *w*, change, the two boundaries and thus
the resilience metric also change with them. How sensitive the resilience
metric is to these settings is used to define robustness. Here it is worth noting that, while it
is possible to examine recovery-based resilience (or the so-called
“engineering resilience”), this paper focuses on regime shift-based
resilience (traditionally called “ecological resilience”) and its
robustness. Quantification of and trade-offs between resilience and
robustness is a novel concept that requires expositional clarity. Presenting
several metrics of resilience, let alone studying their trade-offs with
potentially different metrics of robustness, may confuse the matter and
dilute the key messages we attempt to convey. As such, in what follows, we
will focus on developing a metric for regime shift-based resilience. With the
scope of analysis clarified and suitably bounded, we will now define
resilience and robustness more formally.

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:

We define the resilience of the system against abandonment by PIPs as follows:

$$\begin{array}{}\text{(1)}& {R}_{\mathrm{PPC}}=({\mathit{\pi}}_{\mathrm{P}}/{w}_{\mathrm{P}})-\mathrm{1},\end{array}$$

where *π*_{P} is the net revenue that PIPs collect and
*w*_{P} is the opportunity cost that they will earn if they choose to
work with another system. Positive values of *R*_{PPC} indicate that
the system is resilient against being abandoned by PIPs, while negative
values indicate that the system will eventually collapse due to the PIPs'
abandonment. It is important to note that *π*_{P} results from the
coupled dynamics of the CIS; this means that *R*_{PPC} has already
integrated the dynamics of infrastructure, resource, and RUs (Eqs. A1, A4,
and A5), making it a metric of the system, not of an individual component.

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:

$$\begin{array}{}\text{(2)}& D-T\left({J}_{\mathrm{1},\mathrm{1}}{J}_{\mathrm{2},\mathrm{2}}+{J}_{\mathrm{2},\mathrm{1}}{J}_{\mathrm{1},\mathrm{2}}+{J}_{\mathrm{2},\mathrm{3}}{J}_{\mathrm{3},\mathrm{2}}+{J}_{\mathrm{1},\mathrm{3}}{J}_{\mathrm{3},\mathrm{1}}\right)>\mathrm{0},\end{array}$$

where *D*, *T*, and *J* are determinant, trace, and entries, respectively,
of the Jacobian matrix of the dynamical system (Eqs. A1, A4, and A5)
evaluated at the nontrivial equilibrium point (Eq. A6) – when such an
equilibrium point exists. Here, it is worth noting that we focus on the
equilibrium point related to the nontrivial sustainable long-term outcome.
Analyzing other bifurcations related to other equilibria may be
mathematically interesting, but it could make the study less accessible and
dilute its key message about the resilience–robustness trade-off of different
policies aiming at keeping the system in the basin of attraction of the
non-trivial sustainable long-term outcome.

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:

$$\begin{array}{}\text{(3)}& {R}_{\mathrm{stability}}={\displaystyle \frac{D}{\left|T\left({J}_{\mathrm{1},\mathrm{1}}{J}_{\mathrm{2},\mathrm{2}}+{J}_{\mathrm{2},\mathrm{1}}{J}_{\mathrm{1},\mathrm{2}}+{J}_{\mathrm{2},\mathrm{3}}{J}_{\mathrm{3},\mathrm{2}}+{J}_{\mathrm{1},\mathrm{3}}{J}_{\mathrm{3},\mathrm{1}}\right)\right|}}-\mathrm{1}.\end{array}$$

This formulation is parallel to that of the first resilience metric (Eq. 1);
it possesses the following three properties: *R*_{stability} of zero indicates
the boundary between stability and instability, positive
*R*_{stability} means the system at the equilibrium point is stable,
and the magnitudes of *R*_{stability} are comparable to those of
*R*_{PPC} (see Fig. 1). Note that *R*_{stability}, too, is determined
from the coupled dynamics of the CIS; this means that it has integrated the
dynamics of infrastructure, resource, and RUs (Eqs. A1, A4, and
A5).

This allows us to meaningfully define the overall system resilience as the minimum between the two resilience metrics, namely

$$\begin{array}{}\text{(4)}& {R}_{\mathrm{system}}=\left\{\begin{array}{l}\mathrm{Min}\left[{R}_{\mathrm{PPC}},{R}_{\mathrm{Stability}}\right],{R}_{\mathrm{PPC}},{R}_{\mathrm{stability}}\ge \mathrm{0};\\ \mathrm{0},\phantom{\rule{0.25em}{0ex}}\mathrm{otherwise}.\end{array}\right.\end{array}$$

Equation (4) implies that *R*_{system} is positive only when the
nontrivial equilibrium point (Eq. A6) exists and is stable; otherwise, the
system is considered not resilient and denoted by *R*_{system}=0;
*R*_{system} thus represents the tension between the PIPs being too
greedy (high *C* and low *y*) whereby they get close to the stability boundary
and “not greedy enough” (low *C* and high *y*) whereby they get close
to the PPC, given a particular choice for *w*_{P}. Note that the
values of *w* and *w*_{P} represent the socioeconomic embedding of
the CIS. Therefore, the biophysical structure of the CIS along with the
socioeconomic context in which it is embedded co-determine the maximum
resilience that can be achieved. Given that the nontrivial equilibrium point
exists and is stable, the following applies. If the system is at a greater risk of being abandoned
by the PIPs (and eventually collapsing), *R*_{system}=*R*_{PPC};
if the system is at a greater risk of becoming unstable (and eventually
collapsing), *R*_{system}=*R*_{stability}. Figure 1 illustrates
the relationships between *R*_{PPC}, *R*_{stability}, and
*R*_{system}.

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 *R*_{system}. By choosing *R*_{system} as our
function, we can usefully link these concepts. If we use a ball and cup
metaphor for resilience, the robustness of resilience refers to the degree at
which the geometry of the cup changes as a result of external disturbances
and/or parameter changes. However, as we will argue below, relating high
variance in *R*_{system} to low robustness may be misleading and
should not be used in evaluating a given policy. By definition, the variance
treats “good deviations” and “bad deviations” from the mean equally. For
functions with preferred values, such as resilience or profit, values greater
than the mean and those lower should not be treated in the same way.
Specifically, contribution to a high variance from a heavy tail in the good
direction should not be translated to less robustness. This problem also
arises in assessing financial risk: what makes an asset risky is the values
on the “bad tail” of the distribution (i.e., low or negative profits). This
has motivated more and more analyses to switch to considering other measures
of risk, such as the conditional value at risk, in evaluating their
portfolios of investment (Rockafellar and Uryasev, 2000; Krokhmal et al.,
2002; Sarykalin et al., 2014; AitSahlia et al., 2011; Zymler et al., 2013).
Intuitively, this means that the shape of the “cup” can be asymmetric and
we need to take this into account.

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 *g*, and the payoff that a RU earns from working outside the system
*w*. Thus, we are computing how the resilience of the system to shocks and/or
variation in state variables changes as the parameters *g* and *w* change
(i.e., we are uncertain about the underlying social-ecological setting of the
system). In particular, we assume that *g* is uniformly distributed over the
range [75, 125] and *w* is uniformly distributed over the range [0.75, 1.25].
A social-ecological setting, or setting, is defined as a combination of *g*
and *w*. For a given policy (a *C*–*y* combination), we calculate
*R*_{system} for 10 000 settings (i.e., 10 000 *g*–*w*
combinations, see Fig. 2). Then, from these 10 000 values of the resilience
metric *R*_{system}, we calculate the mean,
${\mathit{\mu}}_{{R}_{\mathrm{system}}}=E\left[{R}_{\mathrm{system}}\right]$, and use it as
the *resilience metric* of the coupled system with a given policy, and
the below-mean mean, ${\mathit{\mu}}_{R<\mathit{\mu}}=E\left[{R}_{\mathrm{system}}|{R}_{\mathrm{system}}<{\mathit{\mu}}_{{R}_{\mathrm{system}}}\right]$, as the *metric for robustness of resilience*. This metric measures the robustness of the
capacity of the system to cope with variation in state variables *I*_{HM},*R*,
and *U* to fundamental uncertainty about the underlying setting of the
system.

3 Results

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The surfaces and contours of the system resilience metric,
${\mathit{\mu}}_{{R}_{\mathrm{system}}}$, and the associated different policies (*C*–*y*)
over the policy space are shown in Fig. 3a and b, respectively. The
policies with sustainable outcomes are located in the middle of the policy
space, with ${\mathit{\mu}}_{{R}_{\mathrm{system}}}$ peaking in the center and declining as
policies become more extreme in either direction. Our analysis also shows
that ${\mathit{\mu}}_{{R}_{\mathrm{system}}}$ is more or less proportional to the fraction
of settings (*g*–*w* combinations) under which the system with that
particular policy (a *C*–*y* combination) results in a sustainable outcome
(*R*_{system}>0). A similar concept has been used in the robust
decision making literature (e.g., Groves and Lempert, 2007; Bryant and
Lempert, 2010).

The surfaces and contours of the robustness of *μ*_{R<μ} associated
with different policies over the policy space are shown in Fig. 3c and d,
respectively. The *μ*_{R<μ} “landscape” is more irregular, having two
local maxima with one being more dominant than the other. The region with
high robustness appears to be in the same general areas as the region with
high resilience. These features reflect the nonlinear interplay between the
model parameters and model structure and may affect the nature of the
trade-off between robustness and resilience reported in Figs. 4 and 5.

We explore the interplay between ${\mathit{\mu}}_{{R}_{\mathrm{system}}}$ and *μ*_{R<μ}
in Fig. 4. Figure 4 shows that there are no perfect policies in the sense
that no policies yield both maximum resilience and maximum robustness. Recall
that the robustness indicates how sensitive *R*_{system} itself is to
uncertainty in the underlying setting of the system (e.g., *g* and *w*). The
best policies are those along the Pareto frontier in the
resilience–robustness space: among this set of Pareto-optimal policies, an
increase in resilience is necessarily accompanied by a decrease in
robustness, clearly illustrating the trade-off between robustness and
resilience. Fig. 5 illustrates where the Pareto-optimal policies are located
in the policy space.

4 Discussion and conclusions

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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 *C* and *y*), is fixed. A natural next step is to
explore if a policy is allowed to change, how one may improve the resilience
and robustness of a CIS, and/or alter the nature of their inherent trade-offs.
For example, if *C* and *y* are to be functions of other factors, e.g.,
resource availability and outside incentives, what functional forms should
they take to improve the system's resilience and robustness? Indeed, in the
absence of transparent metrics, attempts to explore such adaptive policies
are severely limited.

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.

Appendix A

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*Basic model.* Here we briefly describe the basic model presented by
Muneepeerakul and Anderies (2017). The model shows dynamic behavior of three
principal variables, namely the state of the public infrastructure, *I*_{HM},
resource level, *R*, and the fraction of time a user makes use of
infrastructure, *U*, through Eqs. (A1, A4, and A5). The schematic diagram of
this system of equations is shown in Fig. A1.

In this context, *I*_{HM} depends on PIPs in term of maintenance cost
and has a positive relationship with the capacity of users to create resource
flows. Eq. (A1) illustrates the dynamics of *I*_{HM} as follows:

$$\begin{array}{}\text{(A1)}& {\displaystyle \frac{\mathrm{d}{I}_{HM}}{\mathrm{d}t}}=M\left(\mathrm{\dots}\right)-\mathit{\delta}H\left({I}_{HM}\right),\end{array}$$

where *δ* is the infrastructure's depreciation rate and *H*(*I*_{HM}) states functional relationship of public
infrastructure and productivity of each RU. According to
Muneepeerakul and Anderies (2017), many shared infrastructures can be modeled
by threshold functions. Given that *H*(*I*_{HM}) shows
threshold behavior, they used a piecewise linear function to capture such
behavior through Eq. (A2).

$$\begin{array}{}\text{(A2)}& H\left({I}_{HM}\right)=\left\{\begin{array}{l}\mathrm{0},{I}_{HM}<{I}_{\mathrm{0}};\\ h{\displaystyle \frac{{I}_{HM}-{I}_{\mathrm{0}}}{{I}_{m}-{I}_{\mathrm{0}}}},{I}_{\mathrm{0}}\le {I}_{HM}\le {I}_{m};\\ h,{I}_{HM}\ge {I}_{m};\end{array}\right.\end{array}$$

where *h* represents the maximum amount of harvest by each user under no
restriction and *I*_{0} and *I*_{m} are the lower bound and upper bound
thresholds of *I*_{HM}, respectively. Also, *M*(…) is
the maintenance function (Eq. A3) and depends on the social structure of the system.

$$\begin{array}{}\text{(A3)}& M\left(\mathrm{\dots}\right)={\mathit{\mu}}_{\mathrm{2}}yCpRUNH\left({I}_{HM}\right).\end{array}$$

In Eq. (A3), given the number of users *N*, *R**U**N**H*(*I*_{HM}) is the total harvest from the natural infrastructure. The RUs sell total harvest at price *p* to generate revenue. Subsequently, they
assign a proportion *C* of revenue to PIP's for their contribution.
Meanwhile, the PIP's spend proportion *y* of *C* on maintaining public
infrastructure through the maintenance function *M*(…).
Also, *μ*_{2} is the maintenance effectiveness of PIP's investment.

The second variable is the resource level, *R*. They assumed the dynamics of
resource to be

$$\begin{array}{}\text{(A4)}& {\displaystyle \frac{\mathrm{d}R}{\mathrm{d}t}}=G\left(R\right)-RUNH\left({I}_{HM}\right).\end{array}$$

Natural infrastructure is assumed to invoke the conservation law comprising
of regenerating capacity ($G\left(R\right)=g-\mathrm{d}R)$ and total unit
of harvest, *R**U**N**H*(*I*_{HM}). The definition presented for
*G* is the simplest model for natural infrastructure where *g* and *d* are
the natural replenishment and the loss rates, respectively.

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 ${\mathit{\pi}}_{U}=\left(\mathrm{1}-C\right)pRH\left({I}_{HM}\right)$ or leaving the system with the payoff of
*π*_{w}=*w*. According to the replicator equation:

$$\begin{array}{}\text{(A5)}& {\displaystyle \frac{\mathrm{d}U}{\mathrm{d}t}}=rU\left(\mathrm{1}-U\right)\left({\mathit{\pi}}_{U}-w\right).\end{array}$$

The replicator equation represents the fraction of time that RUs assign to
working inside system given *C* and *y*. Like RUs, there is also two
alternatives for PIPs, working inside the system or working for another CIS
which leads to system failure. Meanwhile, *C* and *y* characterize the
strategy or policy of PIPs. The PIPs will participate in this coupled system
only when ${\mathit{\pi}}_{p}=\left(\mathrm{1}-y\right)pCRUNH\left({I}_{HM}\right)\ge {w}_{\mathrm{P}}$. In other words, the PIPs maintain the system when they are
better-off than working outside. This condition is termed the PIP
Participation Constraint (PPC).

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:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{i}_{HM}^{*}={\displaystyle \frac{yC{U}^{*}N{R}^{*}}{g}}H\left({I}_{HM}^{*}\right);{R}^{*}={\displaystyle \frac{g}{d}}\left(\mathrm{1}-{\displaystyle \frac{{i}_{HM}^{*}}{yC}}\right);\\ \text{(A6)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{U}^{*}={\displaystyle \frac{\left(\mathrm{1}-C\right)}{yC}}{\mathit{\varphi}}_{\mathrm{1}}{i}_{HM}^{*};\end{array}$$

where ${i}_{HM}^{*}=\frac{{I}_{HM}^{*}\mathit{\delta}}{{\mathit{\mu}}_{\mathrm{2}}pg}$
(indicates dimensionless) and ${\mathit{\varphi}}_{\mathrm{1}}=\frac{pg}{wN}$, a dimensionless
group representing the relative lucrativeness of the system, namely the ratio
of potential income – with the entire resource flow turned into
income – relative to outside wage. The results reported in this study are
based on the following parameter values: *h*=0.0005; *δ*=0.1;
*I*_{0}=0.3; *I*_{m}=3; *g*=100; *d*=0.02; *N*=1000; *r*=0.15; *p*=10; *w*=1; *w*_{p}=100; and *μ*_{2}=0.001.

Author contributions

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Author contributions.

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.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Social dynamics and planetary boundaries in Earth system modelling”. It is not associated with a conference.

Acknowledgements

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Acknowledgements.

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

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Short summary

For many complex social-ecological systems, robustness and resilience are difficult to quantify and the connections and trade-offs between them difficult to study. In this study, we present an analytical framework to address the linkage between robustness and resilience more systematically. The results reveal the trade-offs between robustness and resilience. They also show how the nature of such trade-offs varies with the choices of certain policies, internal stresses, and external disturbances.

For many complex social-ecological systems, robustness and resilience are difficult to quantify...

Earth System Dynamics

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