Cross-scale cooperation enables sustainable use of a common-pool resource

Abstract

In social-ecological systems (SESs), social and biophysical dynamics interact within and between the levels of organization at multiple spatial and temporal scales. Cross-scale interactions (CSIs) are interdependences between processes at different scales, generating behaviour unpredictable at single scales. Understanding CSIs is important for improving SES governance, but they remain understudied. Theoretical models are needed that capture essential features while being simple enough to yield insights into mechanisms. In a stylized model, we study CSIs in a two-level system of weakly interacting communities harvesting a common-pool resource. Community members adaptively conform to, or defect from, a norm of socially optimal harvesting, enforced through social sanctioning both within and between communities. We find that each subsystem’s dynamics depend sensitively on the other despite interactions being much weaker between subsystems than within them. When interaction is purely biophysical, stably high cooperation in one community can cause cooperation in the other to collapse. However, even weak social interaction can prevent the collapse of cooperation and instead cause collapse of defection. We identify conditions under which subsystem-level cooperation produces desirable system-level outcomes. Our findings expand evidence that collaboration is important for sustainably managing shared resources, showing its importance even when resource sharing and social relationships are weak.

1. Introduction

In social-ecological systems (SESs), social and biophysical dynamics interact within and between the levels of organization at multiple spatial and temporal scales. The Anthropocene is characterized by global changes emerging from local human–environment interactions. Global changes, in turn, cause local impacts and most attempts to manage them also must be deployed locally [1]. The multiscale nature of SES dynamics and governance makes characterization of cross-scale dynamics important, to help understand how changes implemented at one level will propagate across scales [2–9].

According to hierarchy theory, when a complex system’s characteristic levels are organized at different scales, they are ‘quasiseparable’; when modelling one level, variables at other levels may be treated as constant boundary conditions [9,10]. Under this assumption, causes of nonlinear behaviour at one level, such as threshold behaviour, must lie at the same level. However, SESs are not separable in this way. If a variable at some level approaches a critical threshold, even a small perturbation from an interaction with a variable at another level may drive it over the threshold, causing nonlinear behaviour such as a regime shift [9,11,12]. Such sensitive dependence between processes at different scales, which generates behaviour unpredictable from behaviour at single scales, is called a cross-scale interaction (CSI) [12]. Many researchers have qualitatively described CSIs in SESs, conceptually [7,9,11,13–17] and in specific case studies (recently, [2,18–21]). CSIs have been statistically quantified from data at multiple scales. Brondizio et al. [22] analysed Amazon deforestation data at different length scales, showing that understanding deforestation trajectories requires differentiating causes at different scales. Soranno et al. [4] quantified CSIs using multiscale data and Bayesian hierarchical statistical models. Applying this method to a study of lake water quality across different regions of North America, they identified a so-called scale mismatch between quality variation and water management.

To date, however, there have been few attempts to model CSIs in SESs in terms of underlying mechanisms. Peters et al. [8] proposed a mathematical framework for nonlinear dynamics in catastrophic events, in terms of pattern-process relationships, including feedbacks, across scales. They applied this framework to characterize the spread of wildfires, infectious diseases and insect outbreaks, finding that strategies for mitigating risks of such events must account for CSIs and are often counterintuitive. Recently, Lansing and coworkers reported [23] an agent-based model of water and pest control management in Bali’s ancient rice terraces, to help explain shifts from individualistic field-level practices to cooperative regional practices. Modelling agent interactions and management decisions using evolutionary game theory on a spatially embedded lattice showed that long-established spatial patterns observable in the rice terraces can be created by feedback between farmers’ decisions and the ecology of the paddies, triggering a transition from individualistic to cooperative practices. In the theoretical middle ground between generic frameworks and detailed case models lies the potential for stylized, minimal models. These aim to capture essential system features while being simple enough to yield insights into mechanisms. This is helpful for guiding thought experiments relating CSIs to underlying mechanisms and for interpreting empirical case study results.

Here we develop a stylized model of resource management by social pressure in a multilevel SES, accounting for CSIs. Focusing on common-pool resource management, we extend prior work by Tavoni, Schlüter & Levin (TSL) [24], who used evolutionary game theory to model a community of agents harvesting a renewable common-pool resource as a basis for economic production. Agents adaptively chose whether to conform to, or defect from, a norm of socially optimal harvesting, enforced through social sanctioning. Schlüter et al. [25] extended the TSL model, using agent-based methods to test the robustness of norm-driven cooperation to environmental variations. One was the introduction of a neighbouring community and resource pool, with diffusive resource transfer between the pools. Results showed that stably high cooperation in one community could cause cooperation in the other to collapse because added resource availability through the pools’ coupling increased the payoff for defectors. However, the communities did not adjust their norms for added resource availability, and no social interactions between communities were considered.

We extend this analysis by adding two types of social interaction and characterizing how interactions between local-level and system-level behaviours affect outcomes at the system level. First, we consider how communication between communities while setting harvesting norms can adjust the norms for biophysical coupling. Second, we add ongoing social coupling such that norm cooperators can sanction defectors in both communities. We assume that interactions within subsystems are much stronger than interactions between them. This allows us to quantify how nonlinear changes in subsystem behaviour ((dis)appearance of dynamical attractors) arise from biophysical and social CSIs between subsystems. We identify conditions under which subsystem-level cooperation generates desirable system-level outcomes and conditions which prevent this.

The exact system model is analytically insoluble and we use perturbation theory [26] to derive an approximate model in the weak coupling (multiscale) regime. This model closely approximates numerical solutions of the exact model. Solving our model with a combination of analytical and numerical methods characterizes the parameter space more efficiently than purely numerical methods, and the derived equations elucidate mechanisms. This dual approach is also a basis for future development using more advanced multiscale analysis methods from dynamical systems theory and statistical physics, which have demystified numerous complex systems [27–30]. We envisage that, in future, a combination of mathematical and agent-based stylized multiscale models may enable an effective theory of CSIs in SESs, complementing statistical and qualitative analyses from multiscale empirical studies. This may help empirical researchers to quantify CSIs and mechanisms underlying them, revealing new leverage points for governance of real systems.

2. Model

We consider the two-level model schematized in figure 1, comprising two interacting social-ecological subsystems, each of which, in isolation, is described by the TSL model [24]. In each, a structureless community of agents harvests a structureless common-pool resource (i.e. social couplings between all agents are equal and the resource pool is well mixed, so all agents have equal resource access). Each agent adaptively conforms to, or defects from, a norm of socially optimal harvesting, choosing based on the resource availability and the utilities of strategies chosen by other agents. The biophysical variables of interest are the resource stocks, and social variables the norm-cooperative fractions of the subsystem populations. The two are directly coupled only within each subsystem (agents harvest only locally), but the biophysical and social variables are coupled between subsystems. Social interaction between subsystems involves two steps: first, communities may choose to share information about their resources while establishing harvesting norms; second, cooperators sanction defectors to enforce the norms.

Figure 1.

Model system with social and biophysical interactions between symmetrical subsystems. In each, a community of agents extracts a renewable common-pool resource and each adaptively chooses between cooperative (low) and defective (high) extraction effort levels, constrained by a social norm of non-excessive harvesting (quantified by ${E_c^{\ast }}^{(i)}$). Cooperators enforce the norm by sanctioning defectors. Biophysical coupling (strength δ) permits resource transfer between subsystems. Communities may choose to share resource information while setting norms; we assume that this communication is either perfect or nonexistent. Ongoing social coupling (strength λ) enables cooperators to sanction defectors in both communities. (Online version in colour.)

(a). Biophysical dynamics

We model subsystem resource dynamics as the sum of local processes and subsystem interactions. Following TSL [24,25], for each subsystem we use a simple resource model based on an aquifer, which also provides a basis for more generic reasoning about renewable resources. The resource stock of, for example, subsystem 1 (R⁽¹⁾(t)) is replenished by its environment at a constant rate (c⁽¹⁾) and lost at a rate that depends quadratically on the resource occupation [31] (ratio of R⁽¹⁾(t) to its maximum value, $R_{\mathrm{m}}^{(1)}$, determined by limits such as reservoir capacity). The resource is also harvested by the subsystem community with collective effort E⁽¹⁾(t), and transferred diffusively between subsystems at a rate that depends linearly on the resource occupation gradient (e.g. by Darcy’s law for groundwater flow through a porous medium [32]). Under the combination of these processes, subsystem 1’s resource stock evolves according to the first-order, nonlinear ordinary differential equation,

$$\begin{array}{r}{\dot{R}}^{(1)}(t)=c^{(1)}-d^{(1)}{\left(\frac{R^{(1)}(t)}{R_{\mathrm{m}}^{(1)}}\right)}^2-\hspace{0.17em}\hspace{0.17em}q{E}^{(1)}(t)R^{(1)}(t)+r\left(\frac{R^{(2)}(t)}{R_{\mathrm{m}}^{(2)}}-\frac{R^{(1)}(t)}{R_{\mathrm{m}}^{(1)}}\right).\end{array}$$ 2.1

Here, d⁽¹⁾ is a coefficient particular to a given system and q (the ‘technology factor’) is constant. In real systems, resource transfer coupling, r may depend on flow direction due to, for example, constraints of geography or institutional regulations. For simplicity, we assume that r is independent of direction, and that both subsystems have equal resource capacities and replenishment and dissipation coefficients ($R_{\mathrm{m}}^{(i)}\equiv R_{\mathrm{m}}$, c⁽ⁱ⁾ ≡ c, d⁽ⁱ⁾ ≡ d). To fairly compare the effects of biophysical and social subsystem interactions, we define dimensionless variables, ρ⁽ⁱ⁾ = R⁽ⁱ⁾/R_m and τ = (d/R_m)t, and rewrite (2.1) as

$$\begin{array}{r}{\dot{\rho }}^{(1)}(\tau )=\frac{c}{d}-{\rho }^{(1)}{(\tau )}^2-\frac{q{R}_{\mathrm{m}}}{d}{E}^{(1)}(\tau ){\rho }^{(1)}(\tau )+\frac{r}{d}({\rho }^{(2)}(\tau )-{\rho }^{(1)}(\tau )).\end{array}$$ 2.2

In a given system, it may be possible to quantify how coupling strength depends on space and time (e.g. if com-munities harvest water from connected bodies, the flow rate between them depends on distance [32]). However, for generality, we assume that interactions between subsystems are much weaker than interactions within them, and formalize this with a weak coupling parameter, δ ≡ (r/d), 0 ≤ δ ≪ 1. This assumption of multiple scales allows us to treat system-level resource transfer as a small perturbation to subsystem-level dynamics and, therefore, using regular perturbation theory [26], to assume that (2.2) has solutions of the form

$${\rho }^{(1)}(\tau )={\rho }_0^{(1)}(\tau )+\delta {\rho }_1^{(1)}(\tau )+{\delta }^2{\rho }_2^{(1)}(\tau )+\cdots .$$ 2.3

Substituting (2.3) into (2.2) and truncating at first order in δ yields coupled differential equations for the isolated subsystem resource dynamics and a perturbation due to system-level resource transfer, respectively,

$${\dot{{\rho }_0}}^{(1)}(\tau )=\frac{c}{d}-{\rho }_0^{(1)}{(\tau )}^2-\frac{q{R}_{\mathrm{m}}}{d}{E}^{(1)}(\tau ){\rho }_0^{(1)}(\tau )$$ 2.4a

and

$$\begin{array}{r}{\dot{{\rho }_1}}^{(1)}(\tau )={\rho }_0^{(2)}(\tau )-{\rho }_0^{(1)}(\tau )(1+2{\rho }_1^{(1)}(\tau ))-\frac{q{R}_{\mathrm{m}}}{d}{E}^{(1)}(\tau ){\rho }_1^{(1)}(\tau ).\end{array}$$ 2.4b

In §S2 of the electronic supplementary material, we show that this description very closely approximates the complete description (2.2). Perturbation theory makes the system mathematically tractable and links it to the core concepts of social-ecological resilience [5], which may be defined as a SES’s capacity to absorb perturbations while maintaining structure and function [3,5]. Our analysis directly assesses subsystem resilience to cross-scale perturbations.

(b). Social dynamics

Total community extractive effort is the sum of the efforts of all cooperators and defectors. For community 1, comprising n⁽¹⁾ agents,

$$E^{(1)}(\tau )=n^{(1)}{e_c}^{(1)}({\mu }^{(1)}+(1-{\mu }^{(1)}){\varphi }^{(1)}(\tau )).$$ 2.5

Here, 0 ≤ ϕ⁽¹⁾(τ) ≤ 1 is the population fraction made up of cooperators, who each extract with socially optimal (lower) effort, ${e_c}^{(1)}$ while the remaining defectors exert (higher) ${e_d}^{(1)}$. We define the effort factor, ${\mu }^{(1)}={e_d}^{(1)}/{e_c}^{(1)}$. Community productivity is assumed to take Cobb–Douglas form in effort and resource [24]

$$\begin{array}{rl}\hspace{0.17em}{f}^{(1)}(\tau )& =\gamma {[E^{(1)}(t)]}^{\alpha }{[R^{(1)}(t)]}^{\beta }\\ & =\gamma R_{\mathrm{m}}^{\beta }{[E^{(1)}(\tau )]}^{\alpha }{[{\rho }^{(1)}(\tau )]}^{\beta },\end{array}$$ 2.6

where γ is the total factor productivity and α, β are constants. Payoffs for cooperators (c) and defectors (d) are [24]

$${{\pi }_x}^{(1)}(\tau )={e_x}^{(1)}\left(\frac{\hspace{0.17em}{f}^{(1)}(\tau )}{E^{(1)}(\tau )}-w\right),x=c,d,$$ 2.7

where w is the cost of production. Each defector experiences social sanctioning applied by the cooperative fractions of both communities, weighted by community populations and social couplings, λ^(ij),

$${\omega }^{(1)}(\tau )=s(n^{(1)}{\text{e}}^{T{\mathrm{e}}^{g{\varphi }^{(1)}(\tau )}}+{\lambda }^{(21)}{n}^{(2)}{\text{e}}^{T{\mathrm{e}}^{g{\varphi }^{(2)}(\tau )}}),$$ 2.8

where s is the maximal sanctioning that an individual can apply. For consistency with the original TSL model [24,25], we assume that cooperators bear no cost for sanctioning defectors. We assume symmetrical populations and social couplings (n⁽ⁱ⁾ ≡ n; λ^(ij) ≡ λ), but note that they may differ in reality for many reasons. Mechanisms of social coupling are particular to a given system. For generality, we again impose multiple scales by assuming that couplings are much weaker between individuals in different communities than in the same community (0 ≤ λ ≪ 1). Figure 2 plots the sanctioning experienced by a defector in subsystem 1 due to cooperators in both subsystems. The dependencies on ϕ⁽¹⁾ and ϕ⁽²⁾ are both nonlinear, with thresholds at ϕ⁽ⁱ⁾ ∼ 0.35. The curve ω⁽¹⁾ = ω⁽¹⁾(ϕ⁽¹⁾) at ϕ⁽²⁾ = 0 is identical to that in figure 2 of Tavoni et al. [24]. Although the nonlinear factors of both terms in (2.8) are identically parameterized, ω⁽¹⁾ depends more strongly on ϕ⁽¹⁾ than ϕ⁽²⁾ because the second term is scaled by the weak social coupling. The two communities’ sanctioning components add, producing effects not seen when sanctioning is only local. Defectors in community i experience nonzero sanctioning even when ϕ⁽ⁱ⁾ is sub-threshold, provided ϕ^(j) (j ≠ i) is above its threshold (${\varphi }^{(i)}\lesssim 0.35\lesssim {\varphi }^{(j)}$). Moreover, when cooperators dominate both communities (ϕ⁽ⁱ⁾ > 0.5), defectors in each experience more sanctioning than can be applied by their local communities alone.

Figure 2.

Sanctioning experienced by each defector in subsystem 1 due to the cooperative fractions of the subsystem 1 and 2 populations. Parameters: n = 50, λ = 0.2, s = 0.34/n = 6.8 × 10⁻³, T = −150, g = −10. The s value is derived from the value assumed in [24] for the maximum sanctioning applicable by the whole community, h = 0.34. (Online version in colour.)

Each community’s cooperative fraction evolves according to mean-field replicator dynamics [33]. For subsystem 1, this is written as by TSL [24] but using our dimensionless variables

$$\begin{array}{rl}{\dot{\varphi }}^{(1)}(\tau )& =\frac{R_{\mathrm{m}}}{d}{\varphi }^{(1)}(\tau )(1-{\varphi }^{(1)}(\tau ))\left(\frac{{{\pi }_d}^{(1)}(\tau )-{{\pi }_c}^{(1)}(\tau )}{{{\pi }_d}^{(1)}(\tau )}\right)({\omega }^{(1)}(\tau )\\ & -{{\pi }_d}^{(1)}(\tau )).\end{array}$$ 2.9

We assume that whereas sanctioning operates both within and between communities, agents compare harvesting strategy utilities only within their local communities. This is for simplicity and to reflect evidence that, due to in-group bias, people preferentially compare themselves to in-group members [34] while also being likely to sanction out-group members for norm violation [35,36].

(c). Cases for modelling coupled system

When δ = λ = 0, each subsystem is described by the original TSL model [24], which we summarize in electronic supplementary material, §S1 as a benchmark for subsystem behaviour. Its social state space includes three types of dynamical attractor (stable fixed point): purely defective (ϕ⁽ⁱ⁾* = 0), purely cooperative (ϕ⁽ⁱ⁾* = 1) and cooperator-dominated mixed (0.5 < ϕ⁽ⁱ⁾* < 1), which we label D, C and $\overline{\text{C}}$, respectively. Each subsystem’s stability landscape may be characterized by the combination of attractors present, which varies across the parameter space (see electronic supplementary material, §S1). In the sections below, we investigate how biophysical and social perturbations due to subsystem couplings affect the subsystem stability landscapes, relative to an isolated subsystem, in different cases. First, we quantify how biophysical coupling alters each subsystem’s norm of socially optimal harvesting when communities share resource information during norm setting. Two scenarios are obtained (electronic supplementary material, §S3), in which norms are and are not adjusted for coupling. We compute subsystem 1’s biophysical and social fixed points for both scenarios, in cases where coupling is purely biophysical, purely social, or a combination of the two (note that these different coupling configurations span the full range of ‘one-to-one resource access’ SES network motifs proposed by Bodin & Tengö [37]). For logical flow from earlier work, we adopt the parameter values used by TSL [24] for each subsystem, and in electronic supplementary material, §S6 we analyse our findings’ robustness to changes in these parameters. Throughout, we fix coupling strengths such that interactions between subsystems are at least twice as weak as interactions within them (0 ≤ δ ≤ 0.5, 0 ≤ λ ≤ 0.5) and also fix subsystem 2’s effort factor, μ⁽²⁾ arbitrarily at 70% of the range of μ values in an isolated subsystem (μ⁽²⁾ = 0.70(μ_N − 1) + 1 = 2.95). This is because the range of μ⁽ⁱ⁾ depends on the biophysical coupling strength, so varies across the parameter space (electronic supplementary material, §S3). For consistency, therefore, μ⁽²⁾ is fixed relative to the isolated case.

Prior analysis [25] simulated the full coupled subsystem dynamics, for purely biophysical coupling, under each parameter combination. Initial conditions were chosen (ϕ⁽²⁾(0) always 0.9, ϕ⁽¹⁾(0) varied) and the system numerically evolved to equilibrium. Here, since we assume equilibrium and analyse the stability landscape, we instead fix subsystem 2’s long-time behaviour and ignore its transient behaviour, which may originate anywhere in the chosen equilibrium’s basin of attraction. For each scenario of norms and couplings, we consider cases in which subsystem 2 equilibrates at each of its monomorphic fixed points (ϕ⁽²⁾* = 0, 1). Although stabilities of these fixed points vary across the parameter space, they are always accessible through an appropriate choice of initial conditions. This is not true for the mixed fixed points, which vary in both stability and coordinates. Note that when ϕ⁽²⁾* = 1, subsystem 1’s behaviour is independent of μ⁽²⁾ because subsystem 2 has no defectors. For each case, we address two questions: (1) Is subsystem 1’s stability landscape resilient to cross-scale perturbations from subsystem 2? (2) Does subsystem 2’s long-time behaviour promote or prevent similar long-time behaviour in subsystem 1 and, therefore, the overall system (both subsystems together)?

3. Results

(a). Purely biophysical coupling

We first study cases in which 0 < δ ≤ 0.5, λ = 0. Evaluating (S6b) in the electronic supplementary material determines shifts in biophysical equilibria across the parameter space (electronic supplementary material, figure S3). As (S6a) in the electronic supplementary material makes clear, ${{\rho }_1^{(1)}}^{\ast }$ varies in proportion to the difference in zeroth-order resource equilibria. This difference depends on the social fixed point at which subsystem 2 is assumed to equilibrate. When ϕ⁽²⁾* = 0 (electronic supplementary material, figure S3a,b), subsystem 2 is purely defective, the corresponding high level of extractive effort results in lower ${\rho }_0^{(2)\ast }$, and the equilibrium first-order dynamics persistently transfer resource from subsystem 1 to 2. Conversely, when ϕ⁽²⁾* = 1 (electronic supplementary material, figure S3c,d), the lower extractive effort gives higher ${{\rho }_0^{(2)}}^{\ast }$ and the first-order equilibrium persistently transfers resource from subsystem 2 to 1.

Solving (2.9) when ${\dot{\varphi }}^{(1)}=0$ yields monomorphic social fixed points at ϕ⁽¹⁾* = 0, 1, and shifted mixed social fixed points on loci generally defined by ${\omega }^{(1)}({{\varphi }^{(1)}}^{\ast },{{\varphi }^{(2)}}^{\ast })=$ ${\pi }_d^{(1)}(e_c,{\mu }^{(1)},{\mu }^{(2)},{{\varphi }^{(1)}}^{\ast },{{\varphi }^{(2)}}^{\ast })$. More explicitly,

$$\begin{array}{rl}sn\hspace{0.17em}({\text{e}}^{T{\mathrm{e}}^{g{{\varphi }^{(1)}}^{\ast }}}+\lambda {\text{e}}^{T{\mathrm{e}}^{g{{\varphi }^{(2)}}^{\ast }}})& =e_c{\mu }^{(1)}\left(\gamma {[{E^{(1)}}^{\ast }]}^{\alpha -1}\left[\frac{R_{\mathrm{m}}}{2Q({E^{(1)}}^{\ast })}\right.\right.\\ & \times (\delta (2({E^{(1)}}^{\ast }-{E^{(2)}}^{\ast })-Q({E^{(1)}}^{\ast })+Q({E^{(2)}}^{\ast }))\\ & {+2Q({E^{(1)}}^{\ast })(\hspace{0.17em}Q({E^{(1)}}^{\ast })-2{E^{(1)}}^{\ast }))]}^{\beta }-w),\end{array}$$ 3.1

where $Q({E^{(i)}}^{\ast })=\sqrt{1+4{{E^{(i)}}^{\ast }}^2}$. Figure S4 in the electronic supplementary material plots ω⁽¹⁾ and ${\pi }_d^{(1)}$, with curve intersections corresponding to mixed social fixed points because ${\dot{\varphi }}^{(1)}(\tau )=0$ when ${\omega }^{(1)}(\tau )={{\pi }_d}^{(1)}(\tau )$ in (2.9); the defector sanctioning and pay-off balance.

Figure 3a shows shifted biophysical and social fixed points in subsystem 1 under purely biophysical subsystem coupling. Results are shown for the upper bound of the weak coupling range (δ = 0.5, which we call moderate coupling), for both norm scenarios, when subsystem 2 has equilibrated at its purely cooperative social equilibrium. When ϕ^(2)* = 0 instead, subsystem 1 shows behaviour qualitatively similar to an isolated subsystem (electronic supplementary material, §S1), and is therefore resilient to the cross-scale biophysical perturbation, across most of the parameter space (electronic supplementary material, figure S7). However, figure 3a(i) shows a qualitative change (phase transition) in the stability landscape; when ${\mu }^{(1)}\gtrsim 2.9$, there are no mixed social fixed points because defector payoff exceeds sanctioning for all values of ϕ⁽¹⁾ (electronic supplementary material, figure S4b). Intuitively, this is because biophysical coupling gives subsystem 1 agents access to extra resource from subsystem 2, increasing their defector payoff, while without social coupling, subsystem 2 cannot provide extra sanctioning to balance this payoff. Only the D attractor is present in this region so, for any ϕ⁽¹⁾(0) < 1, collapse of cooperation is inevitable. This constitutes a CSI, since the weak perturbation from system-level dynamics generates a phase transition in subsystem 1, from multistable to monostable behaviour, and this cannot be predicted by analysing either subsystem in isolation. This is consistent with Schlüter et al. [25], who described this transition using agent-based methods. (Due to small differences in model formulation, δ in [25] (introduced in equation (4.1)) is equal to one quarter of δ here, e.g. δ = 0.5 here is equivalent to δ = 0.125 in [25].) Notably, for the chosen subsystem parameters, cooperation never collapses in the collaborative norm scenario; operating under a norm set with the benefit of communication between subsystems preserves cooperation where it otherwise fails. Moreover, under both subsystem 2 equilibria (ϕ^(2)* = 0, 1), subsystem 1’s $\overline{\text{C}}$ attractor is more cooperative, for a given μ⁽¹⁾ value, in the collaborative scenario. However, we show in electronic supplementary material, §S6 that suitable changes in n, s, g or T can cause cooperation to collapse even in the collaborative scenario.

Figure 3.

(a) Shifted biophysical (blue, purple) and social (orange) fixed points in subsystem 1 under purely biophysical, moderate (δ = 0.5) subsystem coupling. Results shown for the purely cooperative social equilibrium in subsystem 2 (ϕ^(2)* = 1), in the uncollaborative (i) and collaborative (ii) norm scenarios. Dashed and solid orange loci comprise unstable and stable social fixed points, respectively. Defector payoff overwhelms sanctioning in the red region of (i). (b) Shifted subsystem 1 fixed points under purely social, moderate (λ = 0.5) subsystem coupling. Results shown for the purely defective (i) and purely cooperative (ii) social equilibrium in subsystem 2 (ϕ^(2)* = 0, 1 respectively). Sanctioning overwhelms defector payoff in the green region of (ii). The previously unseen social attractor is labelled: $\overline{\text{D}}$—mixed, defector dominated. (Online version in colour.)

(b). Purely social coupling

We now consider cases in which 0 < λ ≤ 0.5 and δ = 0. Although one may question why communities that do not share resources would sanction each other for violating resource use norms, it is instructive to study such cases before combining social and biophysical coupling. When δ = 0, (2.3) reduces to ${\rho }^{(1)}(\tau )={\rho }_0^{(1)}(\tau )$, the resource also equilibrates as in an isolated subsystem and the uncollaborative and collaborative scenarios are identical. Moreover, the mixed social fixed points are independent of μ⁽²⁾ when δ = 0 in (3.1). Figure S5 in the electronic supplementary material shows the intersections of ω⁽¹⁾ and ${\pi }_d^{(1)}$ (mixed social fixed points) for purely social coupling.

Figure 3b shows how moderate purely social coupling (λ = 0.5, δ = 0) shifts system 1’s fixed points for each of subsystem 2’s monomorphic social equilibria. When ϕ^(2)* = 0, subsystem 1’s stability landscape is qualitatively similar to an isolated subsystem and is, therefore, resilient to the cross-scale social perturbation. However, when ϕ^(2)* = 1, the stability landscape undergoes a phase transition. A mixed, defector-dominated social equilibrium appears for ${\mu }^{(1)}\lesssim 2.6$. We label this attractor $\overline{\text{D}}$, defined such that 0 < ϕ^(1)* < 0.5. For ${\mu }^{(1)}\gtrsim 3.0$, only the $\overline{\text{C}}$ attractor is present. In the intervening region ($2.6\lesssim {\mu }^{(1)}\lesssim 3.0$), sanctioning exceeds defector payoff for all values of ϕ⁽¹⁾ and only the C attractor is present, so the collapse of defection is inevitable for any ϕ⁽¹⁾(0) > 0. Intuitively, this is because social coupling exposes subsystem 1 defectors to additional sanctioning from subsystem 2 cooperators, while the absence of biophysical coupling prevents subsystem 2 from increasing the subsystem 1 defector payoff through added resource availability. We see that the weak perturbation due to social coupling between subsystems can produce two kinds of phase transition in the subsystem stability landscape, from a multistable to a monostable phase and between two different multistable phases.

(c). Combined biophysical and social coupling

Finally, we study interplay between weak biophysical and social couplings (0 < δ ≤ 0.5, 0 < λ ≤ 0.5). Figure S6 in the electronic supplementary material shows the intersections of ω⁽¹⁾ and ${\pi }_d^{(1)}$ for equally moderate biophysical and social couplings (δ = λ = 0.5). Figure 4a plots variations in subsystem 1’s fixed points for subsystem 2’s purely cooperative social equilibrium (ϕ^(2)* = 1). The collapse of subsystem 1 cooperation seen with purely biophysical coupling in the uncollaborative scenario (figure 3a(i)) is prevented. Moreover, within small parameter ranges ($2.9\lesssim {\mu }^{(1)}\lesssim 3.0$ in (i) and $2.5\lesssim {\mu }^{(1)}\lesssim 3.0$ in (ii)), defection collapses instead. Hence, under these conditions, added sanctioning experienced by subsystem 1 defectors overwhelms the extra payoff available due to the added resource access, causing a phase transition in subsystem 1’s stability landscape, which does not occur without social coupling.

Figure 4.

(a) Shifted biophysical (blue, purple) and social (orange) fixed points in subsystem 1 under combined biophysical and social subsystem coupling. Results shown for equally moderate couplings (δ = λ = 0.5), for the purely cooperative social equilibrium in subsystem 2 (ϕ^(2)* = 1). Sanctioning overwhelms defector payoff in the green regions. (b) Phase plots showing how subsystem 1 social stability landscape depends on social and biophysical coupling strengths, when μ⁽¹⁾ = μ⁽²⁾ = 2.95, ϕ^(2)* = 1. Results shown for 500 values, each, of δ and λ (2.5 × 10⁵ combinations) per plot. Five stability landscapes are labelled, each comprising a landscape with either one or two of the four social attractor types described. The monomorphic attractors, C and D are purely cooperative and purely defective, respectively. We define the mixed attractors as $\overline{\text{D}}:\hspace{0.17em}0<{{\varphi }^{(1)}}^{\ast }<0.5$ and $\overline{\text{C}}:\hspace{0.17em}0.5\le {{\varphi }^{(1)}}^{\ast }<1$. The phase structure varies significantly across the (μ⁽¹⁾, μ⁽²⁾, ϕ^(2)*) parameter space (electronic supplementary material, §S6); values chosen here provide an illustrative example. (i) Uncollaborative scenario, containing five phases. The phase containing only the D attractor appears only in a small region at the lower right of panel ($\delta \gtrsim 0.41$) and is magnified in the inset. (ii) Collaborative scenario, containing three phases. (Online version in colour.)

The results reveal important differences between norm scenarios. Subsystem cooperativities are much more mutually sensitive in the uncollaborative scenario. Weak perturbations from subsystem 2 can cause opposing types of change in subsystem 1’s stability landscape; collapse of cooperation when coupling is purely biophysical and collapse of defection when social coupling is present. Conversely, in the collaborative scenario, subsystems do not risk collapse of cooperation due to perturbations from weak biophysical coupling (under the chosen parameters). Social coupling can cause collapse of defection also in this case, however. Furthermore, subsystem 1’s $\overline{\text{C}}$ attractor is more cooperative in the collaborative scenario.

Figure 4b shows how subsystem 1’s social stability landscape depends on social and biophysical coupling strengths across the weak coupling parameter subspace. Five different phases are labelled with the combinations of social attractors present in each. There is a clear asymmetry between how social coupling and biophysical coupling determines the stability landscape. Whereas changes in social coupling cause phase transitions for every value of biophysical coupling in both scenarios, the inverse is not true; for most social couplings, no change in biophysical coupling causes a phase transition. Panel (i) shows a phase plot for the uncollaborative scenario. The collapse of cooperation seen in figure 3a is restricted to a small region at the lower right. At the opposite extreme, where social coupling is moderate, the purely cooperative phase extends across the range of weak biophysical couplings. The spectrum of other phases lies between these extremes. In electronic supplementary material, §S5, we briefly describe mechanisms underlying the behaviour of each phase. The overall trend is that increasing the social coupling causes successive phase transitions, gradually (though not always monotonically) increasing subsystem 1 cooperativity. The collaborative scenario (figure 4b(ii)) shows the same broad features except that the D and $\overline{\text{D}}$C phases are absent, for reasons explained in §3b and electronic supplementary material, §S5. Additionally, whereas the phase boundaries increase monotonically with biophysical coupling in the uncollaborative scenario, they decrease monotonically in the collaborative scenario. Thus, for a given level of resource sharing, communication during norm setting allows lower levels of social coupling to cause phase transitions in subsystem 1’s stability landscape. Collaborative norm setting is, therefore, an investment which can later increase the power of social pressure as a resource-sustaining mechanism. In sum, the results show the importance of collaboration between communities, in both establishing and enforcing harvesting norms, even when biophysical and social coupling are much weaker between subsystems than within each. In electronic supplementary material, §S6 we show that these findings are robust to variations in subsystem parameters.

4. Discussion

Our results show that social and biophysical CSIs can profoundly affect how interacting communities manage a common pool resource, and how their subsystem-level behaviours interact to produce system-level outcomes. Even weak social interactions between communities can significantly increase long-time cooperativity. This reveals a mechanism by which resource managers can help to ensure that subsystem-level efforts produce desired system-level outcomes. When coupling is purely biophysical, at least moderate in strength ($\delta \gtrsim 0.5$), and norms are established without communication, attempts to promote system-level sustainability by pursuing socially optimal resource use at the subsystem level instead drive a group selection mechanism; success for one subsystem guarantees failure for the other and, therefore, the system overall. Enforcing norms between communities resolves this because increased cooperation in one community also encourages cooperation in the other. This system-level cooperation depends strongly on social coupling strength, even in the weak range. In general, strengthened coupling increases cooperation through transitions between stability landscape phases (figure 4b). Moreover, communities that share information during norm setting may protect themselves against the collapse of cooperation at the subsystem level and increase the power of intercommunity social pressure to promote system-level cooperation.

Methodologically, our dual analytical/numerical approach offers a powerful complement to purely numerical, agent-based methods [25]. In the dual approach, finding analytical expressions for subsystem fixed points enables their direct evaluation, bypassing the need for full dynamical simulation to characterize the stability landscape. However, this comes at the cost of statistical behaviour outside the mean field, which is preserved by agent-based methods. There is also a small accuracy cost incurred by the perturbative biophysical model (electronic supplementary material, §S2), though this can, in principle, be reduced by truncating (2.3) at a higher order, notwithstanding mathematical difficulties. As well as efficiently characterizing a two-level system, our dual approach is well suited to the future study of larger systems, including more subsystem interaction mechanisms and/or more hierarchical levels. Powerful methods have been developed in dynamical systems theory and statistical physics, for analysing the interplay between structure and dynamics at different scales in many-body systems. Singular perturbation theory [29] and renormalization [28], for example, were first developed as refinements to the regular perturbation theory on which our biophysical model (2.4a,b) is based. Renormalization was refined into a technique for coarse-graining statistical descriptions of systems with many length and time scales, concisely explaining relationships between behaviour at different scales and providing deep insight into critical transitions. Such multiscale methods have yet to be applied in SES models, even though it is acknowledged that dynamical system methods are useful in this field and that there is a need to better understand multiscale behaviour in SESs and how it relates to their critical transitions. The formalism developed here naturally suits extension through such methods, which we suggest for future research. We also note the conceptual link to multilevel cultural selection theory [38,39], which has been proposed as a tool for sustainability analysis [39]. Formalizing this link is another interesting challenge for future work. We foresee complementary roles for dual analytical/numerical dynamical system methods and agent-based simulation, with the latter providing a computational laboratory in which to test the former in small prototypic systems before application to systems impractically large for full simulation.

Even before extending the model to larger systems, there are many options for further development. One is to modify the social interaction model using results from psychology. Although somewhat mixed [35,40], evidence on in-group bias has suggested that members of a community are likely to sanction out-group members more harshly than in-group members for norm violation [35,36]. This could be modelled as dependence of the parameters controlling the curvature and maximum of the sanctioning experienced by out-group defectors (s, g and T in the interaction term of (2.8)) on the social coupling between communities (λ). Furthermore, psychological limits to human social network size and structure [41] could be incorporated by limiting the social connectedness available to each agent and, therefore, their capacity to sanction and be sanctioned by others. Other interesting results may come from breaking the parametric symmetries assumed here. Augmenting the social dynamics (2.9) to include harvesting strategy utility comparisons between communities would generalize the model to a wider range of contexts. To widen this range even further, the social and biophysical connectivities may be modified within the range of possible four-node SES network motifs [37].

Our model may also be extended with features previously studied in related models. First, adding static [42–44] or adaptive [45] social network structure to each community permits more realistic description of social dynamics. If the social network is spatially embedded and the resource pool not well mixed, agents may not have equal resource access and instead interact strongly only with their local sub-pools [43]. This may introduce additional characteristic scales to the system, requiring a more sophisticated multiscale analysis. Second, top-down institutional behaviour monitoring and enforcement could supplement peer-to-peer sanctioning of defectors [42,46] and/or sanctioning could be made costly to the cooperators who apply it [47]. Third, different biophysical models may be used, such as logistic growth for biological resources [46,47]. Combinations of such features have been shown to produce interesting results. Community cooperativity in a structured social network can depend sensitively on both institutional monitoring effectiveness and sanctioning strength [42] but, at least in the case of logistic resource growth, design of such top-down regulation must be tailored to biophysical features, such as growth rate, to be effective [46]. Ultimately, more realistic representations of SESs will come from using refined subsystem models as building blocks in larger, multilevel, multiscale, networked systems with parameterizations statistically distributed across subsystems. The multiscale analysis methods described above are well suited to such a system. In combination with statistical and qualitative analyses from multiscale empirical studies, such modelling methods offer strong potential for better understanding and managing CSIs in SESs.

Our work raises fundamental questions about what is meant by ‘scale’ in SESs, and how best to model it. Different usage of the term across natural and social sciences influencing SES research [7,9] must be navigated if a coherent theory of CSIs is to be developed. Natural sciences usually define scales as orders of magnitude in spatial and temporal intervals. Social sciences typically define scales as levels of organization in hierarchical social systems, though there has also been interest in how spatial embeddedness affects social systems [48–50]. In case studies, CSIs are commonly posed as interactions between quantities which vary over different spatial and temporal scales, but precisely how such interactions depend on space and time is often not considered. In this study, we considered interactions across scales in the strengths of biophysical and social couplings within and between subsystems. One possibility for interpreting our results in terms of conventional spatio-temporal framing is to determine how biophysical and social couplings depend on space and time. However, given the vast range of resource pool types and social systems which may be studied, this is a nontrivial undertaking. One may expect coupling strengths to vary inversely with spatio-temporal separation and this may indeed be the case in some systems. However, in a globalized, digitally connected world, couplings can depend counterintuitively on, or be effectively independent from, spatio-temporal separations, as in so-called teleconnections [16]. Recent work on social-ecological network analysis [51–53] has begun to quantify how the spatio-temporal embeddedness of social and biophysical dynamics affects SES resilience. We propose that further research in this direction, across a broad range of systems and contexts, may be an important step towards a more general theory of CSIs.

Data accessibility

No data available.

Authors' contributions

All authors jointly designed the research. A.K.R. developed the analysis and interpreted the results under advice from S.J.L. and M.S.; A.K.R. wrote, and S.J.L. and M.S. critically revised, the manuscript. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 682472–MUSES), and Swedish Research Council Formas project grant no. 2014-589.