2.1. The end of the Cold War, complex crises landscapes, and geopolitical conflicts

A prominent example for deep systemic change was the end of the Cold War, when from early October to late December 1989 the Eastern European political regimes of the Warsaw Pact were falling like in a domino chain, while the Soviet Union was dissolved in 1991. These momentous tipping events became possible due to a shift from hostile to friendly attitudes and perceptions between 1985 and 1989 when Gorbachev’s policy of glasnost and perestroika opened new US–Soviet relationships. Simulations by the author shortly before the fall of the Berlin Wall with the VIABLE model showed a turbulent transition from an arms race to disarmament when perceptions changed from worst-case to mutual trust in security-related variables (Bendor & Scheffran, Reference Bendor and Scheffran2019). Threat perceptions further declined after 1989 and nuclear powers reduced their arsenals, even considered abolition of nuclear weapons. At the same time, the United States continued to push for missile defence and military interventions which provoked hostile reaction from Russia, blocking progress in nuclear disarmament in today’s new Cold War period.

The ‘complexity turn’ in international relations after 1990 (Urry, Reference Urry2005) is characterized by multi-scale interactions, among them crises and conflicts in fractal and fragile landscapes at national, subnational and transnational levels, with growing connectivity, number and diversity of agents and overlapping security dimensions that create instability and surprise. Conflicts in the Balkans, in Africa, the Middle East and other parts of the world triggered foreign military interventions. Nuclear and missile proliferation provoked new arms races including outer space and new technologies. New wars and terrorism contributed to cycles of hatred and violence. Since the terror attacks of 11 September 2001 and the financial crisis of 2008, the world has experienced a sequence of crisis events, including the Greek economic crisis and the Arab Spring, the wars in Iraq, Afghanistan, Libya, Syria, Gaza and Ukraine, geopolitical conflicts between EU and Russia, US and China, the refugee crisis and terrorist attacks, populist and nationalist movements, the British Brexit and Trump presidency (Scheffran, Reference Scheffran2017). Environmental disasters, weather extremes and the Coronavirus pandemic came in addition, among others (Figure 1). Such events are not isolated but globally intertwined through compounding connectors and multipliers (Zscheischler et al., Reference Zscheischler, Westra, van den Hurk, Seneviratne, Ward, Pitman, AghaKouchak, Bresch, Leonard, Wahl and Zhang2018). These include energy and economic growth, climate and environmental changes, resource flows and supply chains, financial and commodity markets, mobility and migration, communication and social networks. When the number or density of interconnected events exceeds a threshold and becomes ‘overcritical’, the devastating dynamics runs and spreads by itself like an uncontrolled chain reaction of systemic risks that drive social-ecological systems and infrastructures beyond thresholds of stability if adaptive management and governance capacities are exceeded or disabled, leading to domino effects and cascading crises (Brosig, Reference Brosig2025).

Figure 1. Emerging polycrisis landscape since 1989, connecting events of geopolitical rivalry and globalisation in the upper part with environmental and resource challenges at the bottom and regional conflicts and crises in the middle. Selected linkages between them indicate plausible and relevant connections and relational chains which do not suggest causal or temporal directions, with black events left until 2015, red events after on the right-hand side (modified and updated from Scheffran, Reference Scheffran2017).

Failure to contain crises appears as a loss of control for the liberal world order which is under pressure. One possible explanation is that the expansionist development which emerged from Europe since colonial times and generated comparative advantage to other regions, is reaching multiple limits (ecological, economic, social, political). Increasing marginal costs and risks at the boundaries trigger multiple crises, conflicts and disasters (Scheffran, Reference Scheffran2023) challenging the globalised world order which is meeting resistance of populist movements, civil society, and other powers in geopolitical struggles (Albert, Reference Albert2024; Neumann, Reference Neumann2022). Europe is facing challenges in all geographic directions: Putin's Russia in the East, US nationalism and hegemony in the West, destabilisation of the Mediterranean in the South, climate change, resource competition and rivalries in the Arctic North. The U.S. is struggling to maintain its hegemony and forge alliances in the Indo-Pacific (Brands & Gaddis, Reference Brands and Gaddis2021). The Global South missed the development opportunities of the colonial powers and tries to escape from the spiral of poverty, deprivation, debt, resource exploitation and environmental destruction. Posing an economic, technological and military challenge to the West, China is trying to reshape the international order and expand its global political influence, using the ‘New Silk Road’ to connect East Asia, Europe, and Africa. Geopolitical struggles are framing cyber and hybrid wars, drone and space warfare, anti-globalisation and energy transition, and environmental and climate change (Zuboff, Reference Zuboff2019). Violent conflicts pollute the environment close to ecocide, including CO₂ emissions, destroy the conditions for sustainable peace, distract enormous funds and scarce resources from global problems, and multiply the polycrisis.

In this context, human–environment interactions are associated with tensions, discursive struggles and unequal power relations (Bryant, Reference Bryant1998; Robbins, Reference Robbins2004), critically studied in political ecology which ‘aims to understand how politics and power influence both social and ecological dynamics’ (Büscher et al., Reference Büscher, Dempsey, Lau, Margulies and Massarella2025). As long as a new stable order is not established, the world remains in an interregnum, as Antonio Gramsci wrote in his Prison Notebooks nearly hundred years ago: ‘The crisis consists precisely in the fact that the old is dying and the new cannot be born; in this interregnum a great variety of morbid symptoms appear’ (English translation cited in: Hoare & Nowell-Smith, Reference Hoare and Nowell-Smith1971; see also Babic, Reference Babic2020). The morbid symptoms of his time included economic crisis, fascism and war, which today are imminent again, together with climate change, biodiversity loss and other environmental challenges, compounding in a polycrises of Western hegemony, fossil capitalism and the Anthropocene, creating a ‘climate of complexity’ (Rothe, Reference Rothe2015; Scheffran, Reference Scheffran, Brauch, Oswald Spring, Grin and Scheffran2016).

Analysing the structural roots converging and amplifying in the polycrisis, Albert (Reference Albert2024) suggests an alternative theoretical approach integrating global socioecological relations in a planetary metabolism. In a coevolving landscape of self-organizing systems with competing (counter-) hegemonic projects, crises solutions and possible world futures, frameworks of ‘planetary systems thinking’ are considered as variants of complexity theory, inspired by world-systems theory, ecological Marxism, planetary thinking (Morin & Kern, Reference Morin and Kern1999) and the neo-Gramscian ‘complex hegemony’ approach (Williams, Reference Williams2020).

2.2. The case of the Arab spring and the Mediterranean region

To demonstrate the complex connections and dynamics of systemic risks in the polycrisis, a specific regional case is discussed. In the Mediterranean region, including Southern Europe as well as Middle East and North Africa (MENA), diverse ecological, socio-economic, and political processes are interconnected, including religious movements, violent conflicts, rivalries and military interventions, Arab Spring, terrorism, forced displacement, and divisions between Global North and South. Major challenges to the Mediterranean are posed by global warming, affecting health, agriculture, forestry and fishery, the water–food–energy nexus, rivers and coastal zones, and rural and urban areas. The shrinking resource base undermines living standards and development opportunities, and conflicts with demands of a growing population, economic consumption and irrigation. Climate change contributes to Mediterranean instability, with other vulnerability factors, such as unemployment, poverty, pandemic, economic recession and unstable political regimes. Compared to South Europe, MENA countries are more vulnerable, less able to adapt and mitigate conflict.

In this intricate context, since 2011 a series of protests emerged in the Arab Spring, from Tunisia to Libya, Egypt, Syria, and other MENA countries, multiplying dissatisfaction and spreading protest by social media (Juhola et al., Reference Juhola, Filatova, Hochrainer-Stigler, Mechler, Scheffran and Schweizer2022; Scheffran, Reference Scheffran2017). Some studies argued that the political crisis was aggravated by weather events in China and Russia, which affected the international market price of wheat (Sternberg, Reference Sternberg2012), together with other drivers of food prices, including oil price, bioenergy use, and stock market speculations. This illustrates how in an interconnected world a self-enforcing chain of stressors can trigger international instability and systemic risks, including tipping points (Figure 2).

Figure 2. Causal loop diagram of the Arab Spring and Syrian civil war, with added policy components in black. (Source: Adapted from Lenton et al., Reference Lenton, Armstrong Mckay, Loriani, Abrams, Lade, Donges, Buxton, Milkoreit, Powell, Smith and Constantino2023: 211, redrawn and revised by author.)

In the years before the rebellion, Syria suffered devastating droughts hitting the main growing areas, driving people from rural to urban areas (Kelley et al., Reference Kelley, Mohtadi, Cane, Seager and Kushnir2015). This added to multiple conflict drivers, including dissatisfaction with the Assad regime, the US invasion in Iraq, the Arab Spring, and the Islamic State (Selby et al., Reference Selby, Dahi, Fröhlich and Hulme2017). While the Syrian civil war became a battleground, millions were driven as refugees into neighbouring countries and beyond the region, merging with other migration movements (Figure 2). In the emerging ‘refugee crisis’ reaching the European Union in fall 2015 nationalist and populist movements provoked tensions and authoritarian responses. To govern such events the Mediterranean is lacking effective dialogue and cooperation, e.g. at Euromed, NATO, and OSCE levels, despite calls for multilateral climate governance and solar energy development.

3.1. The climate–conflict–migration–pandemic nexus

To understand the cascading interactions of systemic risks in the polycrisis, in the following the nexus of climate change, conflict, migration, and the Covid-19 pandemic is discussed. Climate change has been described as a risk multiplier for natural resources such as water, food, and energy, and critical infrastructures and supply networks for health and wealth, provoking production losses, price increases, and financial crises. In fragile regions and hot spots, the nexus of climate–conflict risk can be mutually enforcing with environmental degradation and human insecurity, social and political instability, and disease and displacement (Burrows & Kinney, Reference Burrows and Kinney2016; Hoffmann et al., Reference Hoffmann, Dimitrova, Muttarak, Crespo Cuaresma and Peisker2020; Mach et al., Reference Mach, Kraan, Adger, Buhaug, Burke, Fearon, Field, Hendrix, Maystadt, O'Loughlin and Roessler2019). Besides securitising climate policy, ‘climatising’ security policy implies that climate-related practices are introduced into the defence sector, for instance, disaster management, adaptation, mitigation, or sustainable development (Aykut & Maertens, Reference Aykut and Maertens2023). Whether climate stress triggers vicious circles of risk and violence or a positive nexus of governance solutions and synergies in migration, health, peace, and climate policies depends on the effectiveness and acceptance of political and legal frameworks (BMZ, 2021).

Until today many conflicts are related to the ‘security dilemma’, where threats to the security of one agent provoke reactions threatening security of other agents which contribute to ‘vicious circles’ and ‘spirals of violence’ (Buhaug & Von Uexkull, Reference Buhaug and Von Uexkull2021; Scheffran et al., Reference Scheffran, Ide and Schilling2014). For instance, inherently unstable interactions culminated in cascading threats of growing war coalitions before World War 1 (Scheffran, Reference Scheffran, Brauch, Oswald Spring, Grin and Scheffran2016). Today, many conflicts have this escalation potential, including the Russia–Ukraine conflict and the Israeli–Palestine conflict, both attracting support from allies. Violence can transform (e.g. from intercommunal conflict to insurgencies or interstate wars) and spread to neighbouring states or regions, e.g. through (cross-border) migration, ethnic links, natural resource flows, black markets or arms exports. Societies prone to spirals of violence are in transition or on the edge of instability, such as fragile and failing states with social fragmentation, weak governance and inadequate management capacity, such as Kenya and Sudan (Scheffran et al., Reference Scheffran, Ide and Schilling2014). Once critical thresholds of insecurity and violence have been passed, a self-enforcing spiral of violence perpetuates more violent acts. Similar mechanisms may occur in a self-enforcing cycle of cooperation and peace, once a critical threshold or positive tipping point in the opposite direction has passed (Eker et al., Reference Eker, Lenton, Powell, Scheffran, Smith, Swamy and Zimm2024). If not precluded by path dependence, agents can switch from production to destruction (and vice versa) according to individual or collective actions.

In the Covid-19 pandemic a virus spread through a globally connected world, infecting and killing millions of people in an exponentially growing chain reaction. An alliance of science and politics reacted in disaster mode, multiplied by interactions between media and public. In short time, far reaching decisions were made under high uncertainty, including partial shutdown of society and economy worldwide. The crisis connected and separated all humans, from private lives to global economy, and interacted with the climate–conflict–migration nexus (Figure 3).

Figure 3. Transformation from the negative to the positive climate–conflict–migration nexus and the interaction with Covid-19. (Source: Adapted and modified from Scheffran, Reference Scheffran2023.)

3.2. Anticipative and adaptive governance

To address multiple challenges in the climate–conflict–migration–pandemic nexus between collapse and transformation, adequate governance strategies need to maintain stability against complex systemic changes and risks. One approach is to decouple crisis connectors and build synergising connectors. Plausible future pathways consider how stressors, triggers, and risks combine with agent perception, anticipation and (inter-)action in critical transitions of crisis patterns and scenarios. To contain vicious circles, opportunities of virtuous circles can induce positive tipping cascades (Lenton et al., Reference Lenton, Armstrong Mckay, Loriani, Abrams, Lade, Donges, Buxton, Milkoreit, Powell, Smith and Constantino2023). If agents are powerful in capabilities and efficient for action goals, they can withstand, compensate, or counter-act hostility by others, avoiding deviations from stable equilibrium conditions. If the number and intensity of hostile actions exceed critical thresholds, unstable escalation may lead to the breakup of social systems. Stability of social interaction can be maintained if the positive (cooperative) effects of agents exceed their negative (conflicting) effects. Mutual adaptations of actions or institutional control mechanisms can stabilise the interaction and contain conflict.

Humanity can enforce a sustainable transformation, merging solutions and synergies to stabilise human development within available environmental spaces, protecting and preserving the natural resource base. To balance human needs and available natural resources, efficient, sufficient and fair use and distribution of these resources are required.

A key question is whether a transition can be achieved mainly by technical innovations within the existing capitalist economy, catalysed by artificial intelligence to find integrative solutions across different fields of technology, or requires societal innovations and fundamental system change of fossil capitalism, replacing its expansive drivers of growth in a sustainable world. Possible futures are shaped by critical thresholds between pathways of disruption and construction, conflict and cooperation, war and peace, risk and resilience, exclusion and coexistence, identity and diversity, and trade-offs and synergies. How tensions between different policy fields can be reduced or managed determines success of social-ecological transformation.

Adaptive, anticipative, and cooperative governance and agency of stakeholders, states, networks, and institutions use enabling and synergising leverage strategies to contain the polycrisis, establishing conflict-sensitive and resilient climate policies, climate justice and climate matching in North–South cooperation and sustainable energy transition. To integrate innovative concepts in norm-based policies, legal mechanisms and technical solutions involve agents of system change in participatory governance, democratic power distribution, dispute resolution, and sustainable peacebuilding. There is an urgent need for effective responses to anticipated, rapid state changes in the Earth system, focused specifically on tipping-point governance (Milkoreit et al., Reference Milkoreit, Boyd, Constantino, Hausner, Hessen, Kääb, McLaren, Nadeau, O'Brien, Parmentier and Rotbarth2024).

4.1. Sensitivity to change, pathways, and interactions

Expanding the qualitative discussion of the polycrisis, an integrative framework is used to represent the complex interplay of systems, conditions, and actors in the Earth system, including multiple pathways between climate stability C, natural resources N, human security H, and societal stability S (CNHS; Scheffran et al., Reference Scheffran, Link, Schilling, Scheffran, Brzoska, Brauch, Link and Schilling2012). The linkages are characterised by pairwise sensitivities between variables in each of the four compartments of the Earth system, measuring the change of one variable induced by the change of another variable. Effects may be direct (e.g. change in crop yield in response to temperature change) or indirect (e.g. damages caused by an increase in the frequency of coastal flooding due to sea level rise). A prominent example is climate sensitivity, i.e. the global temperature change induced by a doubling of CO₂ concentration in the atmosphere. This narrow definition can be expanded to the sensitivity of climate variables to any other variables of interest, such as sensitivity of climate to natural resources or conflicts. Accordingly, conflict sensitivity could be defined in different ways, for instance how the number of armed conflicts is influenced by global temperature change or precipitation which has been extensively studied with statistical methods, investigated in qualitative field studies, simulated with computer models, and estimated in expert assessments (Mach et al., Reference Mach, Kraan, Adger, Buhaug, Burke, Fearon, Field, Hendrix, Maystadt, O'Loughlin and Roessler2019). The same can be done for any other combination of changes in the climate–conflict–migration–pandemic nexus which requires more empirical research beyond this conceptual paper on fundamental structures and processes in the polycrisis.

The following discusses how sensitivities affect instability in a polycrisis, including compound risks, tipping cascades, and domino effects. One question is how variable changes proliferate through interconnected pathways, such as movements of resources, people, finance, impacts, or market prices, amplifying polycrisis. Models can help to understand these interactions and find governance mechanisms to influence them (Bendor & Scheffran, Reference Bendor and Scheffran2019). The Earth’s four subsystems are characterised by general indicators of their viability and interacting changes (Figure 4):

1. 1. Climate stability C is based on the objective of the UN Framework Convention on Climate Change (UNFCCC) for ‘stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system’. Climatic changes ΔC concern greenhouse gas (GHG) emission and concentration, temperature, precipitation, and weather extremes.

2. 2. Natural resources N indicate the quantity and quality of natural systems (soil, water, forests, biodiversity) providing material and energetic services. They are affected by changes ΔN negatively (death and loss) or positively (growth and regeneration).

3. 3. Human security H protects people from acute threats and facilitates their empowerment and capacity to preserve human life and health, need and well-being, dignity and freedom. Changes ΔH impose impacts and responses to stress, depending on human exposure and vulnerability.

4. 4. Societal stability S is the ability to maintain basic functions and resilience of societies against systemic risks and crises. Changes of societal stability ΔS result from socioeconomic stress, social erosion, tensions and violent conflicts, weakened institutions, and disrupted social networks.

   

   Figure 4. Sensitivities in nature-society interaction and dynamic changes ΔX in a time period inducing changes ΔY′ in the following period, parallel or sequential (expanding work in Scheffran et al., Reference Scheffran, Link, Schilling, Scheffran, Brzoska, Brauch, Link and Schilling2012).

Each input change Δx in a system variable x (cause) may induce an output change Δy′ (effect) of another variable y in the following period Δy′ = y_x Δx. Here, y_x is the sensitivity of changes in y with regard to changes in x (cause–effect relation), which can represent a positive (y_x > 0) or negative coupling (y_x < 0), if not zero. Thus, sensitivities are the connectors (stressors) in the impact chain of the polycrisis while input changes Δx are the triggers, together inducing the output change Δy′ which may result in a critical transition (crisis) if the new state y′ = y + Δy′ moves outside of or into a new stability basin. Similarly, a change in x can affect its own dynamics (self-impact x_x) which determines whether an increase Δx leads to growth (x_x > 0: exponential growth), or to decline (x_x < 0: exponential decay).

For a functional relationship y = f(x) between the variables and sufficiently small variable changes, sensitivity can be approximated by the first-order partial derivative of function f with regard to x; for larger changes higher orders could be included which implies that sensitivities are not necessarily constant for non-linear functions. The product of positive or negative signs of sensitivity and causal change determine the sign of output effect. Whether it leaves the stability basin, depends on the magnitude of induced change and the distance to the stability boundary. For three variables x, y, z the sequential coupling is the product of the sensitivities Δz′ = z_y Δy = z_y y_x Δx corresponding to a domino effect which is interrupted if one sensitivity is zero.

Estimates of sign and magnitude of the relationships can be presented by impact graphs which provide a framework for the network of connections and changes between the variables. Since each of these systems is characterized by a vector of variables X and Y, the links can be represented by a sensitivity matrix (X_Y) (small letters x represent scalar variables, capital letters X are vectors). Key sensitivities are the stress induced in natural resources by climate change (N _C), the impact of energy and environmental change on human security (H _N), and the societal consequences of changes in human security and economic growth (S _H). The coupling between climate stress and societal stability (S _C) captures direct connections between climate change and society as well as indirect linkages through environmental and human impacts. Other linkages are also relevant, such as couplings between human security and climate change H _C, societal and environmental change S _N, and reverse couplings (impacts on climate stability) C _N, C _S, C _H, and so on. Since there is an internal dynamics within each of these systems, there are also internal couplings of variables denoted by C _C, N _N, H _H, and S _S (Figure 4). A discussion of sensitivities is given in Table 1. In general, signs are uncertain and conditional on past data or future scenarios and may increase or decline beyond a threshold (such as 1.5°C global temperature rise).

Table 1. Typical sensitivities in relationships between causes (vertical) and effects (horizontal) of nature-society interaction (revised and updated from Scheffran et al., Reference Scheffran, Link, Schilling, Scheffran, Brzoska, Brauch, Link and Schilling2012)

4.2. Compounding change and cascading chains

The dynamics in the CNHS framework can be represented by a four-dimensional dynamical system of output changes Δy_j′ in the following period induced by the sum of input changes Δx_j in the previous period plus all other external changes Δ_i^y (i, j = 1, …, 4). All changes can combine in a compounding way, either adding in the same positive or negative direction or neutralize each other. Multiple impacts may not only be direct, simultaneous and additive, but also interact in sequential (multiplicative) feedback chains over several time steps, leading to cascading impacts that increase or decay depending on the sign and magnitude of sensitivities.

For instance, temperature increase ΔT > 0 can lead to loss of natural resources ΔN′ = N _T ΔT < 0 for negative sensitivity N _T < 0 which can have a negative impact on human security ΔH′ = H_N ΔN < 0 for H_N > 0. Human responses to this loss can reduce societal stability ΔS′ = S _H ΔH < 0 for S_H > 0. The combined effect of temperature rise ΔT on societal stability along the full pathway ΔC →ΔN →ΔH →ΔS would be negative ΔS’ = S _H H _N N _T ΔT < 0 (Figure 5). For small sensitivities the product is marginal, indicating minor effects on society (green case in the left graph), but if the sensitivities increase beyond a ‘tipping threshold’, the product can lead to an escalating dynamics (red case in the right graph). If the chain is continued, societal change my induce temperature change ΔT′ = T _S S _H H _N N _T ΔT which is positive for T _S < 0 (escalation for T _S S _H H _N N _T > 1) and negative for T _S > 0 (self-limitation). It is also possible that societal stability is directly affected by climate change via ΔC → ΔS (e.g. by a disaster or heatwave), and indirectly by pathway ΔC →ΔH →ΔS and ΔS = S _H H _T ΔT < 0. Alternatively, a loss of human security may induce counteracting responses that foster collaboration between people to compensate for the loss, in which case societal stability may rather be increased ΔS = S _H H _N N _T ΔT > 0 for S _H < 0, resulting in the opposite effect on temperature change ΔT'. This shows that adaptive systems are not determined to fail but able to preserve their existence through feedback cycles that maintain stability within viable limits, either by influencing the direction of sensitivities or the direction of change in system variables. Due to non-linear effects, an increase in global temperature above a certain threshold may trigger instabilities, tipping points and cascading sequences that could exceed the adaptive capacity and resilience of natural and social systems (Lenton et al., Reference Lenton, Armstrong Mckay, Loriani, Abrams, Lade, Donges, Buxton, Milkoreit, Powell, Smith and Constantino2023; Milkoreit et al., Reference Milkoreit, Hodbod, Baggio, Benessaiah, Calderón-Contreras, Donges, Mathias, Rocha, Schoon and Werners2018).

Figure 5. (a) Dampened cascade of moderate temperature rise on chains in the CNHS framework; (b) Escalating tipping cascade beyond critical sensitivity thresholds.

4.3. Stability conditions

System dynamics models are based on equations of the type Δx(t) = F(x, t) where Δx = x(t + 1) − x(t) represents time-discrete change or continuous change dx(t)/dt. Equilibria can be calculated by solving F(x, t) = 0. Linear systems F(x) = Ax have constant coefficients in the interaction matrix A or be a linear approximation of the non-linear case where A contains first-order partial derivatives of function F (Jacobi Matrix). Equilibria are stable if all eigenvalues of matrix A have negative real parts, corresponding to exponential decay. Eigenvalues are solutions of the characteristic equation det |A − λI| = 0 which in the two-dimensional case leads to the quadratic equation λ² + p λ + q = 0 and solutions λ_½ = −p/2 ± sqrt ((p/2)² − q) where p = −(a ₁₁ + a ₂₂) and q = a ₁₁ a₂₂ − a₁₂ a₂₁ which for p < 0 has at least one unstable positive eigenvalue and for p > 0 is stable for q > 0 and unstable for q < 0. For n > 2, stability conditions are more difficult to determine, for instance the Hurwitz criteria or the Lyapunov function. Oscillating behaviour occurs for (a ₁₁ − a ₂₂)² < −4 a ₁₂ a ₂₁ which is possible for opposite signs of a ₁₂ and a ₂₁. For two-dimensional systems, stability is possible for trace A = (a ₁₁ + a ₂₂) < 0 and det A = a ₁₁ a ₂₂ - a ₁₂ a ₂₁ > 0, or for trace A > 0 and det A < 0; thus, the product of self-induced sensitivities should compensate the product of external ones.

In higher-dimensional dynamical systems, the number of eigenvalues increases and thus the likelihood of instability, if the systems were not selected in an evolutionary process that eliminated unstable real-world systems. In the polycrisis, stabilising linkages are exceeded by new unstable interactions with positive eigenvalues until new more stable equilibria are evolving adapting to complexity. Instability in one system can induce instability in other system, potentially spreading harmful changes. While tipping is often associated with non-linearity, stability theory can be applied to linearised dynamic systems where positive and negative eigenvalues separate tipping between exponential growth (instability) and exponential decay (stability). Beyond the tipping point, the exponential dynamics is often influenced by quadratic (logistic) or other non-linear terms which become relevant when approaching another system state. A well-known example is a chair following gravitational acceleration around the tipping point until it is crashing on the ground in a non-linear way.

5.1. Multiple risks, equilibria, and stability

The described framework of nature–society interactions and sensitivities provide the methodological background for analysing systemic risks in the polycrisis. This can build on system dynamics models, for instance those by Lotka and Volterra on population dynamics and Lewis Fry Richardson on the arms race at the beginning of the twentieth century (Gleditsch, Reference Gleditsch2020) or by Forrester, Meadows and the Club of Rome on the state of the world in the 1970s, based on linear dynamical equations. These approaches were continued in the 1990s with syndromes representing undesirable patterns of the world, including qualitative modelling where the sign of coefficients matters.

In multi-risk environments, natural and social systems could reach their limits and capacities of adaptation. This is elaborated for the nexus of climate–conflict–migration–pandemic risks with a focus on Covid-19. In this complex quadrangular relationship (Figure 6) embedded in the larger CNHS framework, individual risks R are discussed and pairwise risk interactions, particularly between climate risk R ₁ and conflict risk R ₂ (Daoudy, Reference Daoudy2021; Mach et al., Reference Mach, Kraan, Adger, Buhaug, Burke, Fearon, Field, Hendrix, Maystadt, O'Loughlin and Roessler2019) which are then connected to migration risk R ₃ and pandemic risk R ₄. Risk is understood as a function of probability and amount of damage, given by proxy variables such as temperature, precipitation and extremes for climate risk, number of conflicts and casualties for conflict risk, displacement numbers for migration risk, and infections for pandemic risk.

Figure 6. Systemic multi-risk framework of the climate–conflict–migration–pandemic nexus with connecting sensitivities and exemplary pairwise dynamic equations of climate–conflict risk.

Risk R ≥ 0 is represented by linear dynamics ΔR = r R + Δ^R = r (R — R*) with equilibrium R* = −Δ^R/r, where r is the growth rate and Δ^R are external drivers of risk change. Four cases are considered in Table 2.

Table 2. Equilibria and stability conditions in single risk R dynamics for possible combinations of growth rate r and external risk change Δ^R

From a risk minimization and containment strategy, most desirable is case (1a) of a stable negative equilibrium when risk stays at zero, most undesirable is (2b) when there is no chance to avoid unlimited risk escalation. Mixed case (1b) implies a stable risk equilibrium R = R* > 0, while in (2a) R = R* > 0 is a threshold separating a stable low risk area (R < R*) and an unstable high risk area. Implications for governance in a polycrisis are to avoid positive risk growth rates r > 0 and externally induced risk increase Δ^R > 0. More realistic than a purely linear approach is a non-linear dynamics where growth rate r is multiplied with a logistic term R (R⁺-R) such that risk cannot move below lower limit R = 0 and above upper limit R = R⁺ while tipping is determined by stability around R = R*. It is important to keep induced risk change as negative as possible and actual risk as low as possible which in a multi-risk world may be increasingly difficult to achieve.

This can be specified for two risk types (climate risk R_l and conflict risk R₂) with the dynamic equations:

(1)\begin{align}\Delta {R_1} &= {\text{ }}{r_{11}}{R_1} + {\text{ }}{r_{12}}{R_2} + {\Delta _1}^R = {r_{11}}\left( {{R_1}-{\text{ }}{R_1}^*} \right) \nonumber\\ \Delta {R_2} &= {\text{ }}{r_{22}}{R_2} + {\text{ }}{r_{21}}{R_1} + {\Delta _2}^R = {r_{22}}\left( {{R_2}-{\text{ }}{R_2}^*} \right) \end{align}

where r_ii and r_ij are the internal and external risk sensitivities and Δ_i^R includes all other (external) drivers of risk change ΔR_i (i = 1, 2). The fixed point conditions ΔR_i = 0 lead to the mutual equilibria:

(2)\begin{align}R_1^\ast&=\,(-\Delta_1^R-\,r_{12}R_2)/r_{11}\,\nonumber\\ R_2^\ast&=\,(-\Delta_2^R-\,r_{21}R_1)/r_{22}\end{align}

Both linear curves intersect for the risk balance:

(3)\begin{align}R_1^\#&=\,(r_{12}\Delta_2^R-\,r_{22}\Delta_1^R)/Z\,\nonumber\\ R_2^\#&=\,(r_{21}\Delta_1^R-r_{11}\Delta_2^R)/Z\end{align}

where Z = r ₁₁ r _22 − r ₁₂ r ₂₁ is the determinant of the sensitivity matrix (Figure 7). Possible interaction dynamics can use the four individual cases as starting points which first of all depend on whether r_ii < 0 (self-stabilising containment of climate risk and conflict risk) or are positive (r_ii > 0), (self-escalating climate risk and conflict risk beyond a tipping point). If the mutual risk forcings are negative (r_ij < 0), climate risk and conflict risk can contain each other below a risk threshold, if they are positive (r_ij > 0) they are mutually escalating either with or without limits (climate change is a stressor to conflict and vice versa in a vicious circle). A qualitative change occurs when Z > 0 (self-effects r ₁₁ r ₂₂ dominate) switches to Z < 0 (mutual-effects r ₁₂ r ₂₁ dominate) while at Z = 0 (r ₁₁ r ₂₂ = r ₁₂ r ₂₁) equilibria are infinite. For symmetric cases (same sign for both r_ii and both r_ij) and Z > 0 the possible combinations are shown in Table 3. For Z < 0 the equilibria and stability conditions are reversed. Asymmetric cases in which r ₁₁ and r ₂₂ have opposite signs as well as r ₁₂ and r ₂₁ correspond to periodic stability conditions which are not explicitly highlighted here.

Figure 7. Two selected cases for equilibria and stability of climate and conflict risk interaction: An unstable saddle point (red) and a stable fixed point (green).

Table 3. Equilibria and stability conditions (eigenvalues EV) in interactions between climate risk R ₁ and conflict risk R ₂ for possible combinations of symmetric cases of r_ii and r_ij, external risk change Δ_i^R and Z > 0 (i, j = 1, 2)

5.2. Interactions between climate, conflict, migration and pandemic risks

The following Table 4 provides a qualitative discussion of the interactions between climate, conflict, migration and pandemic risks.

Table 4. Interactions between climate, conflict, migration and pandemic risks, using Covid-19 as an example