Rate-Induced Transitions in Networked Complex Adaptive Systems: Exploring Dynamics and Management Implications Across Ecological, Social, and Socioecological Systems

Complex adaptive systems (CASs), from ecosystems to economies, are open systems and inherently dependent on external conditions. While a system can transition from one state to another based on the magnitude of change in external conditions, the rate of change -- irrespective of magnitude -- may also lead to system state changes due to a phenomenon known as a rate-induced transition (RIT). This study presents a novel framework that captures RITs in CASs through a local model and a network extension where each node contributes to the structural adaptability of others. Our findings reveal how RITs occur at a critical environmental change rate, with lower-degree nodes tipping first due to fewer connections and reduced adaptive capacity. High-degree nodes tip later as their adaptability sources (lower-degree nodes) collapse. This pattern persists across various network structures. Our study calls for an extended perspective when managing CASs, emphasizing the need to focus not only on thresholds of external conditions but also the rate at which those conditions change, particularly in the context of the collapse of surrounding systems that contribute to the focal system's resilience. Our analytical method opens a path to designing management policies that mitigate RIT impacts and enhance resilience in ecological, social, and socioecological systems. These policies could include controlling environmental change rates, fostering system adaptability, implementing adaptive management strategies, and building capacity and knowledge exchange. Our study contributes to the understanding of RIT dynamics and informs effective management strategies for complex adaptive systems in the face of rapid environmental change.Comment: 25 pages, 4 figures, 1 box, supplementary informatio

Vulnerabilidade estrutural dos hospitais e cemitérios e crematórios da cidade de São Paulo à COVID-19

This is the first report by the COVID19 Observatory - Group: Contagion Networks analyzing mortality data from the city of São Paulo. In this report, we integrated mortality data for the city of São Paulo between 04/02/2020 and 04/28/2020, with information on the flow of victims between hospitals and cemeteries/crematoriums. We included in our analyzes both confirmed and suspected deaths from COVID-19. The main objectives of this report were: (1) to describe the structure of the flow of victims between locations and (2) to suggest changes in the current flow based on geographical distances in order to avoid a potential overload of the mortuary system. We suggest that the city of São Paulo should plan for a potential overload of the mortuary system (that is, the number of burials), based on the presented results. Thus, our results reinforce the need to adopt specific planning for the management of the extraordinary number of victims of this pandemic. Our predictions are based on the structural analysis of the COVID-19 victim flow network, which shows several hotspots with high vulnerability to system overload. These hotspots concentrate with either the greatest number of deaths (hospital) or of burials (cemetery or crematorium), and therefore have high potential to become overwhelmed by receiving many bodies due to the increase in victims of the pandemic. We recommend special attention to be given to localities on the east side of São Paulo, which has both the most vulnerable hospitals in the city, and also houses cemeteries and crematoriums that have a central role in the network and / or are vulnerable. Based on our optimization analysis, we suggest logistical changes in the current flow of bodies from hospitals to cemeteries/crematoriums so as not to overload the funeral system and minimize transportation costs. In this sense, our results are potentially useful for improving the operational planning of the Municipality of São Paulo, ratifying or rectifying actions underway at the municipal level.Este é o primeiro relatório do Observatório COVID19 - Grupo: Redes de Contágio analisando os dados de óbitos da cidade de São Paulo. Neste relatório, integramos os dados de óbitos da cidade de São Paulo entre os dias 02/04/2020 e 28/04/2020 com informações sobre o fluxo de vítimas entre os hospitais e os cemitérios e crematórios da cidade de São Paulo. Incluímos em nossas análises óbitos confirmados e óbitos suspeitos de COVID-19. Os principais objetivos deste relatório são: (1) descrever a estrutura do fluxo de vítimas entre localidades e (2) sugerir mudanças no fluxo com base em distâncias geográficas de maneira a evitar uma potencial sobrecarga do sistema funerário. Sugere-se à prefeitura da cidade de São Paulo que seja realizado um planejamento para uma potencial sobrecarga do sistema funerário (isto é, número de sepultamentos) da cidade de São Paulo com base nos resultados apresentados. Desta forma, nossos resultados reforçam a necessidade de ser adotado planejamento específico para a gestão dos casos extraordinários visualizados no contexto da pandemia. Esta previsão está baseada na análise estrutural da rede de fluxos de vítimas da COVID-19, que indica a concentração de vários locais com alta vulnerabilidade à sobrecarga do sistema. Tais locais concentram a maior quantidade de óbitos (hospitais) ou a maior concentração de sepultamentos (cemitérios ou crematórios) e tem portanto alto potencial de tornarem-se sobrecarregados por receberem muitos corpos devido ao aumento de vítimas da pandemia. Recomenda-se especial atenção à localidades da zona leste de São Paulo, que apresenta os hospitais mais vulneráveis da cidade e abriga cemitérios e crematórios que possuem papel central na rede e/ou encontram-se vulneráveis. Com base em nossa análise de otimização, sugerimos mudanças logísticas no atual fluxo de corpos de hospitais para cemitérios/crematórios de modo a não sobrecarregar o sistema funerário e minimizar os custos de transporte. Neste sentido, nossos resultados são potencialmente úteis ao aperfeiçoamento do planejamento operacional da Prefeitura Municipal de São Paulo, ratificando ou retificando ações em curso no âmbito municipal

Higher rates of change in the driver of decline lead to extinction at lower values of the driver of decline only for perturbed communities.

The figure shows the value of the driver of decline dA at which all pollinators go extinct, , as a function of the rate of change λ of the driver of decline. For a low initial pollinator abundance (left panels), after a critical value of the rate of change λ, has a nonlinear response. This effect disappears for higher resource congestion q, while it increases with stronger adaptation. For a high initial pollinator abundance, increases monotonically with the rate of change λ. The results are averaged over 100 feasible networks per case (lines), shown with the standard deviation across networks (band). ν = 0.7 for the case with adaptive foraging. The initial abundance for all species is Sinit = 0.1 for the low initial abundance condition and Sinit = 2 for the high initial abundance condition. Other parameters in Table A in S1 Text.</p

Adaptability and resource congestion affect hysteretic patterns and the viability of plant-pollinator networks.

For adaptive pollinators, intermediate levels of resource congestion increase the overall persistence of ecological networks. The point of collapse and recovery of pollinator species increases as a function of resource congestion q without (A) and with adaptive foraging (B). For low resource congestion, the system possesses bistable states which disappear after a critical value of the resource congestion. Resource congestion also affects the feasibility of the networks—networks for which all 15 plant and 35 pollinator species survive under no stress. An intermediate level of resource congestion is required for the adaptive model to produce feasible networks. The orange arrows indicate the resource congestion strength q chosen for the simulation of Figs 2 and 3. These values were chosen such that the systems possess bistable states—as observed in the non-overlapping points of collapse and recovery—and have a high fraction of feasibility. For low resource congestion, adaptability increases the range of drivers of decline at which pollinator communities do not collapse, increasing resilience in the Holling sense [41]. (A) ν = 1 and (B) ν = 0.7. The results are averaged over 100 networks per value of resource congestion q with the error bars showing the standard deviation. Other parameters in Table A in S1 Text.</p

The coevolution of foraging effort and species abundances under increasing driver of decline <i>d</i>_<i>A</i>.

The top row shows the evolution of pollinator and plant abundances under linearly increasing driver of decline dA with rate λ = 0.05, starting from a low abundance condition of Sinit = 0.1, with adaptation strength of ν = 0.7 and resource congestion q = 0.2. Each line style and color combination represents a single pollinator species in all graphs (except the plant abundance graph, top right). For example, there is one pollinator species with degree 3. Since the degree is 3, there are three solid blue lines (one for each connection to a plant species). Another example is the two pollinator species with degree 4, thus showing eight lines (four solid lines for one species and four dashed lines for another species). Since the values of the foraging effort α for each individual species add up to 1, the evolution of the foraging effort α is not shown for pollinators with degree one since they have a constant α = 1 to their single connected plant species. Pollinators with high degree rapidly become the most abundant. Furthermore, the foraging effort α drastically changes—especially around 10 time units when most species reach their peak abundance. The two pollinator species with degree 9 survive the longest and also have one plant species in which they invest most of their foraging effort after 10 time units. Other parameters in Table A in S1 Text.</p

Text with supporting information.

The text with supporting information contains additional computational experiments and algorithms. It is divided into five sections. Section A. Simulation set-up. Includes: Table A. Default model parameters. The default parameter values and ranges used in all simulations, unless otherwise specified. AF = Adaptive Foraging, ∼ U(⋅) = drawn from a uniform distribution at the beginning of each simulation. Section B. Network generation. Describes the network generation algorithm and contains: Fig A. Adjacency matrix of a nested network and forbidden links Adjacency matrix of a pollinator network on the left with the corresponding forbidden links matrix on the right. Black squares denote the presence of a link. The connectance is 0.15 and the fraction of forbidden links is 0.3. There is a clear difference visible between generalist species and specialist species, in the sense that there are a few species with high connectivity and many with low connectivity. Section C. Dependence of hysteresis on resource congestion and adaptation. Contains various additional computation experiments. Fig B. Hysteresis for increasing resource congestion q for the non-adaptive model. Equilibrium abundance of pollinator species as a function of the drivers of decline dA for increasing resource congestion q for the non-adaptive model. The blue lines show the equilibrium trajectory for increasing dA and the orange lines show the equilibrium trajectory for decreasing dA. See Table A for the parameters used. Fig C. Hysteresis for increasing resource congestion q for the adaptive model with ν = 0.8. Equilibrium abundance of pollinator species as a function of the drivers of decline dA for increasing resource congestion q for the adaptive model with ν = 0.8. The blue lines show the equilibrium trajectory for increasing dA and the orange lines show the equilibrium trajectory for decreasing dA. See Table A for the parameters used. Fig D. Hysteresis for increasing resource congestion q for the adaptive model with ν = 0.7. Equilibrium abundance of pollinator species as a function of the drivers of decline dA for increasing resource congestion q for the adaptive model with ν = 0.7. The blue lines show the equilibrium trajectory for increasing dA and the orange lines show the equilibrium trajectory for decreasing dA. See Table A for the parameters used. Fig E. Hysteresis for increasing resource congestion q for the adaptive model with ν = 0.6. Equilibrium abundance of pollinator species as a function of the drivers of decline dA for increasing resource congestion q for the adaptive model with ν = 0.6. The blue lines show the equilibrium trajectory for increasing dA and the orange lines show the equilibrium trajectory for decreasing dA. See Table A for the parameters used. Section D. Distribution of pollinator persistence. Describes the distribution of pollinator persisitence in different computational experiments. Fig F. The full distribution of relative pollinator abundance for three different rates of change λ accompanying Fig 2A in the paper (no adaptive foraging).θ is the fraction of the point of collapse dA at which point the relative pollinator persistence is measured. The distributions are bimodal around 0 and 1 which indicates that there is an abrupt collapse of networks at increasing rates of change. See Table A for the parameters used. Fig G. The full distribution of relative pollinator abundance for three different rates of change λ accompanying Fig 2A in the paper (with adaptive foraging).θ is the fraction of the point of collapse dA at which point the relative pollinator persistence is measured. The distributions are mainly bimodal around 0 and 1. However, some networks have a persistence between 0 and 1, indicating partial collapse due to the rate of change. Furthermore, there are a few networks with pollinator persistence significantly above 1, indicating nonlinear effects where sometimes individual networks can profit from higher rates of change. See Table A for the parameters used. Section E. Sensitivity analysis. Contains: Table B. Parameters for the sensitivity analysis on the feasibility of networks. Parameters and their value ranges used for the sensitivity analysis on the feasibility of networks, and plant and pollinator abundances. The fixed parameters can be found in Table A. AF = Adaptive Foraging. Fig H. Sensitivity analysis of the number of plant species alive. Sobol sensitivity analysis of the number of plant species alive depending on five parameters: resource congestion q, nestedness N, connectance D, adaptation strength ν, and migration rate μ. The sample size per parameter was 512. The adaptation strength ν had the strongest effect on the variance of the outcome of the model. Fig I. Sensitivity analysis of the number of pollinator species alive. Sobol sensitivity analysis of the number of pollinator species alive depending on five parameters: resource congestion q, nestedness N, connectance D, adaptation strength ν, and migration rate μ. The sample size per parameter was 512. The adaptation strength ν had the strongest effect on the variance of the number of pollinators alive. The migration rate μ only has a marginal effect. Fig J. Sensitivity analysis of the total number of species alive. Sobol sensitivity analysis of the total number of species alive depending on five parameters: resource congestion q, nestedness N, connectance D, adaptation strength ν, and migration rate μ. The sample size per parameter was 512. The adaptation strength ν had the strongest effect on the variance of the outcome of the model. Fig K. Sensitivity analysis of the abundance of plant species. Sobol sensitivity analysis of the average plant abundance depending on five parameters: resource congestion q, nestedness N, connectance D, adaptation strength ν, and migration rate μ. The sample size per parameter was 512. Fig L. Sensitivity analysis of the abundance of pollinator species. Sobol sensitivity analysis of the average pollinator abundance depending on five parameters: resource congestion q, nestedness N, connectance D, adaptation strength ν, and migration rate μ. The sample size per parameter was 512. Table C. Parameters for the sensitivity analysis on the critical driver of decline of collapse . Parameters and their value ranges used for the sensitivity analysis on the critical driver of decline of collapse . The fixed parameters can be found in Table A. AF = Adaptive Foraging, dA = driver of decline. Fig M. Sensitivity analysis on the driver of decline dA. Sobol sensitivity analysis on the value of driver of decline at which all pollinator are extinct , depending on six parameters: resource congestion q, nestedness N, connectance D, adaptation strength ν, initial abundance per species Sinit, and migration rate μ. The sample size per parameter was 512. (PDF)</p

Pollinator communities with adaptive foraging still collapse at high rates of change but less abruptly in the extent of environmental change.

Figure equivalent to Fig 2, but considering adaptation and resource congestion. Adaptive communities respond to an increasing driver of decline by reweighing their connections. (A) Rate-induced transitions are still present, with some communities exhibiting rate-dependent tipping at 50% of the point of collapse. Non-monotonicity is within the error range, thus, non-significant for the number of simulations. (B) Overall, pollinator persistence is more sensitive to rates of change in a larger domain of changes in the driver of decline, θ, than for communities without adaptive foraging. Some particular networks see an increase in persistence, especially for small changes and low rates of change, leading to distinct relative persistence levels above 1. For all simulations, initial species abundance of Sinit = 0.1. Adaptation strength of ν = 0.7 and resource congestion q = 0.2. The results are averaged over 100 feasible networks, for which all 15 plant and 35 pollinator species survive under no stress, with the bands representing the first to third quartile ranges. Other parameters in Table A in S1 Text.</p

Dynamics of pollinator communities under external stressors.

(A) Causal dynamics of the adaptive foraging model. Plants and pollinators boost each other’s abundance through mutualistic interactions; pollinators receive resources from plants and plants are pollinated by pollinators. Interactions are specified by a bipartite plant-pollinator nested network. The strength of the interactions changes over time through adaptive foraging. Pollinators adapt their investment in connected plant species based on the supply-demand ratio of plant resources. Besides, species experience intraguild competition and an intrinsic growth rate. The system is subjected to a driver of decline (dA)—an external stressor that negatively influences the intrinsic growth rate of all pollinator species. (B) The mutualistic network has a nested structure with a few highly connected species (generalists) and many sparsely connected species (specialists). The edge colors show the weights of the adaptive foraging matrix α, denoting the investment of each pollinator species in plant species. (C) Hysteretic impact of driver of decline dA (without adaptive foraging). At a critical value of the driver of decline, for any increase of the driver of decline, the pollinator community undergoes bifurcation-induced tipping. At this tipping point, the pollinator community collapses—all pollinator species go extinct (very low abundance). By lowering the driver of decline, the system has another bifurcation-induced tipping point, where, for any decrease of the driver of decline, it recovers at the point of recovery where the first species are reintroduced.</p

Influence of adaptive foraging and resource congestion on the bifurcation diagram.

Adaptation changes the bifurcation diagram. The collapse is less abrupt for pollinator communities with adaptive foraging. Resource competition decreases the size of the bistable area if there is no adaptive foraging. S1 Text Figs B-E show the full bifurcation diagrams of all individual pollinator species for different settings of ν and q. The results are averaged over 100 networks per parameter setting, with the error bands showing the standard deviation. Parameters in Table A in S1 Text.</p

Species persistence collapses for high rates of environmental change within environmental ranges that are otherwise presumed safe.

(A) Three scenarios are represented for communities without adaptive foraging, perturbed to start with a low initial species abundance (out of equilibrium). (Inset) Stress in communities increases over time as the driver of decline increases at different rates, λ, up to a maximum value, . The black line represents the point of collapse above which a fixed value of the driver of decline leads to the collapse of all communities and below which some communities are sustained. The maximum value of the driver of decline in each simulation is denoted by the fraction θ of the point of collapse . (A) Dotted orange line represents an increase in the driver of decline up to 90% of the critical value, , dot-dashed green line an increase up to 50%, and dashed pink an increase up to 20%. In this panel, species persistence is calculated as the fraction of pollinator species alive relative to the number of species alive at the lowest rate of change measured (λmin = 10−4). (B) The persistence of species decreases as a function of the maximum value of the driver of decline, represented as a fraction θ of the point of collapse, for a fast rate of change (λ = 1). Communities without adaptive foraging see a critical transition in species persistence when the driver of decline increases to a value close to, but lower than, the point of collapse at a fast enough rate. In this panel, species persistence is calculated as the fraction of pollinator species alive relative to the number of species alive at θ = 0 (no external stressor). Initial species abundance Sinit = 0.1 for all simulations. The results are averaged over 100 feasible networks, for which all 15 plant and 35 pollinator species survive under no stress, with the bands representing the first to third quartile ranges. Other parameters in Table A in S1 Text.</p