Resilience (mathematics)

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Ball-and-valley analogy of resilience. A resilient steady state can be thought of as a ball in a deep valley, and it takes a large perturbation to move the ball away to an alternative steady state. A steady state that is less resilient can be thought of as a ball in a shallow valley, where smaller perturbations can be enough to make the system unstable. Resilience Diagram.svg
Ball-and-valley analogy of resilience. A resilient steady state can be thought of as a ball in a deep valley, and it takes a large perturbation to move the ball away to an alternative steady state. A steady state that is less resilient can be thought of as a ball in a shallow valley, where smaller perturbations can be enough to make the system unstable.

In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. [1] It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible to perturbations and can more easily undergo a critical transition. A common analogy used to explain the concept of resilience of an equilibrium is one of a ball in a valley. A resilient steady state corresponds to a ball in a deep valley, so any push or perturbation will very quickly lead the ball to return to the resting point where it started. On the other hand, a less resilient steady state corresponds to a ball in a shallow valley, so the ball will take a much longer time to return to the equilibrium after a perturbation.

Contents

The concept of resilience is particularly useful in systems that exhibit tipping points, whose study has a long history that can be traced back to catastrophe theory. While this theory was initially overhyped and fell out of favor, its mathematical foundation remains strong and is now recognized as relevant to many different systems. [2] [3]

History

In 1973, Canadian ecologist C. S. Holling proposed a definition of resilience in the context of ecological systems. According to Holling, resilience is "a measure of the persistence of systems and of their ability to absorb change and disturbance and still maintain the same relationships between populations or state variables". Holling distinguished two types of resilience: engineering resilience and ecological resilience. [4] Engineering resilience refers to the ability of a system to return to its original state after a disturbance, such as a bridge that can be repaired after an earthquake. Ecological resilience, on the other hand, refers to the ability of a system to maintain its identity and function despite a disturbance, such as a forest that can regenerate after a wildfire while maintaining its biodiversity and ecosystem services. With time, the once well-defined and unambiguous concept of resilience has experienced a gradual erosion of its clarity, becoming more vague and closer to an umbrella term than a specific concrete measure. [5]

Definition

Mathematically, resilience can be approximated by the inverse of the return time to an equilibrium [6] [7] [8] given by

where is the maximum eigenvalue of matrix .

The largest this value is, the faster a system returns to the original stable steady state, or in other words, the faster the perturbations decay. [9]

Applications and examples

In ecology, resilience might refer to the ability of the ecosystem to recover from disturbances such as fires, droughts, or the introduction of invasive species. A resilient ecosystem would be one that is able to adapt to these changes and continue functioning, while a less resilient ecosystem might experience irreversible damage or collapse. [10] The exact definition of resilience has remained vague for practical matters, which has led to a slow and proper application of its insights for management of ecosystems. [11]

In epidemiology, resilience may refer to the ability of a healthy community to recover from the introduction of infected individuals. That is, a resilient system is more likely to remain at the disease-free equilibrium after the invasion of a new infection. Some stable systems exhibit critical slowing down where, as they approach a basic reproduction number of 1, their resilience decreases, hence taking a longer time to return to the disease-free steady state. [12]

Resilience is an important concept in the study of complex systems, where there are many interacting components that can affect each other in unpredictable ways. [13] Mathematical models can be used to explore the resilience of such systems and to identify strategies for improving their resilience in the face of environmental or other changes. For example, when modelling networks it is often important to be able to quantify network resilience, or network robustness, to the loss of nodes. Scale-free networks are particularly resilient [14] since most of their nodes have few links. This means that if some nodes are randomly removed, it is more likely that the nodes with fewer connections are taken out, thus preserving the key properties of the network. [15]

See also

Related Research Articles

<span class="mw-page-title-main">Ecosystem</span> Community of living organisms together with the nonliving components of their environment

An ecosystem is a system that environments and their organisms form through their interaction. The biotic and abiotic components are linked together through nutrient cycles and energy flows.

<span class="mw-page-title-main">Theoretical ecology</span>

Theoretical ecology is the scientific discipline devoted to the study of ecological systems using theoretical methods such as simple conceptual models, mathematical models, computational simulations, and advanced data analysis. Effective models improve understanding of the natural world by revealing how the dynamics of species populations are often based on fundamental biological conditions and processes. Further, the field aims to unify a diverse range of empirical observations by assuming that common, mechanistic processes generate observable phenomena across species and ecological environments. Based on biologically realistic assumptions, theoretical ecologists are able to uncover novel, non-intuitive insights about natural processes. Theoretical results are often verified by empirical and observational studies, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.

The diversity of species and genes in ecological communities affects the functioning of these communities. These ecological effects of biodiversity in turn are affected by both climate change through enhanced greenhouse gases, aerosols and loss of land cover, and biological diversity, causing a rapid loss of biodiversity and extinctions of species and local populations. The current rate of extinction is sometimes considered a mass extinction, with current species extinction rates on the order of 100 to 1000 times as high as in the past.

In ecology, an ecosystem is said to possess ecological stability if it is capable of returning to its equilibrium state after a perturbation or does not experience unexpected large changes in its characteristics across time. Although the terms community stability and ecological stability are sometimes used interchangeably, community stability refers only to the characteristics of communities. It is possible for an ecosystem or a community to be stable in some of their properties and unstable in others. For example, a vegetation community in response to a drought might conserve biomass but lose biodiversity.

Adaptive capacity relates to the capacity of systems, institutions, humans and other organisms to adjust to potential damage, to take advantage of opportunities, or to respond to consequences. In the context of ecosystems, adaptive capacity is determined by genetic diversity of species, biodiversity of particular ecosystems in specific landscapes or biome regions. In the context of coupled socio-ecological social systems, adaptive capacity is commonly associated with the following characteristics: Firstly, the ability of institutions and networks to learn, and store knowledge and experience. Secondly, the creative flexibility in decision making, transitioning and problem solving. And thirdly, the existence of power structures that are responsive and consider the needs of all stakeholders.

<span class="mw-page-title-main">G. David Tilman</span> American ecologist (born 1949)

George David Tilman, ForMemRS, is an American ecologist. He is Regents Professor and McKnight Presidential Chair in Ecology at the University of Minnesota, as well as an instructor in Conservation Biology; Ecology, Evolution, and Behavior; and Microbial Ecology. He is director of the Cedar Creek Ecosystem Science Reserve long-term ecological research station. Tilman is also a professor at University of California, Santa Barbara's Bren School of Environmental Science & Management.

Ascendency or ascendancy is a quantitative attribute of an ecosystem, defined as a function of the ecosystem's trophic network. Ascendency is derived using mathematical tools from information theory. It is intended to capture in a single index the ability of an ecosystem to prevail against disturbance by virtue of its combined organization and size.

<span class="mw-page-title-main">Ecological resilience</span> Capacity of ecosystems to resist and recover from change

In ecology, resilience is the capacity of an ecosystem to respond to a perturbation or disturbance by resisting damage and subsequently recovering. Such perturbations and disturbances can include stochastic events such as fires, flooding, windstorms, insect population explosions, and human activities such as deforestation, fracking of the ground for oil extraction, pesticide sprayed in soil, and the introduction of exotic plant or animal species. Disturbances of sufficient magnitude or duration can profoundly affect an ecosystem and may force an ecosystem to reach a threshold beyond which a different regime of processes and structures predominates. When such thresholds are associated with a critical or bifurcation point, these regime shifts may also be referred to as critical transitions.

In ecology, the theory of alternative stable states predicts that ecosystems can exist under multiple "states". These alternative states are non-transitory and therefore considered stable over ecologically-relevant timescales. Ecosystems may transition from one stable state to another, in what is known as a state shift, when perturbed. Due to ecological feedbacks, ecosystems display resistance to state shifts and therefore tend to remain in one state unless perturbations are large enough. Multiple states may persist under equal environmental conditions, a phenomenon known as hysteresis. Alternative stable state theory suggests that discrete states are separated by ecological thresholds, in contrast to ecosystems which change smoothly and continuously along an environmental gradient.

In the context of ecological stability, resistance is the property of communities or populations to remain "essentially unchanged" when subject to disturbance. The inverse of resistance is sensitivity.

In ecology, a priority effect refers to the impact that a particular species can have on community development as a result of its prior arrival at a site. There are two basic types of priority effects: inhibitory and facilitative. An inhibitory priority effect occurs when a species that arrives first at a site negatively affects a species that arrives later by reducing the availability of space or resources. In contrast, a facilitative priority effect occurs when a species that arrives first at a site alters abiotic or biotic conditions in ways that positively affect a species that arrives later. Inhibitory priority effects have been documented more frequently than facilitative priority effects. Studies indicate that both abiotic and biotic factors can affect the strength of priority effects. Priority effects are a central and pervasive element of ecological community development that have significant implications for natural systems and ecological restoration efforts.

A social-ecological system consists of 'a bio-geo-physical' unit and its associated social actors and institutions. Social-ecological systems are complex and adaptive and delimited by spatial or functional boundaries surrounding particular ecosystems and their context problems.

The resilience of coral reefs is the biological ability of coral reefs to recover from natural and anthropogenic disturbances such as storms and bleaching episodes. Resilience refers to the ability of biological or social systems to overcome pressures and stresses by maintaining key functions through resisting or adapting to change. Reef resistance measures how well coral reefs tolerate changes in ocean chemistry, sea level, and sea surface temperature. Reef resistance and resilience are important factors in coral reef recovery from the effects of ocean acidification. Natural reef resilience can be used as a recovery model for coral reefs and an opportunity for management in marine protected areas (MPAs).

This is a bibliography of ecology.

<span class="mw-page-title-main">Resilience (engineering and construction)</span> Infrastructure design able to absorb damage without suffering complete failure

In the fields of engineering and construction, resilience is the ability to absorb or avoid damage without suffering complete failure and is an objective of design, maintenance and restoration for buildings and infrastructure, as well as communities. A more comprehensive definition is that it is the ability to respond, absorb, and adapt to, as well as recover in a disruptive event. A resilient structure/system/community is expected to be able to resist to an extreme event with minimal damages and functionality disruptions during the event; after the event, it should be able to rapidly recovery its functionality similar to or even better than the pre-event level.

Climate resilience is defined as the "capacity of social, economic and ecosystems to cope with a hazardous event or trend or disturbance". This is done by "responding or reorganising in ways that maintain their essential function, identity and structure while also maintaining the capacity for adaptation, learning and transformation". The key focus of increasing climate resilience is to reduce the climate vulnerability that communities, states, and countries currently have with regards to the many effects of climate change. Efforts to build climate resilience encompass social, economic, technological, and political strategies that are being implemented at all scales of society. From local community action to global treaties, addressing climate resilience is becoming a priority, although it could be argued that a significant amount of the theory has yet to be translated into practice.

Robustness, the ability to withstand failures and perturbations, is a critical attribute of many complex systems including complex networks.

<span class="mw-page-title-main">Ecosystem collapse</span> Ecological communities abruptly losing biodiversity, often irreversibly

An ecosystem, short for ecological system, is defined as a collection of interacting organisms within a biophysical environment. Ecosystems are never static, and are continually subject to stabilizing and destabilizing processes alike. Stabilizing processes allow ecosystems to adequately respond to destabilizing changes, or pertubations, in ecological conditions, or to recover from degradation induced by them: yet, if destabilizing processes become strong enough or fast enough to cross a critical threshold within that ecosystem, often described as an ecological 'tipping point', then an ecosystem collapse occurs.

Critical transitions are abrupt shifts in the state of ecosystems, the climate, financial systems or other complex dynamical systems that may occur when changing conditions pass a critical or bifurcation point. As such, they are a particular type of regime shift. Recovery from such shifts may require more than a simple return to the conditions at which a transition occurred, a phenomenon called hysteresis. In addition to natural systems, critical transitions are also studied in psychology, medicine, economics, sociology, military, and several other disciplines.

Supply chain resilience is "the capacity of a supply chain to persist, adapt, or transform in the face of change".

References

  1. ↑ Hodgson, Dave; McDonald, Jenni L.; Hosken, David J. (September 2015). "What do you mean, 'resilient'?". Trends in Ecology & Evolution. 30 (9): 503–506. doi:10.1016/j.tree.2015.06.010. hdl: 10871/26221 . PMID   26159084.
  2. ↑ Rosser, J. Barkley (October 2007). "The rise and fall of catastrophe theory applications in economics: Was the baby thrown out with the bathwater?". Journal of Economic Dynamics and Control. 31 (10): 3255–3280. doi:10.1016/j.jedc.2006.09.013.
  3. ↑ Scheffer, Marten; Bolhuis, J. Elizabeth; Borsboom, Denny; Buchman, Timothy G.; Gijzel, Sanne M. W.; Goulson, Dave; Kammenga, Jan E.; Kemp, Bas; van de Leemput, Ingrid A.; Levin, Simon; Martin, Carmel Mary; Melis, René J. F.; van Nes, Egbert H.; Romero, L. Michael; Olde Rikkert, Marcel G. M. (2018-11-20). "Quantifying resilience of humans and other animals". Proceedings of the National Academy of Sciences. 115 (47): 11883–11890. Bibcode:2018PNAS..11511883S. doi: 10.1073/pnas.1810630115 . ISSN   0027-8424. PMC   6255191 . PMID   30373844.
  4. ↑ Engineering Within Ecological Constraints. 1996-03-22. doi:10.17226/4919. ISBN   978-0-309-05198-9.
  5. ↑ Myers-Smith, Isla H.; Trefry, Sarah A.; Swarbrick, Vanessa J. (2012). "Resilience: Easy to use but hard to define". Ideas in Ecology and Evolution. 5 (1). doi: 10.4033/iee.2012.5.11.c . hdl: 20.500.11820/0994de3c-3a91-4a5e-9d94-2d84e05db986 . ISSN   1918-3178.
  6. ↑ PIMM, S. L.; LAWTON, J. H. (July 1977). "Number of trophic levels in ecological communities". Nature. 268 (5618): 329–331. Bibcode:1977Natur.268..329P. doi:10.1038/268329a0. ISSN   0028-0836. S2CID   4162447.
  7. ↑ Chen, X.; Cohen, J. E. (2001-04-22). "Transient dynamics and food–web complexity in the Lotka–Volterra cascade model". Proceedings of the Royal Society of London. Series B: Biological Sciences. 268 (1469): 869–877. doi:10.1098/rspb.2001.1596. ISSN   0962-8452. PMC   1088682 . PMID   11345334.
  8. ↑ Neubert, Michael G.; Caswell, Hal (April 1997). "Alternatives to Resilience for Measuring the Responses of Ecological Systems to Perturbations". Ecology. 78 (3): 653–665. doi:10.1890/0012-9658(1997)078[0653:ATRFMT]2.0.CO;2. ISSN   0012-9658.
  9. ↑ Suweis, Samir; Carr, Joel A.; Maritan, Amos; Rinaldo, Andrea; D’Odorico, Paolo (2015-06-02). "Resilience and reactivity of global food security". Proceedings of the National Academy of Sciences. 112 (22): 6902–6907. Bibcode:2015PNAS..112.6902S. doi: 10.1073/pnas.1507366112 . ISSN   0027-8424. PMC   4460461 . PMID   25964361.
  10. ↑ Willis, Kathy J.; Jeffers, Elizabeth S.; Tovar, Carolina (2018-03-02). "What makes a terrestrial ecosystem resilient?". Science. 359 (6379): 988–989. Bibcode:2018Sci...359..988W. doi:10.1126/science.aar5439. ISSN   0036-8075. PMID   29496865. S2CID   3679255.
  11. ↑ Standish, Rachel J.; Hobbs, Richard J.; Mayfield, Margaret M.; Bestelmeyer, Brandon T.; Suding, Katherine N.; Battaglia, Loretta L.; Eviner, Valerie; Hawkes, Christine V.; Temperton, Vicky M.; Cramer, Viki A.; Harris, James A.; Funk, Jennifer L.; Thomas, Peter A. (September 2014). "Resilience in ecology: Abstraction, distraction, or where the action is?". Biological Conservation. 177: 43–51. Bibcode:2014BCons.177...43S. doi:10.1016/j.biocon.2014.06.008. S2CID   11906742.
  12. ↑ O’Regan, Suzanne M.; O’Dea, Eamon B.; Rohani, Pejman; Drake, John M. (September 2020). "Transient indicators of tipping points in infectious diseases". Journal of the Royal Society Interface. 17 (170): 20200094. doi:10.1098/rsif.2020.0094. ISSN   1742-5689. PMC   7536043 . PMID   32933375.
  13. ↑ Fraccascia, Luca; Giannoccaro, Ilaria; Albino, Vito (2018-08-12). "Resilience of Complex Systems: State of the Art and Directions for Future Research". Complexity. 2018: 1–44. doi: 10.1155/2018/3421529 . hdl: 11573/1242493 . ISSN   1076-2787.
  14. ↑ Guillaume, Jean-Loup; Latapy, Matthieu; Magnien, Clémence (2005), Comparison of Failures and Attacks on Random and Scale-Free Networks, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 186–196, doi:10.1007/11516798_14, ISBN   978-3-540-27324-0, S2CID   7520691 , retrieved 2023-03-01
  15. ↑ Mitchell, Melanie (April 2009). Complexity : a guided tour. Oxford University Press. ISBN   978-0-19-972457-4. OCLC   1164178342.
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