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Resilience (mathematics)

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  • Comment: Interesting topic and notable.It is also well sourced, thank you Ozzie10aaaa (talk) 11:49, 25 June 2023 (UTC)


In mathematical modeling, resilience refers to the ability of a 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. 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.

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.[2] 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.

The concept of resilience is particularly useful in systems the 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.[3][4]

Definition

Mathematically, resilience can be approximated by the inverse of the return time to an equilibrium[5][6][7] 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.[8]

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.[citation needed]

In epidemiology, resilience may refer to the ability of a healthy community to recover from the introduction of infected individuals.

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.[9] 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[10] 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.[11]

Networks that follow the scale-free pattern are able to withstand damage effectively because most of their nodes have only a few connections. Thus, if some random nodes are removed, it is more probable that the nodes with fewer connections are taken out first. This ensures that even if a significant number of random nodes are eliminated, the fundamental characteristics of the network remain unchanged.[citation needed]

See also

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. ^ Engineering Within Ecological Constraints. 1996-03-22. doi:10.17226/4919. ISBN 978-0-309-05198-9.
  3. ^ 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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. ISSN 1076-2787.
  10. ^ 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
  11. ^ Verfasser., Mitchell, Melanie (April 2009). Complexity : a guided tour. ISBN 978-0-19-972457-4. OCLC 1164178342. {{cite book}}: |last= has generic name (help)CS1 maint: multiple names: authors list (link)
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