{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/balls-cups-and-quasi-potentials-quantifying","title":"Balls, cups, and quasi-potentials: quantifying stability in stochastic systems","arxiv_id":"1508.02088","date":"2015-10-23","proceeding":null,"authors":[],"abstract":"When a system has more than one stable state, how can the stability of these\nstates be compared? This deceptively simple question has important consequences\nfor ecosystems, because systems with alternative stable states can undergo\ndramatic regime shifts. The probability, frequency, duration, and dynamics of\nthese shifts will all depend on the relative stability of the stable states.\nUnfortunately, the concept of stability in ecology has suffered from\nsubstantial confusion and this is particularly problematic for systems where\nstochastic perturbations can cause shifts between coexisting alternative stable\nstates. A useful way to visualize stable states in stochastic systems is with a\nball-in-cup diagram, in which the state of the system is represented as the\nposition of a ball rolling on a surface, and the random perturbations can push\nthe ball from one basin of attraction to another. The surface is determined by\na potential function, which provides a natural stability metric. However,\nsystems amenable to this representation, called gradient systems, are quite\nrare. As a result, the potential function is not widely used and other\napproaches based on linear stability analysis have become standard. Linear\nstability analysis is designed for local analysis of deterministic systems and,\nas we show, can produce a highly misleading picture of how the system will\nbehave under continual, stochastic perturbations. In this paper, we show how\nthe potential function can be generalized so that it can be applied broadly,\nemploying a concept from stochastic analysis called the quasi-potential. Using\nthree classic ecological models, we demonstrate that the quasi-potential\nprovides a useful way to quantify stability in stochastic systems.","url_abs":"http://arxiv.org/abs/1508.02088v3","url_pdf":"http://arxiv.org/pdf/1508.02088v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"balls-cups-and-quasi-potentials-quantifying","repo_url":"https://github.com/bmarkslash7/QPot","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}