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Then we compute all the equilibria and study their stability. We also give numerical simulations of some trajectory.<\/jats:p>","DOI":"10.3390\/axioms14060430","type":"journal-article","created":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T10:21:49Z","timestamp":1748859709000},"page":"430","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Interactions Between Wealth and Natural Resources: A Nonlinear ODE Model"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0941-6497","authenticated-orcid":false,"given":"Marino","family":"Badiale","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Turin, 10123 Torino, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5297-6804","authenticated-orcid":false,"given":"Isabella","family":"Cravero","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Turin, 10123 Torino, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"104071","DOI":"10.1016\/j.nonrwa.2024.104071","article-title":"A HANDY-type model with non-renewable resources","volume":"77","author":"Badiale","year":"2024","journal-title":"Nonlinear Anal. 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The Limits to Growth: A Report for the Club of Rome\u2019s Project on the Predicament of Mankind, Universe Books.","DOI":"10.1349\/ddlp.1"},{"key":"ref_7","first-page":"312","article-title":"Update to limits to growth: Comparing the World3 model with empirical data","volume":"26","author":"Herrington","year":"2022","journal-title":"J. Ind. Ecol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1038\/s41567-022-01652-6","article-title":"Limits to economic growth","volume":"18","author":"Murphy","year":"2022","journal-title":"Nat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.ecolecon.2014.02.014","article-title":"Human and nature dynamics (HANDY): Modeling inequality and use of resource in the collapse or sustainability of societies","volume":"101","author":"Motesharrei","year":"2014","journal-title":"Ecol. Econ."},{"key":"ref_10","first-page":"470","article-title":"Modeling sustainability: Population, inequality, consumption, and bidirectional coupling of the Earth and Human Systems","volume":"3","author":"Motesharrei","year":"2016","journal-title":"Natl. Sci. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1736","DOI":"10.1137\/19M1279526","article-title":"Population collapse in Elite-dominated societies: A differential model without differential equations","volume":"19","author":"Akhavan","year":"2020","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1134\/S1560354720020045","article-title":"Revisiting the Human and Nature Dynamics Model","volume":"25","author":"Grammaticos","year":"2020","journal-title":"Regul. Chaotic Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1877","DOI":"10.1137\/22M1494336","article-title":"Sustainability or societal collapse, dynamics and bifurcations of the handy model","volume":"22","author":"Tonnelier","year":"2023","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_14","unstructured":"Al-Khawaja, T. (2021). Mathematical Models, Analysis and Simulations of the HANDY Model with Middle Class. [Ph.D. Thesis, Oakland University]. Available online: https:\/\/our.oakland.edu\/handle\/10323\/11946."},{"key":"ref_15","first-page":"1","article-title":"Analysis and simulation of the HANDY model with social mobility, renewables and nonrenewables","volume":"2023","author":"Shillor","year":"2023","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Castro, R., Fritzson, P., Cellier, F., Motesharrei, S., and Rivas, J. (2014, January 10\u201312). Human-Nature Interaction in World Modeling with Modelica. Proceedings of the 10th International Modelica Conference, Lund, Sweden.","DOI":"10.3384\/ecp14096477"},{"key":"ref_17","unstructured":"Sendera, M. (2019). Data Adaptation in Handy Economy-Ideology Model. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhang, L., Song, X., and Wu, Y. (2016, January 8\u201311). Social Spatial Heterogeneity and System Entrainment in Modeling Human and Nature Dynamics. Proceedings of the Theory, Methodology, Tools and Applications for Modeling and Simulation of Complex Systems, Beijing, China.","DOI":"10.1007\/978-981-10-2672-0_32"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"074020","DOI":"10.1088\/1748-9326\/aa7581","article-title":"Sustainability, collapse and oscillations in a simple World-Hearth model","volume":"12","author":"Nitzbon","year":"2017","journal-title":"Environ. Res. Lett."},{"key":"ref_20","unstructured":"Gantmacher, F.R. (2000). The Theory of Matrices Volume 2, AMS Chelsea Publishing."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0771-050X(80)90013-3","article-title":"A family of embedded Runge-Kutta formulae","volume":"6","author":"Dormand","year":"1980","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S1064827594276424","article-title":"The MATLAB ODE Suite","volume":"18","author":"Shampine","year":"1997","journal-title":"SIAM J. Sci. 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