{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T15:50:47Z","timestamp":1777996247591,"version":"3.51.4"},"reference-count":77,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T00:00:00Z","timestamp":1735776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan","award":["BR21882364"],"award-info":[{"award-number":["BR21882364"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Since private cars and vans accounted for more than 25% of global oil consumption and about 10% of energy-related CO2 emissions in 2022, increasing the share of electric vehicle (EV) ownership is considered an important solution for reducing CO2 emissions. At the same time, reducing emissions entails certain economic losses for those countries whose exports are largely covered by the oil trade. The explosive growth of the EV segment over the past 15 years has given rise to overly optimistic forecasts for global EV penetration by 2050. One of the major obstacles to such a development scenario is the limited availability of resources, especially critical materials. This paper proposes a mathematical model to predict the global EV fleet based on the limited availability of critical materials such as lithium, one of the key elements for battery production. The proposed model has three distinctive features. First, it shows that the classical logistic function, due to the specificity of its structure, cannot correctly describe market saturation in the case of using resources with limited serves. Second, even the use of a special multiplier that describes the market saturation process taking into account the depletion (finiteness) of the used resource does not obtain satisfactory economic results because of the \u201chigh speed\u201d depletion of this resource. Third, the analytical solution of the final model indicates the point in time at which changes in saturation rate occur. The latter situation allows us to determine the tracking of market saturation, which is more similar to the process that is actually occurring. We believe that this model can also be validated to estimate the production of wind turbines that use rare earth elements such as neodymium and dysprosium (for the production of powerful and permanent magnets for wind turbines). These results also suggest the need for oil-exporting countries to technologically diversify their economies to minimize losses in the transition to a low-carbon economy.<\/jats:p>","DOI":"10.3390\/math13010143","type":"journal-article","created":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T06:05:10Z","timestamp":1735797910000},"page":"143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Mathematical Model for Assessing New, Non-Fossil Fuel Technological Products (Li-Ion Batteries and Electric Vehicle)"],"prefix":"10.3390","volume":"13","author":[{"given":"Igor E.","family":"Anufriev","sequence":"first","affiliation":[{"name":"Higher School of Applied Mathematics and Computational Physics, Peter the Great St. Petersburg Polytechnic University, Politekhnicheskaya Ulitsa, 29, 195251 St. Petersburg, Russia"}]},{"given":"Bulat","family":"Khusainov","sequence":"additional","affiliation":[{"name":"Department of Science and Commercialization, Zhetysu University Named After I. Zhansugurov, St. Zhansugurov 187 A, Taldykorgan 040009, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3139-6509","authenticated-orcid":false,"given":"Andrea","family":"Tick","sequence":"additional","affiliation":[{"name":"Keleti Karoly Faculty of Business and Management, Obuda University, Tavaszmez\u0151 Str. 15\u201317, 1084 Budapest, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7993-4919","authenticated-orcid":false,"given":"Tessaleno","family":"Devezas","sequence":"additional","affiliation":[{"name":"Engineering Faculty, Atl\u00e2ntica Instituto Universit\u00e1rio, 2730-036 Barcarena, Portugal"}]},{"given":"Askar","family":"Sarygulov","sequence":"additional","affiliation":[{"name":"Department of Science and Commercialization, Zhetysu University Named After I. Zhansugurov, St. Zhansugurov 187 A, Taldykorgan 040009, Kazakhstan"}]},{"given":"Sholpan","family":"Kaimoldina","sequence":"additional","affiliation":[{"name":"Department of Science and Commercialization, Zhetysu University Named After I. Zhansugurov, St. Zhansugurov 187 A, Taldykorgan 040009, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,2]]},"reference":[{"key":"ref_1","unstructured":"IEA (2023). Emissions from Oil and Gas Operations in Net Zero Transitions: A World Energy Outlook Special Report on the Oil and Gas Industry and COP28, IEA."},{"key":"ref_2","unstructured":"Martin-Amouroux, J.-M. (2022). World Energy Consumption 1800\u20132000: The Results. Encyclopedia of Energy, Association des Encyclop\u00e9dies de l\u2019Environnement et de l\u2019\u00c9nergie. Available online: https:\/\/www.encyclopedie-energie.org\/en\/world-energy-consumption-1800-2000-results\/."},{"key":"ref_3","unstructured":"OPEC (2024). World Crude Oil Production by Country, OPEC."},{"key":"ref_4","unstructured":"IEA (2023). Oil Market Report, IEA."},{"key":"ref_5","unstructured":"World Bank (2024). World Bank Open Data\u2014Fuel Exports (% of Merchandise Exports)\u2014Russian Federation, World, Kazakhstan, Venezuela, RB, World Bank."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rodrigue, J.-P. (2024). The Geography of Transport Systems, Routledge. [6th ed.].","DOI":"10.4324\/9781003343196"},{"key":"ref_7","unstructured":"OICA (2023). 2023 Production Statistics|Www.Oica.Net, OICA. Available online: https:\/\/www.oica.net\/category\/production-statistics\/2023-statistics\/."},{"key":"ref_8","unstructured":"ACEA (2024). Economic and Market Report: Global and EU Auto Industry\u2014Full Year 2023, ACEA."},{"key":"ref_9","unstructured":"Hill, K. (2014). Just How High-Tech Is the Automotive Industry?, Center for Automotive Research (CAR)."},{"key":"ref_10","unstructured":"Stumpf, R. (2024, November 07). Here\u2019s About How Many Cars Are There in the World in 2023. Available online: https:\/\/www.thedrive.com\/guides-and-gear\/how-many-cars-are-there-in-the-world."},{"key":"ref_11","unstructured":"(2024, November 07). IEA Transport\u2014Energy System. Available online: https:\/\/www.iea.org\/energy-system\/transport."},{"key":"ref_12","unstructured":"IRENA (2019). Transforming the Energy System and Holding the Line on Rising Global Temperatures, International Renewable Energy Agency."},{"key":"ref_13","unstructured":"IEA (2024). Global EV Outlook 2024\u2014Moving towards Increased Affordability, International Energy Agency."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4257","DOI":"10.1073\/pnas.1500415112","article-title":"Criticality of Metals and Metalloids","volume":"112","author":"Graedel","year":"2015","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"e1400180","DOI":"10.1126\/sciadv.1400180","article-title":"By-Product Metals Are Technologically Essential but Have Problematic Supply","volume":"1","author":"Nassar","year":"2015","journal-title":"Sci. Adv."},{"key":"ref_16","unstructured":"OECD (2019). Global Material Resources Outlook to 2060: Economic Drivers and Environmental Consequences, OECD."},{"key":"ref_17","unstructured":"Pitron, G. (2021). The Rare Metals War: The Dark Side of Clean Energy and Digital Technologies, Scribe Publications."},{"key":"ref_18","unstructured":"Gielen, D. (2021). Critical Materials For The Energy Transition, IRENA."},{"key":"ref_19","unstructured":"Gielen, D., and Papa, G. (2024). Materials Shortage Will Not Stop the Energy Transition, If We Plan Ahead, IRENA."},{"key":"ref_20","unstructured":"(2024). Center for Sustainable Systems Critical Materials Factsheet|Center for Sustainable Systems, University of Michigan."},{"key":"ref_21","unstructured":"IEA (2023). The Role of Critical Minerals in Clean Energy Transitions\u2014Analysis, IEA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/978-3-031-16470-5_2","article-title":"How Green Is the Green Energy Transition? On the Road to Decarbonization","volume":"Volume 1","author":"Devezas","year":"2022","journal-title":"Global Challenges of Climate Change"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1007\/s40243-019-0146-z","article-title":"The Effect of Critical Material Prices on the Competitiveness of Clean Energy Technologies","volume":"8","author":"Leader","year":"2019","journal-title":"Mater. Renew. Sustain. Energy"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Aiyar, S., Chen, J., Ebeke, C., Garcia-Saltos, R., Gudmundsson, T., Ilyina, A., and Kangur, A. (2023). Geoeconomic Fragmentation and the Future of Multilateralism, International Monetary Fund. Staff Discussion Notes.","DOI":"10.5089\/9798400229046.006"},{"key":"ref_25","unstructured":"Cembalest, M. (2024). Eye on the Market 14th Annual Energy Paper: Electravision, J.P. Morgan. Eye on the Market 14th Annual Energy Paper."},{"key":"ref_26","unstructured":"Clarke, S. (2024, July 09). Energy Transition Will Move Slowly over the Next Decade. Available online: https:\/\/www.eiu.com\/n\/energy-transition-will-move-slowly-over-the-next-decade\/."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Devezas, T., Tick, A., Sarygulov, A., and Rukina, P. (2024). The Slow Pace of Green Transformation: Underlying Factors and Implications. Energies, 17.","DOI":"10.3390\/en17194789"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Tick, A., Akaev, A., Devezas, T.C., Sarygulov, A., Petryakov, A., and Evgenevich, A.I. (2024). Assessing Decarbonization Approaches across Major Economies. Energies, 17.","DOI":"10.3390\/en17174381"},{"key":"ref_29","unstructured":"IEA (2013). Global EV Outlook 2013\u2014Understanding the Electric Vehicle Landscape to 2020, Intrenational Energy Agency."},{"key":"ref_30","unstructured":"IEA (2017). Global EV Outlook 2017\u2014Two Million and Counting, International Energy Agency."},{"key":"ref_31","unstructured":"Editorial Board EVBoosters (2024). Global Investment in Battery Technology Booms Amid EV Surge, EV Market Reports."},{"key":"ref_32","unstructured":"Bloomberg NEF (2023). Lithium-Ion Battery Pack Prices Hit Record Low of $139\/kWh, Bloomberg NEF."},{"key":"ref_33","unstructured":"Knupfer, S.M., Hensley, R., Hertzke, P., Schaufuss, P., Laverty, N., and Krmaer, N. (2017). Electrifying Insights: How Automakers Can Drive Electrified Vehicle Sales and Profitability | McKinsey, McKinsey & Company."},{"key":"ref_34","unstructured":"Grelier, F., Poliscanova, J., Ambel, C.C., Bannon, E., and Alexandridou, S. (2019). Electric Surge: Carmakers\u2019 Electric Car Plans Across Europe 2019\u20132025, European Federation for Transport and Environment AISBL."},{"key":"ref_35","unstructured":"Lienert, P. (2022). $1.2 Trillion in Electric Vehicles and Batteries Through 2030, Reuters."},{"key":"ref_36","unstructured":"Lepre, N., Burget, S., and Gabriel, N.U.S. (2023). Investment in Electric Vehicle Manufacturing (2023)\u2014$210 Billion of Announced Investments in Electric Vehicle Manufacturing Headed for the U.S., ATLAS Public Policy."},{"key":"ref_37","unstructured":"IEA (2021). Japan 2021 Energy Policy Review, IEA."},{"key":"ref_38","unstructured":"Kwon, H. (2023). Essays on Electric Vehivle Policies, Faculty of the Graduate School of Cornell University."},{"key":"ref_39","unstructured":"(2024, November 07). Panion EV Incentives 2023: These 6 Countries Have the Best | PANION. Available online: https:\/\/www.panion.org\/ev-incentives-2023-these-6-countries-promote-e-mobility-the-most\/."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Jurlin, K. (2023). How Efficient and Socially Sensitive Are Fiscal Incentives for Electric Cars in Europe?. J. Risk Financ. Manag., 16.","DOI":"10.3390\/jrfm16060283"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jeem.2010.05.004","article-title":"Giving Green to Get Green? Incentives and Consumer Adoption of Hybrid Vehicle Technology","volume":"61","author":"Gallagher","year":"2011","journal-title":"J. Environ. Econ. Manag."},{"key":"ref_42","unstructured":"Giswold, J., Dong, M., Giswold, J., and Dong, M. (2023). Costing Support for EV Battery Manufacturing, Office of the Parliamentary Budget Officer."},{"key":"ref_43","unstructured":"Walton, R. (2024, November 07). Federal Incentives Help Drive $52B of Investments in 17 EV Battery Production Facilities: BofA. Available online: https:\/\/www.utilitydive.com\/news\/federal-incentives-ev-battery-production-report\/643370\/."},{"key":"ref_44","unstructured":"Cazzola, P., Paoli, L., and Teter, J. (2023). Trends in the Global Vehicle Fleet: Managing the SUV Shift and the EV Transition, GFEI."},{"key":"ref_45","unstructured":"Ritchie, H., and Roser, M. (2024, July 09). Tracking Global Data on Electric Vehicles. Available online: https:\/\/ourworldindata.org\/electric-car-sales."},{"key":"ref_46","unstructured":"(2024, July 09). Statista Electric Vehicles Worldwide-Statistics & Facts. Available online: https:\/\/www.statista.com\/topics\/1010\/electric-mobility\/."},{"key":"ref_47","unstructured":"IEA (2024). Batteries and Secure Energy Transitions, International Energy Agency."},{"key":"ref_48","unstructured":"Petavratzi, E., and Josso, P. (2021). Global Material Flows of Lithium for the Lithium-Ion and Lithium Iron Phosphate Battery Markets."},{"key":"ref_49","unstructured":"U.S. Geological Survey (2024). Geological Survey. Mineral Commodity Summaries 2024."},{"key":"ref_50","unstructured":"(2024, November 08). Statista Global Lithium Production 2023. Available online: https:\/\/www.statista.com\/statistics\/606684\/world-production-of-lithium\/."},{"key":"ref_51","unstructured":"U.S. Geological Survey (2022). Geological Survey. Mineral Commodity Summaries 2022."},{"key":"ref_52","unstructured":"GlobalData (2023). Critical Raw Materials Shaping Energy Transition, GlobalData."},{"key":"ref_53","unstructured":"IEA (2018). Global EV Outlook 2018\u2014Towards Cross-Modal Electrification, IEA."},{"key":"ref_54","unstructured":"Gifford, S. (2022). Lithium, Cobalt and Nickel: The Gold Rush of the 21st Century | Cornish Lithium Plc., The Faraday Institution. Faraday Insights."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Battery University Group (2023). BU-205: Types of Lithium-Ion, Battery University.","DOI":"10.1201\/9781003387879-24"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.trpro.2022.01.033","article-title":"Improvements in Post-Release Customs Control of Goods Moved by Different Modes of Transport","volume":"61","author":"Fuchs","year":"2022","journal-title":"Transp. Res. Procedia"},{"key":"ref_57","first-page":"21","article-title":"The Slow Lane: A Study on the Diffusion of Full-Electric Cars in Italy","volume":"19","author":"Bonacina","year":"2024","journal-title":"FEEM Work. Pap."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"7077","DOI":"10.1038\/s41467-023-42893-0","article-title":"Decarbonization Potential of Electrifying 50% of U.S. Light-Duty Vehicle Sales by 2030","volume":"14","author":"Woody","year":"2023","journal-title":"Nat. Commun."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"113276","DOI":"10.1016\/j.enpol.2022.113276","article-title":"Electric Vehicle Deployment & Fossil Fuel Tax Revenue in Mexico to 2050","volume":"171","author":"Bonilla","year":"2022","journal-title":"Energy Policy"},{"key":"ref_60","unstructured":"Gutierrez Energy Management Institute (2018). Driving the Future A Scenario for the Rapid Growth of Electric Vehicles, UH Energy."},{"key":"ref_61","unstructured":"Bloomberg NEF (2017). Electric Vehicle Outlook 2017\u2014Bloomberg New Energy Finance\u2019s Annual Long-Term Forecast of the World\u2019s Electric Vehicle Market, Bloomberg NEF."},{"key":"ref_62","unstructured":"Sitty, G., and Taft, N. (2016). What Will the gLoballLight-Duty Vehicle Fleet Look like through 2050?, Fuel Freedom Foundation."},{"key":"ref_63","unstructured":"EIA (2021). EIA Projects Global Conventional Vehicle Fleet Will Peak in 2038, Energy Today."},{"key":"ref_64","unstructured":"Fulton, L., Myers Jaffe, A., and McDonald, Z. (2019). Internal Combustion Engine Bans and Global Oil Use, Institute of Transportation Studies (ITS), University of California."},{"key":"ref_65","unstructured":"Global Fuel Economy Initiative (2020). GFEI 2020 Vehicle Efficiency and Electrification: A Global Status Report, Global Fuel Economy Initiative. Global Fuel Economy Initiative."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"103005","DOI":"10.1016\/j.trd.2021.103005","article-title":"Alternative Electrification Pathways for Light-Duty Vehicles in the European Transport Sector","volume":"99","author":"Rottoli","year":"2021","journal-title":"Transp. Res. Part Transp. Environ."},{"key":"ref_67","unstructured":"Nocedal, J., and Wright, S.J. (2006). Numerical Optimization, Springer."},{"key":"ref_68","unstructured":"Bloomberg NEF (2019). Electric Vehicle Outlook 2019, Bloomberg NEF."},{"key":"ref_69","unstructured":"Bloomberg NEF (2023). Electric Vehicle Outlook 2023, Bloomberg NEF."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"106611","DOI":"10.1016\/j.resconrec.2022.106611","article-title":"Regionalized Life Cycle Assessment of Present and Future Lithium Production for Li-Ion Batteries","volume":"187","author":"Schenker","year":"2022","journal-title":"Resour. Conserv. Recycl."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"106488","DOI":"10.1016\/j.resconrec.2022.106488","article-title":"Electric Vehicle Lithium-Ion Battery Recycled Content Standards for the US\u2014Targets, Costs, and Environmental Impacts","volume":"185","author":"Dunn","year":"2022","journal-title":"Resour. Conserv. Recycl."},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Gohlke, D., Rush, L., Kelly, J., and Dai, Q. (2021). Lithium-Ion Battery Supply Chain for E-Drive Vehicles in the United States: 2010\u20132020.","DOI":"10.2172\/1778934"},{"key":"ref_73","unstructured":"Racu, A. (2023). Battery Metals Demand from Electrifying Passenger Transport, Transport & Environment."},{"key":"ref_74","first-page":"53","article-title":"The Preponderance of Electric Vehicles and the Availability of \u201cGreen\u201d Electricity","volume":"Volume 2","author":"Devezas","year":"2024","journal-title":"Global Energy Transition and Sustainable Development Challenges\u2014Scenarios, Materials, and Technology"},{"key":"ref_75","unstructured":"Bloomberg NEF (2023). A Route for Solid-State Battery Adoption: Europe and U.S., Bloomberg NEF."},{"key":"ref_76","unstructured":"Lampinen, M. (2024). Silicon Anodes: A Pragmatic Path to Battery Transformation, Automotive World."},{"key":"ref_77","unstructured":"Poliscanova, J. (2024). Northvolt Collapse Shows Stamina and Leadership Are Lacking for Europe to Realise Its Battery Ambition. Transp. Environ., Available online: https:\/\/www.transportenvironment.org\/articles\/northvolt-collapse-shows-stamina-and-leadership-are-lacking-for-europe-to-realise-its-battery-ambition."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/13\/1\/143\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T15:23:39Z","timestamp":1759850619000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/13\/1\/143"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,2]]},"references-count":77,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["math13010143"],"URL":"https:\/\/doi.org\/10.3390\/math13010143","relation":{},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,2]]}}}