, and 2100 than in the 610 business-as-usual scenarios, so the average OG of the 611 remaining reserves is slightly higher. As a result, ? FD does 612 not have to decrease over time at the same rate as ? OG, 2020.

, it 614 eventually decreases rapidly and becomes equal to ? OG when 615 the average OG and reserve stock are equal to those calculated 616 in the business-as-usual scenarios (Figure 8a). The rapid 617 decline of ? FD is illustrated by the equally rapid decline of 618 primary production in 2120?2180 (Figure 9b,c) or 2240? 619 2280 (Figure 9d), FD cannot remain larger than ? OG for very long

, In the case of high recycling and regeneration rates, ? FD is 644 found to decrease after 2030 to follow the reference price p TI at 645 constant per-tonne profit of 600 US-$ 1998 /tonne. However, this 646 drop in ? FD is less pronounced than in the business-as-usual 647 scenarios and the increase in price to keep ? FD constant after 648 2020 is much lower. The price reaches, vol.8000, 1998.

, The comparison of Figure 9b,c shows that increasing CRRR mitigation and adaptation. In 660 addition, the cumulative amounts of metal lost would be 661 significantly reduced, from 4000 Mt in 2100 or 7000 Mt in 662 2300 (Figure 9b) to 2500 or 4500 Mt (Figure 9c), respectively. 663 These considerations are an urgent call for the implementation 664 of an efficient metal collecting, processing

, Interests and Limitations of the Prey?Predator

, The prey?predator dynamics used in the present 668 study is able to reproduce the 1900?2015 evolutions of copper 669 production, reserves, price, costs of production, revenues, Dynamics

, In particular, the demand was assumed to be inelastic, which is 690 not realistic and constitutes an obvious limitation of our 691 modeling. Moreover, all of the discussed scenarios assume a 692 constant long-term energy price, the range of, vol.25, 1998.

. Brent, Should the price of energy increase 694 significantly in the future, the production costs and price of 695 copper would increase more rapidly

, Another important source of uncertainty concerns the rate 698 of reserve regeneration. In the long run, Arndt et al. 19 recently 699 argued that the distribution of copper in the crust is not 700 bimodal but unimodal

A. Elshkaki, T. Graedel, L. Ciacci, and B. K. Reck, Copper demand, 759 supply, and associated energy use to 2050. Global Environ, Change, vol.760, pp.305-315, 2016.

A. Elshkaki, T. E. Graedel, L. Ciacci, B. K. Reck, and . Resource,

, Demand Scenarios for the Major Metals, Environ. Sci. Technol, vol.763, issue.52, pp.2491-2497, 2018.

M. K. Hubbert, Nuclear Energy and the Fossil Fuel. In Drilling 765 and Production Practice

D. H. Meadows, D. Meadows, J. Randers, and W. W. Behrens, , 1972.

, The oil drum: Europe. Posted by Cfris Vernon, 2007.

, The oil drum: Europe. Posted by L, 2010.

R. A. Kerr, The Coming Copper Peak, Science, vol.343, pp.722-774, 2014.

H. U. Sverdrup, D. Koca, and K. V. Ragnarsdottir, Peak metals, 776 minerals, energy, wealth, food and population: urgent policy 777 considerations for a sustainable society, J. Environ. Sci. Eng, issue.2, p.778, 2013.

H. U. Sverdrup, K. V. Ragnarsdottir, and D. Koca, On modelling 780 the global copper mining rates, market supply, copper price and the 781 end of copper reserves, Resour. Conserv. Recycl, vol.87, pp.158-174, 2014.

H. U. Sverdrup and K. V. Ragnarsdottir, Natural resources in a 783 planetary perspective, Geochem. Persp, vol.3, pp.129-130, 2014.

S. Northey, S. Mohr, G. Mudd, Z. Weng, and D. Giurco,

, Modelling future copper ore grade decline based on a detailed 786 assessment of copper resources and mining, Resour. Conserv. Recycl, vol.787, 2014.

J. E. Tilton and G. Lagos, Assessing the long-run availability of 789 copper, Resour. Policy, vol.32, pp.19-23, 2007.

V. Steinbach and F. Wellmer, Consumption and use of non-791 renewable mineral and energy raw materials from an economic 792 geology point of view, 1408.

F. Wellmer, Reserves and resources of the geosphere, terms 794 so often misunderstood. Is the life index of reserves of natural 795 resources a guide to the future?, Z. Dtsch. Ges, vol.159, pp.575-590, 2008.

L. D. Meinert, G. R. Robinson, and N. T. Nassar, Mineral 797 resources: Reserves, peak production and the future. Resources, vol.798, 2016.

M. Henckens, E. Van-ierland, P. Driessen, and E. Worrell,

, Mineral resources: Geological scarcity, market price trends, and future 801 generations, Resour. Policy, vol.49, pp.102-111, 2016.

N. T. Arndt, L. Fontbote, J. W. Hedenquist, and S. E. Kesler,

J. F. Thompson and D. G. Wood, Future global mineral resources

. Geochem and . Persp, , vol.6, pp.1-171, 2017.

T. Prior, D. Giurco, G. Mudd, L. Mason, and J. Behrisch,

, Resource depletion, peak minerals and the implications for sustainable 807 resource management, Global Environ. Change, vol.22, pp.577-587, 2012.

J. Harmsen, A. Roes, and M. K. Patel, The impact of copper 809 scarcity on the efficiency of 2050 global renewable energy scenarios. 810 Energy, vol.50, pp.62-73, 2013.

E. Nickless, A. Bloodworth, L. Meinert, D. Giurco, and S. Mohr, 812 Littleboy, A. Resourcing Future Generations White Paper: Mineral 813 Resources and Future Supply, 2014.

G. M. Mudd, Global trends in gold mining: Towards 816 quantifying environmental and resource sustainability, Resour. Policy, vol.817, pp.42-56, 2007.

G. M. Mudd, The environmental sustainability of mining in

, Australia: key mega-trends and looming constraints, Resour. Policy, vol.820, pp.98-115, 2010.

S. H. Ali, Mineral supply for sustainable development 822 requires resource governance, vol.543, 2017.

O. Vidal, B. Goffe, and N. Arndt, Metals for a low-carbon society
URL : https://hal.archives-ouvertes.fr/hal-01426278

, Nature Geosci, vol.6, issue.894, p.825, 2013.

O. Vidal, H. Leboulzec, and C. Francois, Modelling the material 826 and energy costs of the transition to low-carbon energy, EPJ Web, vol.827

. Conf, , vol.189, p.828, 2018.

D. May, T. Prior, D. Cordell, and D. Giurco, Peak minerals: 829 theoretical foundations and practical application, Nat. Resour. Res, vol.830, pp.43-60, 2012.

M. K. Hubbert and . Energy-resources,

J. Muller and V. Dirner, Using sigmoid functions for modelling 834

, South African gold production, Geosci. Eng, vol.56, pp.44-58, 2010.

J. Muller and H. Frimmel, Abscissa-transforming second-order 836 non-renewable resources, Math. Geosci, vol.43, p.838, 2011.

H. E. Frimmel and J. Muller, Estimates of mineral resource 840 availabilityHow reliable are they? Akademie fr Geowissenschaften und 841

. Geotechnologien, , vol.28, pp.39-62, 2011.

R. Bleischwitz, V. Nechifor, G. Saturation, and . Over-time, , p.843

, When Demand for Minerals Peaks

. Prisme, , vol.34, p.844, 2016.

R. Bleischwitz, V. Nechifor, M. Winning, and B. Huang,

, Extrapolation or saturation-Revisiting growth patterns, development 846 stages and decoupling, Global Environ. Change, vol.48, issue.35, p.847, 2018.

K. M. Johnson, J. M. Hammarstrom, M. L. Zientek, and . Dicken, , p.848

C. L. , Estimate of Undiscovered Copper Resources of the World

U. , , p.849

, Geological Survey, issue.36, p.850, 2013.

D. A. Singer, Future copper resources, Ore Geol. Rev, vol.86, p.851, 2017.

S. E. Kesler and B. H. Wilkinson, Earth's copper resources 853

, Geology, vol.36, pp.255-258, 2008.

A. J. Lotka, Elements of Mathematical Biology

. Publications, , vol.39, p.857, 1956.

V. Volterra, Variazioni e fluttuazioni del numero d'individui in 858 specie animali conviventi

C. Ferrari, , p.859, 1927.

U. Bardi and A. Lavacchi, A simple interpretation of Hubbert's 860 model of resource exploitation, vol.2, pp.646-661, 2009.

M. D. Gerst, Revisiting the cumulative grade-tonnage 862

, relationship for major copper ore types, Econ. Geol, vol.103, p.863, 2008.

M. D. Vieira, M. J. Goedkoop, P. Storm, and M. A. Huijbregts, , vol.865

, Ore grade decrease as life cycle impact indicator for metal scarcity: the 866 case of copper, Environ. Sci. Technol, vol.46, pp.12772-12778, 2012.

S. Gloser, M. Soulier, and L. A. Tercero-espinoza, Dynamic 868 analysis of global copper flows. Global stocks, p.869

, flows, recycling indicators, and uncertainty evaluation, Environ. Sci, vol.870

. Technol, , vol.47, pp.6564-6572, 2013.

, International Copper Study Group, The World, vol.872, 2017.

M. Soulier, S. Gloser-chahoud, D. Goldmann, and L. Espinoza,

A. , Dynamic analysis of European copper flows, Resour. Conserv, vol.876

. Recycl, , vol.129, pp.143-152, 2018.

, World Bank Indicators, vol.47, p.879, 2015.

, United Nations. UN Data, 2015.

J. Bolt and J. L. Van-zanden, The Maddison Project: collaborative 881 research on historical national accounts, Econ. Hist. Rev, vol.67, p.882, 2014.

J. N. Rauch, Global mapping of Al, Cu, Fe, and Zn in-use stocks 884 and in-ground resources, Proc. Natl. Acad. Sci. U.S.A, vol.106, p.885, 2009.

R. B. Gordon, M. Bertram, and T. E. Graedel, Metal stocks and 887 sustainability, Proc. Natl. Acad. Sci. U.S.A, vol.103, issue.51, p.888, 1209.

D. Singer and W. D. Menzie, Quantitative Mineral Resource, issue.53, p.894

G. Govett, World Mineral Supplies; Developments in Economic 895 Geology

R. Schodde, The Key Drivers Behind Resource Growth: An 897 Analysis of the Copper Industry over the Last 100 years, MEMS 898 Conference Mineral and Metal Markets over the Long Term

, Copper Statistics

D. L. Edelstein and . U. Ed, Geological Survey, vol.901, 2014.

T. Norgate and S. Jahanshahi, Low grade ores-smelt, leach or 903 concentrate? Miner. Eng, vol.23, pp.65-73, 2010.

R. L. Rudnick and S. Gao,

, Treatise on Geochem, vol.3, issue.659, 2003.

D. A. Singer, The lognormal distribution of metal resources in 907 mineral deposits, Ore Geol. Rev, vol.55, pp.80-86, 2013.

M. Pwc, PwCs annual review of global trends in the mining 909 industry, 2003.

O. C. Herfindahl, Copper Costs and Prices, pp.1870-1957

M. E. Schlesinger, M. J. King, K. C. Sole, and W. Davenport,

, Extractive Metallurgy of Copper, issue.62, 2011.

P. F. Chapman, The energy cost of producing copper and 916 aluminium from primary sources, Met. Mater, vol.8, pp.107-111, 1974.

P. F. Chapman, The energy costs of materials, Energy Policy, vol.918, pp.47-57, 1975.

G. M. Mudd and M. Diesendorf, Sustainability of uranium mining 920 and milling: toward quantifying resources and eco-efficiency

. Sci, , vol.42, 2008.

J. Johnson, E. Harper, R. Lifset, and T. E. Graedel, Dining at the 923 periodic table: Metals concentrations as they relate to recycling

, Environ. Sci. Technol, vol.41, pp.1759-1765, 2007.

T. G. Gutowski, S. Sahni, J. M. Allwood, and M. F. Ashby,

E. Worrell, The energy required to produce materials: constraints on 927 energy-intensity improvements, parameters of demand, Phil. Trans. R

. Soc, , vol.371, 2013.

G. Calvo, G. Mudd, A. Valero, and A. Valero, Decreasing ore 930 grades in global metallic mining: a theoretical issue or a global reality? 931 Resources, vol.5, 2016.

O. Vidal, F. Rostom, C. Francois, and G. Giraud, Global trends in 933 metal consumption and supply: the raw material-energy nexus, vol.13, pp.319-324, 2017.

M. Yellishetty, P. Ranjith, and A. Tharumarajah, Iron ore and steel 936 production trends and material flows in the world: Is this really 937 sustainable? Resour, Conserv. Recycl, vol.54, pp.1084-1094, 2010.

R. D. Rosenkranz, Energy Consumption in Domestic Primary 939 Copper Production, 1976. (71) 940 COCHILCO: Comision Chilena del Cobre, Statistical database 941 on production and energy use, 2014.

T. E. Graedel, E. M. Harper, N. T. Nassar, and B. K. Reck, On the 944 materials basis of modern society, Proc. Natl. Acad. Sci. U.S.A, vol.945, issue.112, pp.6295-6300, 2015.