『Abstract
　The intricate dynamics of world oil production are traditionally modeled by a single differential equation, the Hubbert model, arguably too simple in nature. In this article, the Hubbert model is extended to higher-order models which break the symmetry of the Hubbert curve, one of its main controversial features. Our numerical results exhibit a shift of the peak into the future followed by a steeper decline in production, resembling recent production curves of some natural gas fields. Finally, a new supply-demand-reserves model is introduced which is capable of reproducing the US peak, based on more realistic assumptions for production changes.

Keywords: Hubbert; oil peak; supply-demand model』

Introduction
The Hubbert model
Supply-demand model
Supply-demand-reserves model (A)
Supply-demand-reserves model (B)
Discussion and conclusions
Acknowledgments
References