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The Banff International Research Station will host the "New Mathematical Methods for Complex Systems in Ecology" workshop in Banff from July 28, 2019 to August 2, 2019. Mathematical modelling in ecology has a long and illustrious history in explaining various aspects of complex dynamics of ecosystems. Mathematical tools used in modelling have been constantly evolving to become more and more efficient. New challenges and ever increasing demand for improving the predictive power of models and for a better understanding of the complexity of the underlying systems have inspired the emergence of novel research areas. An important example is given by the theory of non-autonomous dynamical systems under slow variation of parameters. Developing this theory is expected to help predicting sudden regime shift or critical transitions in ecosystems, in particular as a result of the global climate change. An alternative approach to understanding critical transitions focuses on long-living transients: dynamical regimes mimicking asymptotical behaviour for a long but finite time. In spite of its obvious relevance and importance, mathematics of long-living transients is in its infancy. Another emerging area is mathematics of multiscale monitoring which is required in order to link the theoretical findings to the dynamics of real ecosystems. A huge practical challenge is designing efficient monitoring systems allowing for the existence of multiple scales and environmental complexity. Finally, new mathematical tools are needed to analyse the functional sensitivity of models prediction with respect to the choice of specific mathematical formulations. This workshop brings together applied mathematicians, researchers in mathematical ecology and theoretical ecologists to address these issues in order to clarify the state-of-the-art in this field, to refine the existing challenges and problems, to highlight important recent findings, and to develop a roadmap to facilitate fast future progress in mathematics of complex ecosystems. Along with the presentations made by the leading experts, an important part of the workshop will be group discussions aimed to formulate the precise mathematical statements of relevant problems and consolidate scattered facts, and “brainstorm sessions” where we are hoping to make a breakthrough in some of the above topics. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada-s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta-s Advanced Education and Technology, and Mexico-s Consejo Nacional de Ciencia y Tecnología (CONACYT).