Current Challenges for Mathematical Modelling of Cyclic Populations (15frg202)


(University of Bordeaux)

Rebecca Claire Tyson (University of British Columbia Okanagan)

(University of Miami)

Jonathan Sherratt (Heriot-Watt University)


The Banff International Research Station will host the "Focused Research Group on Current Challenges for Mathematical Modelling of Cyclic Populations: Final writing and submission of a multi-author review paper arising from a previous BIRS workshop" workshop in Banff from November 8 to November 15, 2015.

The high-amplitude natural oscillations of animal populations, or population cycles, occurring in species as diverse as lynx, moth or salmon, is a widespread ecological phenomenon that has stimulated a century of research at the interface between mathematics and biology. Its implications range from the management of insect pests to the fundamental understanding of boreal and arctic ecosystems. Using mathematical models, applied mathematicians and ecological modellers have made significant advances in explaining how population cycles form and persist, and how these cycles arise and interact in time and space.

Mathematical work has established that trophic interactions (e.g. interactions between predators and prey, parasites and hosts) often underlie observed population cycles. There are, however, many other potential causes to cycles, and in numerous cases biotic interactions between individuals are heightened by seasonal or climatic variation. In recent years, the role attributed to such environmental variation has increased notably, which is all the more important now that scientists are trying to predict the consequences of climate change. There are many mathematical and ecological approaches that address multi-annual population cycles and spatio-temporal patterns of abundance, but efforts to merge the two are challenging because of the need to address the complex spatial and temporal dynamics simultaneously. Challenges include the sheer time required for simulations, discontinuities resulting from seasonal changes in behaviour, sensitive dependence on the functional forms used in the models, and complex bifurcation structures that arise in forced and unforced oscillatory systems.

The paper we are writing is a valuable compendium of state-of-the art knowledge on cyclic populations, and current mathematical challenges for future management. Our work summarizes the known causes and consequences for whole ecosystem population cycles, with a special focus on how environmental variation can affect population cycling, what generates current changes in the dynamics of cyclic species, as well as the spatial aspects of population cycling, including the puzzling phenomenon that nearby populations frequently tend to fluctuate in synchrony. The result is a comprehensive reference of current mathematical explanations for observed behaviours in cyclic populations, and a clear agenda for future mathematical research on the many unanswered questions that remain.

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).