Current Challenges for Mathematical Modelling of Cyclic Populations (13w5151)

Arriving Sunday, November 10 and departing Friday November 15, 2013


Jonathan Sherratt (Heriot-Watt University)
Rebecca Tyson (University of British Columbia Okanagan)
Hao Wang (University of Alberta)


We propose to bring together ecologists and mathematicians with expertise in cyclic populations to discuss recent advances in our theoretical understanding of the causes and implications of population cycles from both the ecological and mathematical points of view. The main scientific objectives of the workshop are:

1. To bring mathematicians and ecologists together to address, from theoretical and experimental points of view, the mathematical questions surrounding persistence, spread and management of cyclic populations in dynamic landscapes.
2. To stimulate mathematical investigations and field experiments addressing these questions in ways that maximize the potential synergies between the disciplines of mathematics and ecology.

A 5-day workshop at the Banff International Research Station is an ideal format for the intense and highly interdisciplinary collaboration needed to address cyclicity in complex spatio-temporal landscapes effectively. There are many interesting 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. In addition, while new mathematical approaches have been developed in these areas, they are relatively new to ecologists. We see this workshop as an invaluable opportunity to foster cross-fertilization of mathematical ideas with ecological theory.

Each day of the 5-day workshop will consist of four 60-minute plenary talks, two in the morning and two in the evening. Talks will be grouped by topic, pairing mathematical and ecological talks, so as to stimulate observation of (1) the connection between new mathematical approaches and ecological theory, and (2) previously unsuspected links between the questions being asked by the two groups. Discussion sessions are an extremely important part of scientific workshops, and we propose to include several different discussion formats in order to maximize the opportunity for participants at all levels to express and exchange ideas. First, substantial time will be reserved for plenary discussions after each pair of talks in which the entire group of workshop participants engages in a dialogue about the research presented. This format has worked well in previous workshops, and the junior participants (students, postdocs, beginning faculty) benefit especially from exposure to the ideas of the whole group . In addition, there will be small group discussions in which groups of 5-10 participants gather for a formal discussion of subtopics suggested by the plenary lectures. Finally, there will be time for even smaller discussions between participants who wish to discuss current and future research collaborations. The list of topics will be finalised nearer to the time to take into account recent research developments, but will include (i) Seasonal forcing of cyclic populations (ii) Climate and latitude based gradients in cycles (iii) The role of space in population cycling.

Population cycling is a ubiquitous phenomenon, applying across a number of animal, insect and bird populations in a wide variety of ecosystems (Murdoch 2002, Turchin 2003). The dynamics of cyclic populations generate events of significant management and economic concern. For example, important cyclic events include periodic insect outbreaks, (e.g. spruce budworm - Bouchard & Pothier 2010, mountain pine beetle - Hicke 2006), population lows in economically valuable fish stocks (e.g. salmon - Krkosek 2011), closure of grouse moors during troughs in population cycles (Hudson et al 1998) and cycles in the observed effectiveness of biocontrol agents (Bruggen 2006). Our ability to manage, anticipate, and mitigate the effects of these cyclic populations rests heavily on our mathematical understanding of the processes that generate or drive the observed cyclicity.

Three recent developments have raised the profile and importance of research on population cycles. Firstly it has become clear that climate change is having a major effect on the demographic parameters of many cyclic populations (Ims 2008). Recently, it has been shown that season length can affect the onset or stability of cycles in stable populations (Frithjof & Tyson, in preparation), as can habitat fragmentation and loss (Strohm 2009). These point to potentially large effects of climate change that are, at present, only very superficially understood. Secondly an increasingly large amount of high quality spatiotemporal data on cyclic populations is becoming available, and in many cases this reveals complicated spatial dynamics, layered on top of the temporal cycles. Thirdly there is an increasing awareness of the environmental importance of population cycles, due for instance to the risk of extinctions during periods of low density, or damage to economically valuable crops during periods of high density.

This increase in ecological interest and importance coincides with recent mathematical developments that have the potential to confront the ecological challenges in a robust manner. These developments include new methods for studying the dynamics of periodically forced oscillators; the theory of absolute and convective stability and its study via the absolute spectrum; new methods linking invasion dynamics with spatiotemporal behaviour behind invasion; new work on averaging techniques which enable analytic study of seasonally complex models; methods in spatiotemporal statistics that enable newly available spatiotemporal data sets to be used effectively; and new methods in spatial optimal control theory that provide a firm and robust foundation for parameter estimation.