# Impact of climate change on biological invasions and population distributions (13w5095)

## Organizers

Henri Berestycki (L Ecole des Hautes Etudes en Sciences Sociales Paris)

Alan Hastings (University of California Davis)

Mark Lewis (University of Alberta)

Péter Molnár (Princeton University)

## Objectives

The issues outlined are broad and important, but initially progress must be made on more specific questions that can be defined in ways amenable to mathematical treatments. The purpose of this BIRS workshop is to generate, develop and apply new tools for the analysis of invasions and population distributions under environmental change. Thus the mathematical problems that we will address are intrinsically spatial. Not only do temperatures vary with latitude, the very processes of population spread and shifts in range boundary are modelled using equations with spatial operators. Hence the mathematical and scientific challenges are significant, but the potential payoffs are large, making this worthy of being held as a BIRS workshop.

The workshop will focus on the following issues:

Shifts in species range boundaries under climate change: As temperatures continue to increase, temperature isoclines will move quickly, particularly in northern countries, at rates predicted to be in excess of 1000 meters per year. Which species will keep up with shifting temperature isoclines and which will fall behind, and how can this be determined using dynamical systems? Initial attempts to solve this problem have used nonlinear reaction diffusion equations in a fixed moving coordinate frame for temperature. However, the meeting would provide the opportunity to discuss and develop more realistic models that include integral operators that allow for long-distance dispersal, variable-speed coordinate frames for temperature, and changing external environmental conditions. Other possibilities include slowly decaying initial conditions, which can appear as the result of a colonization-retraction event, and then may lead to accelerating rates of spread. Mathematical analysis of such systems requires gradually varying travelling waves. However, most current approaches focus on asymptotic behavior with constant parameters and these applications will require time varying parameters and responses on similar time scales. Applications range from understanding the spread of mountain pine beetle from western to central Canada, to analyzing the potential for loss of endangered plant species due to climate warming to tracking the advance of pine processionary moth as it moves across Europe.

Thus, many types of biological invasions are induced by climate change. Several challenging mathematical questions are related to such biological invasions in general and require new, sophisticated, mathematical developments. The proposed meeting will be the occasion to evaluate the state of the art in mathematical modelling as well as to identify the key mathematical challenges that arise or that are required for further progress. One of the aims is to bring together modelers, mathematicians and ecologists with this goal in mind.

Dynamics of invasive species under variable environments: One hallmark of recent global warming is an increase in climatic variability. How will this increased climatic variability affect spreading speeds for biological invaders? Analyses of integrodifference equations in temporally random environments have shown that spreading speeds depend crucially upon the precise form variability takes and upon statistical correlations in the joint processes of growth and dispersal. This analysis relies upon the abstract ergodic theory for nonlinear infinite-dimensional operators. At the same time practical experiments in growth chambers are underway to examine dynamics like this in a highly replicated fashion using flour beetles (Tribolium). The meeting will allow mathematicians and ecologists to compare theory with experiment and move the understanding of species' responses to global change forward.

Multispecies interactions under changing environmental conditions: Historical records show vast changes in the competitive interactions of plants as they shifted behind retreating ice sheets after the last ice age. One possible explanation is that some plants may have escaped spatially from their natural predators during vegetation shifts, allowing them to become "supercompetitors." Initial models for such systems have involved spreading speeds for multispecies integrodifferential equations, but these have not been fully analyzed. Indeed, mathematical analysis of such systems is particularly challenging due to the lack of a comparison theorem for estimating spreading speeds. This is a very active area of mathematical research.

This is but one of many ways that spatial interactions between species can be affected by climate change. Climate may also influence competitive relationships of animals, as evidenced by the northward shift of the red fox and the simultaneous range contraction of its competitor, the Arctic fox. Predator-prey impacts may be as simple as the decline of local polar bear populations due to the bears' limited ability to access their prey in a spatially varying melting sea ice habitat, or as complex as the climate-driven collapse of population cycles in a lemming-four-predator system in Greenland. Mutualistic relationships may also be disrupted by climate change. For example, changes in blooming phenology have resulted in temporal mismatches between the availability of plants and their pollinators. One of the goals of the workshop is to develop a framework to categorize the types of perturbations that climate change can induce on such multispecies spatial interactions and to investigate the kinds of mathematical analysis that are possible. This will be achieved by bringing together ecological theoreticians and mathematicians adept in the analysis of multispecies population models to share recent results and chart a course for future research.

Shifting patterns of vegetation under climate change: As rainfall patterns change, some regions are predicted to come under increasing drought. One response of vegetation to dry conditions is to form distinct spatially heterogeneous vegetation bands. These bands of "tiger bush" can be found in many semi-arid regions throughout the globe. Formation of vegetation bands can be understood, with the aid of mathematics, as a pattern formation problem in the context of plant growth and competition for water. Here nonlinear advection-reaction-diffusion models, that describe the processes, can predict the wavelength of vegetation bands and also explain how the vegetation bands slowly shift across the landscape. Hysteresis effects are also present: the nature of the bands depends not only on rainfall levels, but also on whether the level is increasing or decreasing. More significantly the process of desertification itself appears to exhibit hysteresis, making it difficult to revegetate once a threshold has been passed. Here mathematical models can actually be used to identify factors to predict imminent desertification. This workshop will further develop the theme of vegetation patterns as indicators of environmental change with the use of mathematical models.

The interplay between emerging environmental questions and new mathematics: The meeting will bring together researchers in dynamical systems with researchers in ecological and environmental modelling.

Not only will the mathematical analysis inform the science. The science itself should lead to new mathematical challenges and insights, particularly in the following areas:

* Long range dispersal and non standard diffusion are topics that clearly are important perspective for modelling in ecology processes under climate change. Ecological problems represent a wealth of mathematical challenging problems that will require both insights in modeling and mathematical developments at the forefront of research.

* Likewise, careful analysis of the role of heterogeneity in dispersal shape and speed in the presence of environmental change is an important topic for ecology.

* In many instances it can be argued that dispersal involves several scales. Thus, developing mathematical approaches that combine multiscale diffusion will be important for having more realistic models.

* The models that have been introduced to study the impact of climate shift often rely on reaction- diffusion models with forced speed. In such models the ability of a population to survive mostly depends on its growth and dispersal capabilities. However, it is known that populations can also adapt themselves to new environments. Models that takes adaptation phenomena into account has been proposed. A rigorous mathematical analysis of this promising approach should be carried out. This would enable us to understand the intertwined effects of growth, dispersal and adaptation in the success of a biological invasion.

* New mathematical developments are required to effectively study of reaction-diffusion systems with forced speeds for interacting species.

* Questions of model calibration and parameter identification much need to be set on more firm ground. Models that combine stochastic terms with evolution equations will be much relevant in this perspective.

* The genetic consequences of range expansions have not yet received much attention from mathematicians and modelers. However, range expansions are known to have an important effect on the shaping of genetic diversity of many species, and generally lead to a loss of genetic diversity along the expansion axis. More general model will involve systems that take into account a population with varying traits. In mathematical terms this brings a new degree of complexity that combine reaction-diffusion equations with genetically evolving populations. Various classes of models can be envisioned in this framework for instance integro- differential equations modeling long-distance dispersal events.

In summary, the meeting will provide a synergistic research environment, where researchers that do not usually interact will come together and experience cross-disciplinary opportunities for developing and applying methods of dynamical systems to population dynamics under environmental change.