Integrodifference Equations in Ecology: 30 years and counting (16w5121)

Arriving in Banff, Alberta Sunday, September 18 and departing Friday September 23, 2016

Organizers

(University of Washington)

(University of Ottawa)

(University of Glasgow)

(University of Alberta, Canada)

Objectives

Statement of Objectives

The main purpose of this proposed BIRS workshop is to instigate the development of novel theory and tools related to the mathematical analysis of IDEs and their applicability to pressing ecological and environmental challenges of our times. The workshop will foster a synergistic research environment where researchers from different areas of dynamical systems, stochastic processes, non-local operators, invasion and conservation biology who do not usually interact will exchange ideas and results and to develop joint ideas and projects for the future. The workshop will include experienced scientists as well as young researchers worldwide. The workshop will focus on the following aspects.

  • Spatial Heterogeneity: Traveling wave solutions often serve as descriptions of species invasion and range expansion processes. To study spreading speeds one typically assumes that the habitat is spatially homogeneous. In reality, most landscapes are heterogeneous on many scales. Organisms often have clear habitat preferences and adjust their movement according to resource availability, landscape features or conspecific density. Prevention programs against invasive species might create additional heterogeneity through targeted removal of resources (e.g., host plants of forest invasive insects) or localized application of pesticides. How then does one model dispersal in such environments? And what are the effects of landscape variation and spatially localized intervention on the spread of an organism?

    Few papers have dealt with IDEs in heterogeneous landscapes. They all assume temporally static, spatially periodic heterogeneity, employ relatively simple dispersal kernels and assume spatially continuous solutions of the IDE [4, 9, 22, 25]. Recently, novel dispersal kernels were derived from random walk models in patchy landscapes. These kernels are discontinuous, as are the resulting solutions of the corresponding IDE [19, 20]. A novel analytical framework is needed to study discontinuous solutions, emergent traveling waves and related spreading speeds. Temporally varying landscapes are models for global change of climatic conditions. Initial research for IDEs with moving habitat patches unveils how dispersal may facilitate or hinder a species' ability to keep up with climate change [26].

    The workshop will bring together modelers and analysts with shared interest in these questions, review the state of the art of models and devise a way forward to deal with their analysis. Particular discussion questions include: How can we model dispersal in heterogeneous landscapes? Which of the resulting dispersal kernels are supported by empirical data? And how can we collect data for heterogeneous dispersal processes when data collection for homogeneous landscapes is already difficult? How does spatial heterogeneity affect ecological communities?

  • Multispecies models: Most theory on IDEs considers single-species models, but realistic ecosystem descriptions need to include several species and their interactions. Such models are relatively simple to formulate but their analysis poses great challenges. Simple two-species competition models have monotonicity properties, so that comparison theorems allow analytical results. Yet, even those models show some surprisingly anomalous spreading speeds [15]. Three-species competition or predator-prey relationships generally do not allow comparison theorems, and little is known about spreading speeds and traveling waves in these models. Novel phenomena arise, such as the formation of spatial stable patterns [21] or cyclic and chaotic behavior in the wake of an invasion [10, 23]. Some of these emergent patterns are well understood in RDEs, and currently being developed for IDEs, but many phenomena are still unexplored. An additional difficulty arises when sessile species (or species with sessile life stages) are included in these models as the next-generation operator in the IDE model fails to be compact, and classical existence theorems for traveling waves fail. There is much recent interest in traveling wave theory for non-compact and non-monotone operators [3]. A completely new challenge from a modeling and analysis perspective is to include movement behavior that depends on the presence of other species. In RDE systems, such questions lead to cross-discussion models that are notoriously difficult to analyze. For IDEs only a single numerical study exists. Hence, the workshop is a very timely opportunity for analysts and modelers to share these results and explore ways to tackle the next round of challenging problems in this area.

  • Stochasticity and data: Although deterministic models were successful in predicting the speed of invading populations they do not capture the patchy spread and variation in invasion speeds observed in real systems. Stochastic models are crucial for quantifying the variability in spread rates, yet despite their importance, there is relatively little work on stochastic IDEs. Extrinsic stochasticity (caused by environmental factors) and intrinsic variability (based on demographic processes) affect spread rates in different ways: the latter typically reduces invasion speed while the former may increase spread [6, 12, 17]. These results are based on branching random walks, a stochastic analogue of IDEs, as well as moment closure techniques and perturbation analysis. These tools need to be extended and novel tools developed to better understand the behavior of stochastic IDEs and to quantify the variation in spreading speed and other ecologically significant quantities.

    A great challenge for all mathematical models is to reconcile theory with empirical studies since large temporal and spatial scales are involved in invasion processes and conservation measures and since we typically observe only one realization of the process. The recent experimental work of replicating invasions in a laboratory setting offers a unique opportunity to better understand stochastic invasion empirically~[18]. The high variability in spread rates found in these experiments could not be explained by demographic stochasticity alone. This advance in the empirical understanding offers a timely opportunity to guide a new push in the much needed theoretical developments of stochastic IDEs and bring those researchers working on the empirical and analytical side of the problem together. Open topics include: extending existing stochastic IDE theory to multi-species systems (initial work has been done for stage-structured populations) and expanding our understanding of non-linear stochastic IDEs. Non-linear stochastic IDEs remain largely unexplored to date. The few existing studies show that they exhibit quite different behaviour from their linear counter parts. Yet, since non-linear equations are most applicable for describing real ecological systems, novel theory needs to be developed to understand these systems and use them in ecological applications.

  • Non-local operators: Several other mathematical modeling frameworks are closely related to IDEs, yet researchers from these fields rarely interact. One of the aims of the workshop is bring these diverse groups together in the general setting of non-local operators and to provide a stimulating environment for exchange of ideas, tools and results.

    Reaction-diffusion equations with non-local terms appear in various places in the literature [8]. The non-local operator may describe non-local interaction or movement, depending on the ecological question. No systematic theory of these equations is currently available, however, there is much recent interest in studying the qualitative dynamics of these equations, for example, the study of accelerating waves through tracking of level sets or the generalization of the theory of lambda-omega systems from RDEs.

    Integral projection models (IPMs) project the density of a population forward in discrete generations, while individuals are continuously structured by state (e.g., size) [5]. IPMs are formulated very similarly to IDEs but details (e.g., typical shapes of the kernels) and research questions are quite different. In a spatial setting, when individuals are structured by continuous state and location, IPMs and IDEs are merged, and the resulting model has both aspects, e.g., kernels that represent progression through states and kernels that represent movement in space.

    Impulsive reaction-diffusion equations are the most recent class of models presented here [14]. In the simplest relevant form, a reaction-diffusion process acts for a certain period of time (e.g., the time of a year in which an organism is active), and an impulse describes the outcome of the inactive phase or may capture the effects of harvesting. Hence, the setting is very similar to that of a simple IDE with the movement process modeled explicitly. In fact, certain linear impulsive RDEs are equivalent to certain linear IDEs, but their nonlinear extensions typically are not. The study of impulsive RDEs in ecology is only in its infancy, and therefore the workshop could act as a catalyst to launch this exciting new research direction.



    We envision several specific outcomes from this workshop that will help popularize IDEs and make their theory and application more accessible to a wider range of mathematical and ecological researchers.

  • Publications: We aim to publish a summary of open research problems and challenges as complied during the workshop in two journal articles, one centered around the mathematical analytical challenges and one outlining a research program for relating IDEs to empirical work and data. To this end, we will invite participants to submit their thoughts on the most pressing issues prior to the workshop, and we will hold group discussion sessions during the workshop to categorize and annotate the input. We will identify potential lead authors for these two publications prior to the workshop among the invited participants.

  • Electronic resources: We aim to set up a global, electronic resource for the theory and application of IDEs, for example in the form of a website, wiki or other as appropriate. A similar initiative in the field of Adaptive Dynamics is already hugely successful, see http://mathstat.helsinki.fi/~kisdi/addyn.htm. While this initiative is managed by a single individual (Eva Kisdi) we envision a community-based resource that also links to videos of presentations (e.g., the BIRS talks and previous conferences) and other material such as relevant online course notes, conference posters and computational tools.


  • Training opportunities: We aim to invite a substantial number of junior researchers to the workshop and invite them to present their current research to the audience of experienced scientists early on during the workshop. The daily schedule during the workshop will allow for impromptu meetings and discussions where junior researchers can receive feedback and new inspiration. We might also approach carefully selected junior researchers as potential lead authors for the two journal articles.



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