# Emerging Challenges at the Interface of Mathematics, Environmental Science and Spatial Eco (11w5106)

Arriving in Banff, Alberta Sunday, July 3 and departing Friday July 8, 2011

## Organizers

Stephen Cantrell (University of Miami)

Robert Holt (University of Florida)

Mark Lewis (University of Alberta, Canada)

## Objectives

Traditional mathematical models in ecology, epidemiology, evolution, and related areas typically assume that the environment is uniform in time and space and that the transport of organisms, chemicals, genes, and pathogens through the environment is random. In reality, many environments are heterogeneous in time and/or space and many physical processes and behavioral responses involve nonrandom transport.

Incorporating heterogeneity and nonrandom transport into models for biological processes and then analyzing such models leads to significant new mathematical challenges. There are three unifying features that many of those challenges have in common: (i) The first is that quantities of interest that can be computed readily for random transport in homogeneous environments are, at present, difficult or even impossible to compute or analyze for parameter dependence when there is heterogeneity and/or biased movement; (ii) The second, which often gives rise to the first, is that the mathematical methods and physical or biological insights that allow us to understand simple models cannot be used in the presence of heterogeneity, anisotropy, or nonrandom transport; for example, it is easy to obtain an explicit formulae for the principal eigenvalues of the relevant Laplace operators in simple geometries and uniform space, and these immediately lead to a simple formulation for the minimal size required for a region to sustain a diffusing population with given growth and diffusion rates. On the other hand, there are no such formulae for the principal eigenvalue of a periodic parabolic operator with advective terms and variable coefficients, and there are few results on how the eigenvalues of such operators depend on parameters. Thus, it is much harder to determine the minimal habitat size needed for persistence in environments with semi-realistic patterns of spatial variation. A significant reason for this limitation in our current understanding is that orthodox analytic methods (such as the variational formulation of eigenvalues) can no longer be used, so progress requires the development of new methods; (iii) The third feature concerns the nonlocal nature of many transport processes which can include long-distance displacement or jumps. When transport is nonlocal, local partial differential equation models must be extended to include nonlocal interactions, resulting in integrodifferential or related integrodifference equations. Here, standard mathematical tools for classical spatial models based on partial differential equations, such as regularity theory, maximum principles or the existence of principal eigenvalues, do not necessarily apply. There is considerable empirical evidence that nonlocal dispersal occurs in some populations, but its significance has rarely been addressed outside of the context of invasion theory. New mathematics will be needed for the analysis of nonlocal dispersal models in other contexts. We envisage that the workshop help define and develop the new mathematics needed for such analysis.

Properly addressing these important issues will not only require new mathematical ideas but also close and on-going interaction between researchers who analyze models (mainly mathematicians) and those who pose questions and formulate models and attempt to link model to data (mainly biologists). We propose a workshop bringing both groups together to identify important analytic challenges associated with models for environmental heterogeneity and nonrandom transport and to develop new mathematical approaches to addressing them. New mathematical challenges come to the fore in various ways; to illustrate the nature of these challenges and to frame the problems more concretely, we the following describes a number of specific examples.

The rate of spread of a diffusing population into a uniform environment can, typically, be determined by a direct calculation based on a linear analysis, and such an analysis can often be extended to interacting populations. There are some abstract mathematical extensions of these notions to variable and stochastic environments, and there has been some empirical work on the topic, but analytic computations or estimates of spread rates are available only in a few special cases. Similarly, the basic reproduction number for epidemiological models, R0, can be computed explicitly by matrix theoretic methods, assuming a nonspatial context and a temporally constant environment. Recently, there have been extensions of R0 theory to include infinite-dimensional models and temporally periodic environments. However, in practice, it remains a challenge as how to calculate the quantity in variable or spatially complex environments. Quantities such as R0 and spread rates that play a critical role in our understanding of models can sometimes be characterized in terms of eigenvalues for certain operators or matrices or as Floquet coefficients, but determining those analytically in terms of model parameters in the context of spatio-temporal heterogeneity presents many difficulties that will require new mathematical ideas to resolve. By mixing top mathematicians and quantitatively skilled environmental scientist together in the workshop, we expect to not only address the above challenges mathematically but to also provide insight back to the questions of environmental concern that originally inspired the mathematical models.

Additional new mathematical challenges have recently arisen from issues of pressing environmental concern with anisotropic and directional movement. Examples include streams and rivers, where diffusion is augmented by downstream advection plus anthropogenic water flow manipulation, and terrestrial environments where structures such as roads can lead to movement anisotropy. Advection has been treated in connection with various purely physical processes, and anisotropic diffusion has been studied in the context of image processing, but neither of those areas of research addresses how spatial effects interact with nonlinear population or community dynamics. In the case of models with advection, standard methods based on variational principles do not apply directly, so alternative methods must be developed.

Finally, we will also consider the impacts of environmental change on populations from the context of optimal dispersal strategies. How should individuals within a population adjust their movement behavior in response to increasing environmental heterogeneity (both spatial fragmentation and temporal fluctuations in environmental conditions)? The issue of the evolution of dispersal strategies under changing environmental conditions is a question of very substantial interest in theoretical biology. It is also of urgent applied interest given the growing evidence for rapid evolutionary shifts in dispersal rates in organisms subsequent to land use changes and alterations in climatic range limits. There has been work on this problem from a number of diverse mathematical viewpoints, ranging from game theory, adaptive dynamics, to optimization to mathematical quantitative genetics, but each of those approaches emphasizes certain aspects of the phenomenon and ignores others. A major mathematical challenge in this area is the development of a synthesis of the different modeling approaches that has the capacity to address topics ranging from mechanistic descriptions of movement under environmental heterogeneity, to the evolutionarily and convergent stability of dispersal strategies, all within a unified framework.

Incorporating heterogeneity and nonrandom transport into models for biological processes and then analyzing such models leads to significant new mathematical challenges. There are three unifying features that many of those challenges have in common: (i) The first is that quantities of interest that can be computed readily for random transport in homogeneous environments are, at present, difficult or even impossible to compute or analyze for parameter dependence when there is heterogeneity and/or biased movement; (ii) The second, which often gives rise to the first, is that the mathematical methods and physical or biological insights that allow us to understand simple models cannot be used in the presence of heterogeneity, anisotropy, or nonrandom transport; for example, it is easy to obtain an explicit formulae for the principal eigenvalues of the relevant Laplace operators in simple geometries and uniform space, and these immediately lead to a simple formulation for the minimal size required for a region to sustain a diffusing population with given growth and diffusion rates. On the other hand, there are no such formulae for the principal eigenvalue of a periodic parabolic operator with advective terms and variable coefficients, and there are few results on how the eigenvalues of such operators depend on parameters. Thus, it is much harder to determine the minimal habitat size needed for persistence in environments with semi-realistic patterns of spatial variation. A significant reason for this limitation in our current understanding is that orthodox analytic methods (such as the variational formulation of eigenvalues) can no longer be used, so progress requires the development of new methods; (iii) The third feature concerns the nonlocal nature of many transport processes which can include long-distance displacement or jumps. When transport is nonlocal, local partial differential equation models must be extended to include nonlocal interactions, resulting in integrodifferential or related integrodifference equations. Here, standard mathematical tools for classical spatial models based on partial differential equations, such as regularity theory, maximum principles or the existence of principal eigenvalues, do not necessarily apply. There is considerable empirical evidence that nonlocal dispersal occurs in some populations, but its significance has rarely been addressed outside of the context of invasion theory. New mathematics will be needed for the analysis of nonlocal dispersal models in other contexts. We envisage that the workshop help define and develop the new mathematics needed for such analysis.

Properly addressing these important issues will not only require new mathematical ideas but also close and on-going interaction between researchers who analyze models (mainly mathematicians) and those who pose questions and formulate models and attempt to link model to data (mainly biologists). We propose a workshop bringing both groups together to identify important analytic challenges associated with models for environmental heterogeneity and nonrandom transport and to develop new mathematical approaches to addressing them. New mathematical challenges come to the fore in various ways; to illustrate the nature of these challenges and to frame the problems more concretely, we the following describes a number of specific examples.

The rate of spread of a diffusing population into a uniform environment can, typically, be determined by a direct calculation based on a linear analysis, and such an analysis can often be extended to interacting populations. There are some abstract mathematical extensions of these notions to variable and stochastic environments, and there has been some empirical work on the topic, but analytic computations or estimates of spread rates are available only in a few special cases. Similarly, the basic reproduction number for epidemiological models, R0, can be computed explicitly by matrix theoretic methods, assuming a nonspatial context and a temporally constant environment. Recently, there have been extensions of R0 theory to include infinite-dimensional models and temporally periodic environments. However, in practice, it remains a challenge as how to calculate the quantity in variable or spatially complex environments. Quantities such as R0 and spread rates that play a critical role in our understanding of models can sometimes be characterized in terms of eigenvalues for certain operators or matrices or as Floquet coefficients, but determining those analytically in terms of model parameters in the context of spatio-temporal heterogeneity presents many difficulties that will require new mathematical ideas to resolve. By mixing top mathematicians and quantitatively skilled environmental scientist together in the workshop, we expect to not only address the above challenges mathematically but to also provide insight back to the questions of environmental concern that originally inspired the mathematical models.

Additional new mathematical challenges have recently arisen from issues of pressing environmental concern with anisotropic and directional movement. Examples include streams and rivers, where diffusion is augmented by downstream advection plus anthropogenic water flow manipulation, and terrestrial environments where structures such as roads can lead to movement anisotropy. Advection has been treated in connection with various purely physical processes, and anisotropic diffusion has been studied in the context of image processing, but neither of those areas of research addresses how spatial effects interact with nonlinear population or community dynamics. In the case of models with advection, standard methods based on variational principles do not apply directly, so alternative methods must be developed.

Finally, we will also consider the impacts of environmental change on populations from the context of optimal dispersal strategies. How should individuals within a population adjust their movement behavior in response to increasing environmental heterogeneity (both spatial fragmentation and temporal fluctuations in environmental conditions)? The issue of the evolution of dispersal strategies under changing environmental conditions is a question of very substantial interest in theoretical biology. It is also of urgent applied interest given the growing evidence for rapid evolutionary shifts in dispersal rates in organisms subsequent to land use changes and alterations in climatic range limits. There has been work on this problem from a number of diverse mathematical viewpoints, ranging from game theory, adaptive dynamics, to optimization to mathematical quantitative genetics, but each of those approaches emphasizes certain aspects of the phenomenon and ignores others. A major mathematical challenge in this area is the development of a synthesis of the different modeling approaches that has the capacity to address topics ranging from mechanistic descriptions of movement under environmental heterogeneity, to the evolutionarily and convergent stability of dispersal strategies, all within a unified framework.