### Organizers

The globe is warming because humans are altering the global cycling and distribution of carbon. Fossil fuel burning, land management transfer carbon, and other nutrients formerly in relative stable pools into the atmosphere as CO2 and other gasses. These gasses in turn trap heat and alter the heat/energy budget of the earth, which in turn feeds back and alters element cycles further.

Global element cycles and energy flows present several problems to both ecologists and mathematicians. The most salient feature of the globe as a system is that it is closed to element cycles but open to energy fluxes. What happens when we close a dynamical system by coupling component open systems and still maintain the constraint of conservation of matter?

Element cycles are also not independent of one another but are coupled through relatively constant stoichiometries of elements for specific fluxes or specific compartments. How do changes in these constants alter the stabilities and trajectories of the closed global ecosystem as opposed to the more open sub-ecosystems that comprise it?

Feedbacks between ecosystem components can result in alternative stable states of material cycles. Changes in global control parameters (e.g., temperature, precipitation, and their spatial distributions) could cause rapid shifts between these stable states. What kind of bifurcations might underlie a closed system like the globe?

These are a few of many representative problems of global ecology with interesting biological and mathematical aspects. This workshop will bring together ecologists and mathematicians to explore these or other problems.

### Accepted Speakers

Monday, June 26, 2006 | |
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Time | Session |

09:00 AM 09:45 AM | John Pastor - Mathematical Challenges of Global Change The global climate system and the global cycles of carbon and nutrients are changing rapidly due to large inputs of fossil fuels from industrialized societies and land clearing. This is an unreplicated and uncontrolled experiment on a vast scale, and so mathematical approaches are essential for understanding the effect of this human-induced perturbation on the stability and transient responses of the global system. Various biotic feedbacks to the carbon cycle may amplify or dampen the response of the global climate system to fossil fuel combustion, but estimating the strength of these feedbacks remains an unsolved problem. The globe almost certainly has multiple states which may be separated by bifurcations that depend on such parameters as atmospheric temperature, carbon dioxide concentration, and uptake and decay rates of the earth's biota, but the nature of these bifurcations is poorly understood. Moreover, the cycle of carbon is linked to that of other elements (most especially nitrogen and phosphorus) in stoichiometric proportions and so perturbations to one cycle ramify through the others. Stochastic processes further complicate the system. Finally, changes at small spatial scales and short time scales may be propagated to those at larger and longer scales or may be overridden by large scale and slower processes, but mathematical techniques to handle these problems are just being developed. This Workshop will explore these and other questions by demonstrating mathematical approaches to answering these ecological questions and identifying new mathematical problems that arise from considerations of the biology and geology of energy and material fluxes at global scales. |

09:45 AM 10:30 AM | John Harte - Biotic Feedbacks to Global Climate Change Climate change can transform ecosystems and ecosystems affect climate; hence ecosystem-mediated feedbacks may influence future climate change. I present evidence from paleoclimatic change, from a recent drought, from ongoing climate-warming experiments, and from observational work along climate gradients, for strong feedback effects that are not incorporated in our current climate models. I show that a number of simplifying assumptions currently made in climate-ecosystem investigations are inadequate to the task of incorporating scaling and feedback into earth-system models, stressing the confounding role of species-level and even population-level, responses of vegetation to climate change. I conclude by suggesting several approaches to the daunting conceptual, empirical and mathematical challenges posed by the necessity of merging macroecology and climate science. |

11:00 AM 11:45 AM | Rob Armstrong - Carbon Fluxes in the Ocean: From Productivity at the Surface to Fluxes in the Deep Layers Here I examine several ways in which mathematical modeling might aid in solving problems in the global carbon cycle. (i) The ocean is a huge reservoir for carbon; the role of the deep ocean (the "bottom") must be appreciated before one can seriously address surface interactions (the "up"). (ii) Primary productivity drives the ocean carbon cycle; but it is the amount of carbon that sinks into the deep ocean that matters for carbon sequestration. And here taxonomic structure of the makes a crucial difference, since some taxa have mineral tests that sink rapidly, and others do not. (iii) Both spatial and taxonomic descriptions of the oceanic carbon cycle must start at their base from evolutionary principles, most notably optimization. |

Tuesday, June 27, 2006 | |
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Time | Session |

09:00 AM 09:45 AM | Raymond Pierrehumbert - Snowball Earth and a Warm Earth The human control exerted over climate in the form of anthropogenic global warming is novel only in that the agent of the climate change is sentient and can be reasonably expected to anticipate the consequences and (hopefully) act accordingly. Throughout the history of the Earth, it has in fact been quite routine for the biosphere to cause massive changes in the Earth's climate,which in turn alter the conditions in which life subsequently evolves. There is no general principle upon which one should expect these changes to be benign to the agents causing the change. I examine two signature cases of such an interaction. The first is the transition from a methane world on the Early Earth to an oxygen/CO2 world, precipitated by evolution of oxygenic photosynthesis; this transition is believed to have precipitated one or more Paleoproterozoic snowball Earth episodes, which may in turn have influenced the oxygenation of the atmosphere. The second is the evolution of vascular land plants, which have a profound effect on silicate weathering and lead to a cooler climate. I will discuss some aspects of these problems in terms of simplified coupled climate/biogeochemical models, with a particular emphasis on the role of vegetation cover feedback in the Mesozoic CO2 evolution. |

09:45 AM 10:30 AM | Bruce Peckham - Bifurcations, Alternative States, and Nutrient Fluxes in Peatlands Peatlands are significant ecologically because they contain a large percentage of the world's soil carbon and nitrogen. Some peatlands are dominated by moss, while others have moss coexisting with other shrubs. From a modeling point of view, moss is treated differently from shrubs because moss receives most of its nutrients directly from rainwater, rather than in uptake through roots. This leads to differential equations models which allow the possibility of either moss monoculture or coexistence equilibrium states. This talk will include discussion of the development of the models, their behavior, and bifurcations which can change the long term behavioir of the ecosystem. Main collaborator: John Pastor |

11:00 AM 11:45 AM | Barbara Bailey - Quantifying the Predictability of Noisy Nonlinear Biogeochemical Systems Statistical modeling of dynamical systems makes the estimation and construction of confidence intervals for interesting quantities from data possible. When noise is an integral part of the system's dynamics, a nonlinear time series approach can be used to quantify the dynamics and predictability of the system. This involves fitting nonlinear models and estimating dynamical systems quantities of interest such as global and local Lyapunov exponents, along with measures of uncertainty for these estimates. This approach will be used quantify the predictability of the effects of different types of noise on a simple biogeochemical model of plankton dynamics. The models consist of nonlinear systems of first-order differential equations for the flows or intercompartmental exchanges among nutrients, phytoplankton, zooplankton and detritus. |

Wednesday, June 28, 2006 | |
---|---|

Time | Session |

09:00 AM 09:45 AM | Robert Sterner - Stochiometric Element Cycling at Large Scales Global change routinely is analyzed via "single currency" models - for instance when scientists study the global carbon or global nitrogen budgets. However, interactions among these separate currencies are powerful, constraining features of element cycling at all scales from the micro- to the macroscopic. Future progress in ecological theory at the global scale will depend on improving the mathematical treatment of such stoichiometric interactions. Large scale stoichiometric relationships include the Redfield ratio of Earth's oceans. Redfield's observations linked the C:N:P:O ratios dissolved in ocean deepwater to the relevant ratios in the particulate matter in the surface ocean. He noted the close correspondence in these ratios and proposed that over geologic time scales, the cycles of C, N and O came into balance with P. Other stoichiometric relationships relate to "Liebigian" relationships among potentially limiting elements - global feedbacks to increased C and N are not independent but are linked. A key concept in stoichiometry at all scales is the degree of homeostatic regulation of organism element content. I will present a summary of key findings about dynamics of stoichiometrically constrained ecological systems, with a strong eye toward what we have to say now about global-scale dynamics and where we still have more work to do. |

09:45 AM 10:30 AM | Christopher Klausmeier - The Origin and Stability of Redfield Ratios Because phytoplankton live at the interface between the abiotic and the biotic compartments of ecosystems, they play an important role in coupling multiple nutrient cycles. The quantitative details of how these multiple nutrient cycles intersect is determined by phytoplankton stoichiometry. Here we review some classic work and recent advances on the determinants of phytoplankton stoichiometry and their role in determining ecosystem stoichiometry. First, we use a model of growth with flexible stoichiometry to reexamine Rhee and Goldman's classic chemostat data. Second, we discuss a recent data compilation by Hall and colleagues that illustrates some limits to phytoplankton flexibility and a model of physiological adaptation that can account for these results. Third, we discuss Redfield's mechanism for the homeostasis of the oceans' nitrogen-to-phosphorus stoichiometry and show its robustness to additional factors such as iron-limitation and temporal fluctuations. Finally, we use a model of resource allocation to determine the how the optimal nitrogen-to-phosphorus stoichiometry depends on the ecological conditions under which species grow and compete. |

11:00 AM 11:45 AM | Irakli Loladze - Stoichiometry of Wild Plants and Crops in the High CO2 World One of the most certain aspects of global change is the rising concentrations of atmospheric CO2 - the major nutrient for plants. Field experiments have shown that plants exposed to elevated CO2 frequently reduce transpiration but boost photosynthesis. Enhanced photosynthesis can increase carbon (C) assimilation, while decreased transpiration can reduce mass flow of nutrients to rhizosphere. This raises a question of how C:Nutirent content in plants will respond to elevated CO2. This question is important because several chemical elements such as iron (Fe), iodine (I), and zinc (Zn) are already deficient in the diets of the half of human population, which derives 84% of its calories from plant products. Apart from a ubiquitous decline of nitrogen (N) concentration in plant tissues, little is known about the effects of high CO2 on overall plant stoichiometry. Here, to the best of our knowledge, we present the most complete, albeit limited, database of CO2 effects on plant stoichiometry. In addition, we develop a dynamical model of a plant that reflects changes in C assimilation and transpiration as CO2 levels change. The analysis of the database and the model suggests that despite huge diversity of plant responses to elevated CO2, certain patterns in the changes of plant stoichiometry may prevail in the high CO2 world. |

Thursday, June 29, 2006 | |
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Time | Session |

09:00 AM 09:45 AM | David Schimel - Time and Space Scale Issues in Global Carbon Cycles Time and Space Scale Issues in Global Carbon Cycles |

10:15 AM 11:00 AM | Gabriel Katul - Scaling and the Analysis of Long Time Series Scaling and the Analysis of Long Time Series |

Friday, June 30, 2006 | |
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Time | Session |

Name | Affiliation | |
---|---|---|

Ågren, Göran | Goran.Agren@eom.slu.se | Ecology and Environmental Research, Swedish University of Agricultural Sciences |

Aguda, Baltazar | bdaguda@gmail.com | Mathematical Biosciences Institute, The Ohio State University |

Armstrong, Rob | rarmstrong@notes.cc.sunysb.edu | Department of Ecology and Evolution, State Univ. of New York @ Stony Brook |

Bailey, Barbara | babailey@stat.uiuc.edu | Mathematics, University of Illinois at Urbana-Champaign |

Berliner, Mark | mb@stat.osu.edu | Statistics, The Ohio State University |

Best, Janet | jbest@mbi.osu.edu | |

Cohen, Yosef | cohen006@umn.edu | Fisheries and Wildlife Conservation, University of Minnesota |

Djordjevic, Marko | mdjordjevic@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Eaturu, Anuradha | aeaturu@yahoo.com | Mathematical Sciences, Kent State University |

Enciso, German | German_Enciso@hms.harvard.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Goel, Pranay | goelpra@helix.nih.gov | Mathematical Biosciences Institute (MBI), The Ohio State University |

Grajdeanu, Paula | pgrajdeanu@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Guttal, Vishwesha | vguttal@princeton.edu | Physics, The Ohio State University |

Harte, John | jharte@berkeley.edu | Energy and Resources Group, University of California, Berkeley |

Jenerette, Darrel | gdj@email.arizona.edu | Ecology and Evoultionary Biology, University of Arizona |

Just, Winfried | just@math.ohio.edu | Mathematical Biosciences Institute, The Ohio State University |

Katul, Gabriel | gaby@duke.edu | Nicholas School Environment&EarthSciences, Duke University |

Kerkhoff, Andrew | kerkhoffa@kenyon.edu | Biology and Mathematics, Kenyon College |

Klausmeier, Christopher | klausme1@msu.edu | Kellogg Biological Station, Michigan State University |

Kuang, Yang | kuang@asu.edu | Mathematics and Statistics, Arizona State University |

Lim, Sookkyung | limsk@math.uc.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Lin, Shili | lin.328@osu.edu | Statistics, The Ohio State University |

Loladze, Irakli | iloladze@math.unl.edu | Mathematical Biosciences Institute, The Ohio State University |

Lou, Yuan | lou@math.ohio-state.edu | Mathematics, The Ohio State University |

Michalak, Anna | amichala@engin.umich.edu | Civil and Environmental Engineering, University of Michigan |

Moyer, Elizabeth | moyer@huarp.harvard.edu | Chemistry and Chemical Biology, Harvard University |

Nagy, John | john.nagy@sccmail.maricopa.edu | Life Sciences, Scottsdale Community College |

Nevai, Andrew | anevai@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Pastor, John | jpastor@nrri.umn.edu | Natural Resources Research Institute, University of Minnesota |

Peckham, Bruce | bpeckham@d.umn.edu | Mathematics and Statistics, University of Minnesota |

Pierrehumbert, Raymond | RTP1@geosci.uchicago.edu | Geophysical Sciences, University of Chicago |

Post, Wilfred | wmp@ornl.gov | Environmental Sciences Division, Oak Ridge National Laboratory |

Rietkerk, Max | m.rietkerk@geog.uu.nl | Environmental Sciences, Utrecht University |

Schimel, David | schimel@ucar.edu | Climate and Global Dynamics Division, National Center for Atmospheric Research |

Schugart, Richard | richard.schugart@wku.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Srinivasan, Partha | p.srinivasan35@csuohio.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Sterner, Robert | stern007@umn.edu | Ecology, Evolution and Behavior, University of Minnesota |

Stigler, Brandy | bstigler@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Tian, Paul | tianjj@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Torn, Margaret | mstorn@lbl.gov | Geochemistry Dept., Lawrence Berkeley Laboratory |

Uriarte, Maria | mu2126@columbia.edu | Ecology, Evolution & Env. Biology, Columbia University |

Vadrevu, Krishna | vadrevu.2@osu.edu | Entomology, The Ohio State University |

Wang, Hao | wanghao@math.gatech.edu | Mathematics and Statistics, Arizona State University |

Here I examine several ways in which mathematical modeling might aid in solving problems in the global carbon cycle. (i) The ocean is a huge reservoir for carbon; the role of the deep ocean (the "bottom") must be appreciated before one can seriously address surface interactions (the "up"). (ii) Primary productivity drives the ocean carbon cycle; but it is the amount of carbon that sinks into the deep ocean that matters for carbon sequestration. And here taxonomic structure of the makes a crucial difference, since some taxa have mineral tests that sink rapidly, and others do not. (iii) Both spatial and taxonomic descriptions of the oceanic carbon cycle must start at their base from evolutionary principles, most notably optimization.

Statistical modeling of dynamical systems makes the estimation and construction of confidence intervals for interesting quantities from data possible. When noise is an integral part of the system's dynamics, a nonlinear time series approach can be used to quantify the dynamics and predictability of the system. This involves fitting nonlinear models and estimating dynamical systems quantities of interest such as global and local Lyapunov exponents, along with measures of uncertainty for these estimates. This approach will be used quantify the predictability of the effects of different types of noise on a simple biogeochemical model of plankton dynamics. The models consist of nonlinear systems of first-order differential equations for the flows or intercompartmental exchanges among nutrients, phytoplankton, zooplankton and detritus.

Climate change can transform ecosystems and ecosystems affect climate; hence ecosystem-mediated feedbacks may influence future climate change. I present evidence from paleoclimatic change, from a recent drought, from ongoing climate-warming experiments, and from observational work along climate gradients, for strong feedback effects that are not incorporated in our current climate models. I show that a number of simplifying assumptions currently made in climate-ecosystem investigations are inadequate to the task of incorporating scaling and feedback into earth-system models, stressing the confounding role of species-level and even population-level, responses of vegetation to climate change. I conclude by suggesting several approaches to the daunting conceptual, empirical and mathematical challenges posed by the necessity of merging macroecology and climate science.

Scaling and the Analysis of Long Time Series

Because phytoplankton live at the interface between the abiotic and the biotic compartments of ecosystems, they play an important role in coupling multiple nutrient cycles. The quantitative details of how these multiple nutrient cycles intersect is determined by phytoplankton stoichiometry. Here we review some classic work and recent advances on the determinants of phytoplankton stoichiometry and their role in determining ecosystem stoichiometry. First, we use a model of growth with flexible stoichiometry to reexamine Rhee and Goldman's classic chemostat data. Second, we discuss a recent data compilation by Hall and colleagues that illustrates some limits to phytoplankton flexibility and a model of physiological adaptation that can account for these results. Third, we discuss Redfield's mechanism for the homeostasis of the oceans' nitrogen-to-phosphorus stoichiometry and show its robustness to additional factors such as iron-limitation and temporal fluctuations. Finally, we use a model of resource allocation to determine the how the optimal nitrogen-to-phosphorus stoichiometry depends on the ecological conditions under which species grow and compete.

One of the most certain aspects of global change is the rising concentrations of atmospheric CO2 - the major nutrient for plants. Field experiments have shown that plants exposed to elevated CO2 frequently reduce transpiration but boost photosynthesis. Enhanced photosynthesis can increase carbon (C) assimilation, while decreased transpiration can reduce mass flow of nutrients to rhizosphere. This raises a question of how C:Nutirent content in plants will respond to elevated CO2. This question is important because several chemical elements such as iron (Fe), iodine (I), and zinc (Zn) are already deficient in the diets of the half of human population, which derives 84% of its calories from plant products. Apart from a ubiquitous decline of nitrogen (N) concentration in plant tissues, little is known about the effects of high CO2 on overall plant stoichiometry. Here, to the best of our knowledge, we present the most complete, albeit limited, database of CO2 effects on plant stoichiometry. In addition, we develop a dynamical model of a plant that reflects changes in C assimilation and transpiration as CO2 levels change. The analysis of the database and the model suggests that despite huge diversity of plant responses to elevated CO2, certain patterns in the changes of plant stoichiometry may prevail in the high CO2 world.

The global climate system and the global cycles of carbon and nutrients are changing rapidly due to large inputs of fossil fuels from industrialized societies and land clearing. This is an unreplicated and uncontrolled experiment on a vast scale, and so mathematical approaches are essential for understanding the effect of this human-induced perturbation on the stability and transient responses of the global system. Various biotic feedbacks to the carbon cycle may amplify or dampen the response of the global climate system to fossil fuel combustion, but estimating the strength of these feedbacks remains an unsolved problem. The globe almost certainly has multiple states which may be separated by bifurcations that depend on such parameters as atmospheric temperature, carbon dioxide concentration, and uptake and decay rates of the earth's biota, but the nature of these bifurcations is poorly understood. Moreover, the cycle of carbon is linked to that of other elements (most especially nitrogen and phosphorus) in stoichiometric proportions and so perturbations to one cycle ramify through the others. Stochastic processes further complicate the system. Finally, changes at small spatial scales and short time scales may be propagated to those at larger and longer scales or may be overridden by large scale and slower processes, but mathematical techniques to handle these problems are just being developed. This Workshop will explore these and other questions by demonstrating mathematical approaches to answering these ecological questions and identifying new mathematical problems that arise from considerations of the biology and geology of energy and material fluxes at global scales.

Peatlands are significant ecologically because they contain a large percentage of the world's soil carbon and nitrogen. Some peatlands are dominated by moss, while others have moss coexisting with other shrubs. From a modeling point of view, moss is treated differently from shrubs because moss receives most of its nutrients directly from rainwater, rather than in uptake through roots. This leads to differential equations models which allow the possibility of either moss monoculture or coexistence equilibrium states. This talk will include discussion of the development of the models, their behavior, and bifurcations which can change the long term behavioir of the ecosystem.

Main collaborator: John Pastor

Other collaborators: Scott Bridgham, Jake Weltzin, Jiquan Chen

The human control exerted over climate in the form of anthropogenic global warming is novel only in that the agent of the climate change is sentient and can be reasonably expected to anticipate the consequences and (hopefully) act accordingly. Throughout the history of the Earth, it has in fact been quite routine for the biosphere to cause massive changes in the Earth's climate,which in turn alter the conditions in which life subsequently evolves. There is no general principle upon which one should expect these changes to be benign to the agents causing the change. I examine two signature cases of such an interaction. The first is the transition from a methane world on the Early Earth to an oxygen/CO2 world, precipitated by evolution of oxygenic photosynthesis; this transition is believed to have precipitated one or more Paleoproterozoic snowball Earth episodes, which may in turn have influenced the oxygenation of the atmosphere. The second is the evolution of vascular land plants, which have a profound effect on silicate weathering and lead to a cooler climate. I will discuss some aspects of these problems in terms of simplified coupled climate/biogeochemical models, with a particular emphasis on the role of vegetation cover feedback in the Mesozoic CO2 evolution.

Time and Space Scale Issues in Global Carbon Cycles

Global change routinely is analyzed via "single currency" models - for instance when scientists study the global carbon or global nitrogen budgets. However, interactions among these separate currencies are powerful, constraining features of element cycling at all scales from the micro- to the macroscopic. Future progress in ecological theory at the global scale will depend on improving the mathematical treatment of such stoichiometric interactions. Large scale stoichiometric relationships include the Redfield ratio of Earth's oceans. Redfield's observations linked the C:N:P:O ratios dissolved in ocean deepwater to the relevant ratios in the particulate matter in the surface ocean. He noted the close correspondence in these ratios and proposed that over geologic time scales, the cycles of C, N and O came into balance with P. Other stoichiometric relationships relate to "Liebigian" relationships among potentially limiting elements - global feedbacks to increased C and N are not independent but are linked. A key concept in stoichiometry at all scales is the degree of homeostatic regulation of organism element content. I will present a summary of key findings about dynamics of stoichiometrically constrained ecological systems, with a strong eye toward what we have to say now about global-scale dynamics and where we still have more work to do.