SOME TOPICS, THOUGHTS, AND SUGGESTED PARTICIPANTS
FOR A WORKSHOP ON GLOBAL ECOLOGY
Overall Theme
What would be the attributes of a minimal mathematical model for the global ecosystem? What
mathematical properties would follow from these attributes? What are the biological
analogues/implications?
Possible Topics and Some Thoughts
1. What are the biological and geological structures of input-output and source-sink
relationships of global element cycles and how do the mathematical properties of
models depend on resolution of these source-sink relationships and input-output
budgets?
The Biological Problem: The development of global models has proceeded by first developing
detailed models of one of three compartments (oceans, atmosphere, terrestrial biosphere). In the
first phase of development, transfers between these compartments were treated as inputs and
outputs, values for which were specified from measurement data with great uncertainty. More
recently, models that couple two or more of these compartments have been developed in great
spatial detail (especially coupled atmosphere-ocean models), leaving transfer with the biosphere
to be put in “by hand” with great uncertainty. Thus, coupling of compartments begins to
effectively close material cycles as transfers that were previously treated as inputs-outputs are
now subsumed in feedback loops, some of which may be non-linear, leaving fewer inputs and
outputs to be freely specified. Eventually, a complete global model will be open to fewer
pathways of material inputs and outputs and most of the mass balance will be subsumed in a set
of linear and nonlinear feedbacks. Furthermore, what constitutes an input and output depends to
some extent on the time scales of interest. For example at the ecological time scale, the ocean
sediments are a sink and the final “output” of carbon, but at geological time scales, volcanic
activity, tectonic uplift, and carbonate weathering put this sink back into the atmosphere. Thus
ecologically the global carbon cycle is open (with an ocean sediment sink) but geologically it is
closed. These different time scales pose interesting questions on what measurements are
important to answer the question of what ultimately controls the global carbon cycle.
Mathematical Problems: What happens when we close a dynamical system by coupling open
systems and still maintain the constraint of conservation of matter? This does not appear to be as
simple as taking limits as I, O → 0. Some work suggests that for some model ecosystems the
constraints on stability of interior equilibria switch signs when the cycle is closed (Williams
1972, Pastor and Cohen 1997, Loreau and Holt 2004). Is this a general result? If so, then an open
ecological model of the globe (with an ocean sink) will necessarily behave mathematically very
differently from a geologically closed model. Recent work by Loreau and Holt (2004) raises
interesting mathematical questions about how input-output ratios constrain stabilities of sourcesink relationships in model ecosystems. How might these apply to global models?
2. What accounts for the seeming constancy of ratios of elemental fluxes between
compartments of the global ecosystem?
The Biological Problem: In 1958, Alfred Redfield reported a remarkable constancy of elemental
ratios of organic debris from marine organisms settling on the ocean floor of 80C: 15N: 1P. But
upwelling waters contain these elements in the ratios 800C: 15N: 1P. Redfield suggested that
phytoplankton control seawater stoichiometry partly by importing nitrogen from the atmosphere
via N fixation. Broecker (1974) suggested the excess C can be accounted for by downward
settling of CaCO3. His recalculated “Redfield ratios” for sinking particles are 120C: 15N: 1P:
40Ca, but the ratios for upwelling waters are 800C: 15N: 1P: 3200Ca. Clearly, constancy of the
Redfield ratios requires that transfers take place between the ocean and other compartments of
the global ecosystem. Similar, albeit not as constant, ratios have been suggested for terrestrial
biota as well (eg., one set of ratios for woody plants, another set for grasses, etc). Why are these
ratios so constant? What does these different ratios imply for stability and recovery of global
element cycles after perturbation?
Mathematical Problems: Here we must couple a dynamical system of one element (eg. carbon,
often assumed to be a surrogate for energy) with dynamical systems of other elements (eg., N and
P). How might such ratios of fluxes arise in coupled dynamical systems? Alternatively, what
happens to the properties of a dynamical system when one imposes stoichiometric constraints?
Recent modeling studies (Tyrrell 1999, Lenton and Watson 2001, Klausmeier et al. 2004) shed
some light on these questions. Are constancies of ratios a consequence of certain stable equilibria
or are there multiple optimal ratios corresponding to multiple stable equilibria or to different
transient phases (Klausmeier et al 2004)? How do changes in these ratios alter the stabilities and
trajectories of ecosystems? How do the ratios relate to the openness of the system? How are they
related to the structure of food webs? Some early work by May (1973) following on prior work
by Ulanowicz (1972) indicates that the stability of trophic systems with recycling requires that
energy content per unit biomass must increase as one climbs the trophic ladder. Does this result
also apply to element ratios?
3. How rapidly and in what manner can the global ecosystem change in response to
changes in control parameters?
The Biological Problem: Changes in global control parameters (temperature, precipitation, and
their spatial distributions) could cause rapid shifts in the stability and distribution of ecosystems.
Examples include vegetation-precipitation feedbacks, vegetation-nutrient feedbacks, and water
chemistry-density-temperature feedbacks in the oceanic thermohaline circulation system. Thus,
slow changes in a control parameter (e.g., temperature, which is a result of the open energy flux)
could result in rapid changes in material fluxes and their distributions at the global scale.
Mathematical Problems: Feedbacks between sub-ecosystems of the globe can cause shifts
between multiple or alternative stable states. Such rapid changes in ecosystem behavior and
distributions with changes in control parameters appear suspiciously like bifurcations in
dynamical systems (Strogatz 1994). Bifurcation theory is not well-known in the biological
sciences, but could account not only for different equilibria but also their domains of stability
with regard to material fluxes (Hilbert et al. 2000, Loladze et al. 2000, Pastor et al. 2002). Some
recent work suggests that bifurcations demarcate different ecosystems with different carbon and
nutrient balances in model ecosystems (Hilbert et al. 2000, Pastor et al. 2002). Can we use
bifurcation theory to describe different stable states of global element cycles? Can bifurcation
theory help predict rapid changes in ecosystem and global carbon cycles as the temperature
warms gradually? How do the bifurcations relate to the underlying symmetries imposed by mass
balance? How do they relate to the (topological?) arrangement of positive and negative feedbacks
in the dynamical system? Do they relate to differences in time constants (turnover rates) of
compartments and rate constants of processes? How do stoichiometric ratios affect the nature of
the bifurcations (see especially Loladze et al. 2000)?
References
Broecker, W.S. 1974. Chemical Oceanography. Harcourt Brace Jovanovich, NY.
Hilbert, D.W., N. Roulet, and T, Moore. 2000. Modeling and analysis of peatlands as dynamic
systems. J. Ecol. 88: 230-242.
Klausmeier, C.A., E. Lichtrman, T. Daufresne, and S.A. Levin. 2004. Optimal nitrogen-tophosphorus stoichiometry of phytoplankton. Nature 429: 171-174.
Lenton, T.M. and A.J. Watson. 2001. Redfield revisited 1. Regulation of nitrate, phosphate, and
oxygen in the ocean. Global Biogeochemical Cycles 14: 225-248.
Loladze, I., Y. Kuang, and J.E. Elser. 2000. Stoichiometry in producer-grazer systems: linking
energy flow with element cycling. Bulletin of Mathematical Biology 62: 1137-1162.
Loreau, M. and R. Holt. 2004. Spatial flows and the regulation of ecosystems. American
Naturalist 163: 606-615.
May, R.M. 1973. Mass and energy flow in closed ecosystems: a comment. J. Theoretical Biology
39: 155-163.
Pastor, J. and Y. Cohen. 1997. Herbivores, the functional diversity of plants species, and the
cycling of nutrients in ecosystems. Theoretical Population Biology 51: 165 -179.
Pastor, J., B. Peckham, S.D. Bridgham, J.F. Weltzin, and J. Chen. 2002. Plant community
composition, nutrient cycling, and alternative stable equilibria in peatlands. American Naturalist
160: 553-568.
Redfield, A.C. 1958. The biological control of chemical factors in the environment. American
Scientist 46: 205-221.
Strogatz, S.H. 1994. Nonlinear Dynamics and Chaos. Perseus Books, Reading, Mass.
Tyrrell, T. 1999. The relative influences of nitrogen and phosphorus on oceanic primary
production. Nature 400: 525-531.
Ulanowicz, R.E. 1972. Mass and energy flow in closed systems. J. Theoretical Biology 34: 239253.
Williams, F.M. 1972. The mathematics of microbial populations, with emphasis on open systems.
Pp. 395-427 in E.S. Deevey, ed. Growth By Intussusception: Essays in Honor of G. Evelym
Hutchinson. Transactions of the Connecticut Academy of Sciences.
Possible Structures for the Workshop

Each day devoted to one (or possibly two) major topics
If two, one in the morning and one in the afternoon

Begin each session with a tutorial lecture in either mathematical approaches or biological
problem (perhaps 2 tutorial lectures) followed by 2-3 “case studies” exploring an aspect
of the problem in detail.

One or two commentators present thoughts at the end of each session.

Leave plenty of time for discussion – finish formal talks by 4PM? Long lunches?

Tutorial sessions on computers in models/programs/software? One afternoon devoted to
this?

An evening plenary talk?

Social functions in evening?

Summary paper for AMS Notices? Other outlet?
Co-organizers:
John Harte
The Department of Environmental Science, Policy and Management
305 Hilgard Hall
University of California
Berkeley, CA 94720
jharte@socrates.berkeley.edu
David Schimel
Climate and Global Dynamics Division
The National Center for Atmospheric Research
1850 Table Mesa Dr.
Boulder, CO 80307
schimel@ucar.edu
Possible invitees:
Göran Ågren
Swedish University of Agricultural Sciences
Box 7072, SE-750 07 Uppsala Sweden
Goran.Agren@eom.slu.se
Ernesto Bosatta
Swedish University of Agricultural Sciences
Box 7072, SE-750 07 Uppsala Sweden
Ernesto.Bosatta@eom.slu.se
Frank Berendse
Wageningen Universiteit en Research Centrum (DLO)
Departement Omgevingswetenschappen
Nature Conservation and Plant Ecology Group
Bornsesteeg 69
NL-6708 PD Wageningen
Arnhem/Nijmegen
The Netherlands
Frank.Berendse@STAF.TON.WAU.NL
Stephen Carpenter
Center for Limnology
University of Wisconsin
680 North Park St.
Madison WI 53706
srcarpen@wisc.edu
Carlos Castillo-Chavez
Arizona State University
Dept. of Mathematics and Statistics
PO Box 871804
Tempe, AZ 85287-1804
chavez@math.asu.edu
Yosef Cohen
Dept. of Fisheries and Wildlife Conservation
University of Minnesota
St. Paul MN 55108
Cohen006@tc.umn.edu
Roddy Dewar
INRA
Unité de Bioclimatologie
Bordeaux, France
dewar@bordeaux.inra.fr
William Fagan
Dept. of Biology
University of Maryland
College Park, MD 20742
bfagan@glue.md.edu
Chris Field
Carnegie Institution of Washington
Department of Global Ecology
260 Panama Street
Stanford, CA 94305
cfield@globalecology.stanford.edu
George Hornberger
Dept. of Statistics
University of Virginia
Halsey Hall, P O Box 400135
Charlottesville, VA 22904-4135
bc5g@virginia.edu
Darrel Jenerette
Dept. of Plant Biology
Arizona State University
Tempe AZ 85287
jenerette@asu.edu
Christopher Klausmeier
School of Biology
Georgia Institute of Technology
310 Ferst Drive
Atlanta, GA 30332-0230
christoper.klausmeier@biology.gatech.edu
Yang Kuang
Dept. of Mathematics
Arizona State University
Tempe AZ 85287-1804
kuang@asu.edu
Simon A. Levin
Dept. of Ecology and Evolutionary Biology
Princeton University
Princeton, NJ 08544
levin@princeton.edu
Irakli Loladze
Dept. of Mathematics
Arizona State University
Tempe AZ 85287-1804
loladze@asu.edu
Michel Loreau
Laboratoire d'Ecologie
UMR 7625
Ecole Normale Superieure
46, rue d'Ulm
F-75230 Paris Cedex 05, France
Loreau@ens.fr
Donald Ludwig
Mathematics Department
Ecology Hut B8 Room 120H
UBC, Vancouver, BC V6T 1Z2
ludwig@math.ubc.ca
Ross McMurtrie
Room 406A
Biological Science Building
University of New South Wales
Sydney, Australia
r.mcmurtrie@unsw.edu.au
Steve Pacala
Department of Ecology and Evolutionary Biology
Princeton University
Princeton, NJ 08544-1003
pacala@princeton.edu
Bruce Peckham
Dept. of Mathematics and Statistics
University of Minnesota Duluth
Duluth MN 55812
bpeckham@d.umn.edu
W.M. Post
Oak Ridge National Laboratory
Environmental Sciences Division
PO Box 2008, Bldg 1509
Oak Ridge TN 37831
wmp@ornl.gov
Ed Rastetter
The Ecosystems Center
Marine Biological Laboratory
Woods Hole, MA 02543
erastett@mbl.edu
Marten Scheffer
Dept. of Aquatic Ecology and Water Quality Management
Wageningen University
PO Box 8080
6700 DD Wageningen
The Netherlands
Marten.Scheffer@wur.nl
Harlan Stech
Dept. of Mathematics and Statistics
University of Minnesota Duluth
Duluth MN 55812
hstech@d.umn.edu
Robert Sterner
Dept. of Ecology, Evolution and Behavior
University of Minnesota
St. Paul MN 55108
stern007@tc.umn.edu
Jianguo Wu
Dept. of Plant Biology
Arizona State University
Tempe AZ 85287
jingle@asu.edu