Alain Vzina

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Inverse modeling and the parsimony principle in
investigations of microbial food webs

• Alain Vézina
• with contributions from Olivier Berreville,
  Nathalie Niquil and Delphine Leguerrier,

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MICROBIAL FOOD WEB DYNAMICS IN THE OCEAN
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OUTLINE

• A bit of philosophy inverse problems and
  parsimony
• Quick primer on inverse analysis of microbial
  food web dynamics
• Different parsimony criteria and what they do
• How to stop worrying and learn to love parsimony
• Alternatives to parsimony?

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What is an inverse problem?

• Direct problem (forward problem)
• Given the cause, find its effect
• Given the question, find its answer
• Inverse problem (backward problem)
• Given the effect, find its cause
• Given the answer, find its question

Source Tuncay Aktosun,Mississipi State University
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An inverse problem
Answer Washington, George
Question Barbara, what is the state whose
capital city is Olympia? This question was asked
by George Bush, the former president of the U.S.,
while solving a crossword puzzle in the Sunday
paper
Source Tuncay Aktosun,Mississipi State University
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Problem of underdetermination

• For every set of data, there are infinitely many
  theories which are consistent with it

Source Wikipedia
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Ockams razor to the rescue?
Entia non sunt multiplicanda praeter necessitatem

• Physics
• Copernican vs Ptolemaic model of the solar system
• Mechanical vs caloric theory of heat
• Einsteinian vs Ether theory of electromagnetism
• Biology
• Individual vs group selection in evolutionary
  theory
• Cladist vs pheneticist systematics

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BUT
There are often different and clashing criteria
for what is the simplest hypothesis, and it is
not clear whether a simpler hypothesis is pro
tanto more likely to be true and if not, what
justification other than laziness there is for
adopting it. Philosophy of Science (1961)
While Occam's razor is a useful tool in the
physical sciences, it can be a very dangerous
implement in biology. It is thus very rash to use
simplicity and elegance as a guide in biological
research. Francis Crick
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MICROBIAL FOOD WEB DYNAMICS IN THE OCEAN
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DYNAMIC MODEL OF MICROBIAL FOOD WEB
Dependent on knowledge of parameters and
functional relationships
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INVERSE MODEL OF MICROBIAL FOOD WEB
Cut out the middle man (i.e. parameters,
functional forms)!
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SETTING UP THE INVERSE MODEL MASS BALANCE

• Flows along columns
• Compartments along rows

Am
bm
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SETTING UP THE INVERSE MODEL DATA

• Flows along columns
• Compartments along rows

Ad
bd
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SOLVING THE INVERSE MODEL
x vector of all flows
Minimize
Subject to
Infinity of plausible solutions
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PARSIMONY CRITERIA
Crit
Description
Rationale
Reference
minimize
Vézina and Platt 88
MN
Minimize squared
solution size
flows
minimize solution structure
SM
Minimize
Vézina et al. 2004
squared differences
between flows
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ARCTIC POLYNYA FOODWEB
MN
SM
SP
LP
SP
LP
FL
SC
FL
SC
LC
B
LC
B
A
A
CD
CD
Det
Det
300 - 2700
25 - 300
mgC/m2/d
2 - 25
0 - 2
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• Parsimonious solutions introduce distortions in
  reconstructed food webs
• zero/extreme flows (active constraints)
• equalizes flows over similar path lengths
• shortens the food chain, underestimates recycling
  processes

What to do, what to do?
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Potential approaches

• Get more data
• Get more solutions
• Calculate solutions differently

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Get more data
Multiple tracers
CarbonNitrogen or Phosphorus (Vézina and Pahlow
2003) - tend to be correlated -
effect on solution is small Stable isotopes
(Eldridge et al. 2005) - larger impact
on solutions - reduces the number of active
constraints (less zero/extreme flows)
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Incorporation of stable isotope data into inverse
model

• Flows along columns
• Compartments along rows

Am
bm
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Percent change in flow after adding stable
isotope constraint (from Eldridge et al. 2005)
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Get more data
Obvious approach Raises the complexity of the
solution Limitations Specialized data often not
available - reduces number of data sets for
comparisons Tracer data incomplete - requires
non-linear optimization Still dependent on
parsimony principle
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Get more solutions
Monte Carlo approaches

• Use stochastic simulations to generate a large
  number of alternative solutions
• Describe the space of possible solutions (mean,
  variance, correlation structure)
• Quantify the uncertainty of the solutions
• Ex. Savenkoff et al. 2004
• Richardson et al. 2004
• Kones et al. In press

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From Savenkoff et al. 2004 Can. J. Fish. Aquat.
Sci. 61 2194-2214
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1000 random solutions for a model with 23 flows
From Kones et al. J. Mar. Syst., in press
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Parsimonious solution not significantly different
from ensemble of random solutions
From Kones et al. J. Mar. Syst., in press
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Patterns in the correlation structure of the flows
From Kones et al. J. Mar. Syst., in press
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Get more solutions

• No reason not to do this!
• Can lead to Monte Carlo Markov Chain techniques
  to estimate the flows.
• Highlights Bayesian nature of these inverse
  analyses

Still based on minimizing least-squares
differences with the data (or maximizing the
likelihood of the data given the model) remains
tied to parsimony
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Calculate solutions differently

• Are there organizing principles that apply to
  ecosystems and that can be used to calculate
  inverse solutions?

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How much difference does it make?
Simulated inverse analyses
Start with a toy model
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3
Fix some of the flows as data and invert for the
rest
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Calculate inverse solutions that simultaneously
balance the flows and maximize/minimize
properties of the whole network (goal functions)
Direct optimization
Simulation
Minimize flows Maximize flows Maximize
productivity ? MEP (sensu Kleidon 2004)
Maximize ascendancy Maximize resilience
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Simulation of alternative solutions
Toy model has 5 compartments and 16 flows. 2 of
the flows are fixed as inputs and output to and
from the food web. Generate all possible
combinations of the remaining 14 flows without
replacement 16383 combinations Minimize and
maximize each combination of flows 32766
solutions
Screen solutions for those that maximize
ascendancy, resilience, etc
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3
Least-squares (parsimony)
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Max. Asc/Prod. (MEP)
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Max. Resil.
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Correlations among solutions
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Calculate solutions differently

• Use of ecosystem-level goal functions has a
  substantial impact on the solutions
• Max. ASC/Prod. (MEP)/Resil. tend to group
  together in contrast to parsimonious solutions
• Parsimonious solution approximates true solution
  best (although only one theor. model was tried
  but see Vezina et al. 2004. Prog. Oceanogr.)
• Pursue alternatives to parsimony
• Loose ecosystem-level rules rather than goal
  functions?

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Weak / strong interactions in food webs?
McCann. 2000. The diversity-stability debate.
Nature 405 228-233

• Food web theory traditionally based on analysing
  structural features
• - topology of food web links
• - diverse (complex) model food webs are less
  stable
• - contradicts observations and experimental
  evidence
• Recent shift towards using energy flux models
• - real world constraints on food web
  interactions
• - increasing complexity does not lead to
  instability
• - distribution of interaction strengths must be
  skewed towards weak interactions
• Parsimonious solutions spread the flows around
  more than alternatives (have more balanced
  distribution of interaction strengths)
• Possible to define a constraint space that
  excludes unstable distributions of interactions
  strengths?

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Conclusions

• Using parsimony is an epistemological stance
  has nothing to do with a biological or ecological
  basis
• Not just one definition of parsimony some may
  be better than others
• Whos afraid of parsimony? Improvements in
  observational capabilities and statistical
  techniques reduce our dependency on it.
• Alternatives to parsimony in filling in the
  blanks in food webs are unclear.

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Why bother?

• Simple to implement
• Information to run dynamic models is often not
  available
• Inversions of dynamic models are complex and
  operator dependent
• Linear inversions provide information on flows
  that are difficult or impossible to measure
  directly

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Has it been useful?

• Influence the development of dynamic models
• Importance of diatom grazing by microzooplankton
  (Fasham et al. 1994)
• Importance of detritus in material cycling
  (Vézina and Savenkoff 1999, Richardson et al.
  2004)
• Provides information on how flows change in
  response to experimental perturbations
• Metabolic flows across a nutrient gradient
  enrichment (Olsen et al. 2006)
• Net production of DMS in response to Fe
  enrichment (LeClainche et al. 2006)

Box model approach in fisheries (ECOPATH) has
been widely used and has resulted in syntheses of
large scale patterns in food web processes
(comparative aspect).