Connectance Modification and Eigenvector Analysis of Food Webs

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Connectance Modification and Eigenvector Analysis
of Food Webs

• Jonathan L. Bowers, Meridith Bartley, Dr. Albert
  J. Meier
• Center for Biodiversity Studies, Department of
  Biology,
• Western Kentucky University

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Why Study Food Web Ecology?

• Best method for depicting feeding relationships
• Provide, although complex, models of species
  interactions and biodiversity
• Whats not interesting about this ?????

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Major Contributors

• Charles Elton (1927)
• Lindeman, R.L. (1942)
• May (1973)
• Pimm Lawton (1975)
• Also, Joel Cohen, Bernard Patten, Gary Polis

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Concepts of Research Interest

• Trophic Levels of Food Webs
• Pyramid Scheme or Semi-Cyclic Flow
• Compartmentalization of Systems
• Energy Flows
• Applications of properties (connectance, linkage
  density) toward trophic pathways in food webs
• But.

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What about Indirect Pathways?

• Importance of Indirect Pathway Study
• Mathematical Modeling
• Study Greater Effects of Predator and Prey Gain
  and Loss
• Use of Theoretical Connectance Modification

Species B Consumes A
A
B
VERSUS
C
A
B
INDIRECT RELATIONSHIP BETWEEN SPECIES A AND
C
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Predator and Prey Connectance Modification

• Artificial introduction, not of new species, but
  change in feeding patterns of existing species
• Limits on variability
• Test Predators and Preys effect on increases of
  indirect pathways and the variable distribution
  of such

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Method for Depicting Food Webs

• Food Webs depicted mathematically by adjacency
  matrices
• Binary matrices
• Zeros denote no direct link
• Ones denote feeding link
• In the matrices, the predator is in the column
  consuming prey in row
• Column Eats Row
• So, energy transfer is from row to column in
  these adjacency matrices

A
B
C
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Simple Food Web Construct vs. Lavigne
Spaghetti Model (1992)
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A Method For Obtaining Indirect Links

• Since the adjacency matrix shows direct links and
  the length of those paths are simply one, to get
  paths of length two, you would square the matrix
  (multiply by itself)

X

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Methods, Tests, and Materials

• 12 Food Webs ranging from Maine, North Carolina,
  and New Zealand
• Pine Forest, Tussock and Pasture Grassland,
  Broadleaf Forest
• Diversity of habitat and climate
• Presence of ooze or detrital organic matter in
  original web was criterion for selection
  (Lindeman 1942)
• Common and highly significant in the nutrient
  cycling of systems

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Methods (Cond)

• Eigenvalues and Eigenvectors
• In square matrices, there exists eigenvalues and
  eigenvectors (together named eigenpairs) that
  satisfy the following equation
• Ax ?x
• Where A is a square matrix, ? is the eigenvalue,
  and x is the assoicated eigenvector
• By taking each eigenvector and dividing by the
  sum of all eigenvectors, the relative
  distribution of direct and indirect pathways are
  obtained as a percentage
• The largest of these shows the compartment with
  the most pathway potential (dominant eigenpair of
  the matrix)

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Artificial Selections

• Each food web had modified adjacency matrices for
  predator and prey
• Chosen artificial introductions are
  super-predators and universal prey either
  consuming or being consumed by all in the system
• Second analysis has connectance modification such
  that a chosen predator and prey have connectance
  C 50 in column
• Selected by linkage density
• Reasoning for
• Each adjacency (original or modified) taken to
  powers two and three (squared and cubed)
• Returns indirect pathways (potential energy flow)
  of length two and three, respectively

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Grouping and Graphing

• Grouping of indirect pathways
• 3 Groups
• Top Predators
• Middle Predators
• Bottom Trophic Organisms
• Distribution and relative changes in total
  indirect links

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Results
Table 1 Matrix A2
Table 2 Matrix A3
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Results through Eigenvector AnalysisPredator/Prey
Modification 100 C
Table 3 Comparative Regionalization of Indirect
Links Of Length 2
Table 4 Comparative Regionalization of
Indirect Links Of Length 3
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Results through Eigenvector AnalysisPredator/Prey
Modification 50 C
Table 3 Comparative Regionalization of Indirect
Links Of Length 2
Table 4 Comparative Regionalization of
Indirect Links Of Length 3
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Conclusions

• Changes in the sums of indirect links showed
  increases with the introduction of universal
  predator and prey species
• Introductions of universal predators tended to
  have localized indirect effects across the twelve
  webs whereas universal prey species experienced
  much more diverse indirect links in the three
  groupings.
• Although to a lesser degree, similar results were
  observed when predator and prey were modified
  such that C 50
• Eigenvector analysis can be used to observe the
  relative spatial distribution of direct and
  indirect pathways as percentage values

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Conclusions (Cond)

• Modifying the connectance of these food webs in
  this fashion introduces cycles known as closely
  connected components (K) (Borrett and Patten)
• The introduction of these cycles yields an
  exponential increase in pathways known as pathway
  proliferation (Borrett and Patten)

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Future Direction

• This study looks not at the weight of indirect
  effects but rather the potential indirect
  pathways
• Adjacency matrices are not weighted (all links
  treated as equal) and represent potential, rather
  than realized, energy flow
• Future analysis will incorporate weighted graphs,
  taking the eigenvectors of the matrix to explain
  the distribution of indirect effects in the
  system
• This, in a trophic sense, would show (through the
  dominant eigenpair), which node in the system
  experiences the most energy through-flow
• Can view this as a net energy exchange or as
  separate entities of energy input and output to
  any given node
• May contribute in part to the quantification of
  keystone species in a trophic cascade

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Special Thanks

• Dr. Stuart Whipple, Institute of Ecology,
  University of Georgia
• Dr. Stuart Borrett, Dept. of Biology, University
  of North Carolina-Wilmington
• Dr. Claus Ernst, Dept. of Mathematics, WKU