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Connectance Modification and Eigenvector Analysis of Food Webs: 

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

Why Study Food Web Ecology?: 

Why Study Food Web Ecology? Best method for depicting feeding relationships Provide, although complex, models of species interactions and biodiversity What’s not interesting about this ?

Major Contributors: 

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

Concepts of Research Interest: 

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……….

What about Indirect Pathways?: 

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 A B C VERSUS A B INDIRECT RELATIONSHIP BETWEEN SPECIES “A” AND “C” Species “B” Consumes “A”

Predator and Prey Connectance Modification: 

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 Prey’s effect on increases of indirect pathways and the variable distribution of such

Method for Depicting Food Webs: 

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

Slide8: 

Simple Food Web Construct vs. Lavigne “Spaghetti” Model (1992)

A Method For Obtaining Indirect Links: 

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 =

Methods, Tests, and Materials : 

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

Methods (Con’d): 

Methods (Con’d) Eigenvalues and Eigenvectors In square matrices, there exists eigenvalues and eigenvectors (together named eigenpairs) that satisfy the following equation Ax = λx Where A is a square matrix, λ 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)

Artificial Selections: 

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

Grouping and Graphing: 

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

Results: 

Results Table 1: Matrix A2 Table 2: Matrix A3

Results through Eigenvector Analysis Predator/Prey Modification = 100% C: 

Results through Eigenvector Analysis Predator/Prey Modification = 100% C Table 3: Comparative Regionalization of Indirect Links Of Length 2 Table 4: Comparative Regionalization of Indirect Links Of Length 3

Results through Eigenvector Analysis Predator/Prey Modification = 50% C: 

Results through Eigenvector Analysis Predator/Prey Modification = 50% C Table 3: Comparative Regionalization of Indirect Links Of Length 2 Table 4: Comparative Regionalization of Indirect Links Of Length 3

Conclusions: 

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

Conclusions (Con’d): 

Conclusions (Con’d) 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)

Future Direction: 

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

Special Thanks: 

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

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