*Kevin S. McCann*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691134178
- eISBN:
- 9781400840687
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691134178.003.0010
- Subject:
- Biology, Ecology

This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a ...
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This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a “statistical universe” of interaction strengths for a given set of n species. It then describes some matrix approaches to see if context-dependent techniques can be applied to matrix theory, along with the simple graphical techniques of Gershgorin discs employed as an intuitive approach to eigenvalues. It argues that there are some rather intriguing “gravitational-like” properties of Gershgorin discs for some important biologically motivated matrices. The chapter proceeds by discussing some classic whole-matrix results that highlight the connections between the stability of lower-dimensional modules and whole food webs. Finally, it shows how the ideas derived from classic whole-system matrix approaches generally agree with the results of modular theory.Less

This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a “statistical universe” of interaction strengths for a given set of *n* species. It then describes some matrix approaches to see if context-dependent techniques can be applied to matrix theory, along with the simple graphical techniques of Gershgorin discs employed as an intuitive approach to eigenvalues. It argues that there are some rather intriguing “gravitational-like” properties of Gershgorin discs for some important biologically motivated matrices. The chapter proceeds by discussing some classic whole-matrix results that highlight the connections between the stability of lower-dimensional modules and whole food webs. Finally, it shows how the ideas derived from classic whole-system matrix approaches generally agree with the results of modular theory.

*Kevin S. McCann*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691134178
- eISBN:
- 9781400840687
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691134178.003.0003
- Subject:
- Biology, Ecology

This chapter explains the use of modular or motif-based theory to interpret the dynamics of whole food webs. According to Robert Holt, modules are “as motifs with muscles.” Holt's modular theory ...
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This chapter explains the use of modular or motif-based theory to interpret the dynamics of whole food webs. According to Robert Holt, modules are “as motifs with muscles.” Holt's modular theory focuses on the implications of the strength of the interactions on the dynamics and persistence of these units. In this book, the term “module” means all motifs that include interaction strength, whereas the term “motif” represents all possible subsystem connections, including the trivial one-node/species case to the n-node/species cases. Part 2 considers the dynamics of important ecological modules or motifs such as populations, consumer–resource interactions, food chains, and omnivory, while Part 3 uses the logic attained from this modular or motif-based theory in order to elucidate the dynamics of whole food webs. The book argues that ecologists must make a concerted effort to understand how coupling different modules ultimately modifies flux within each individual module.Less

This chapter explains the use of modular or motif-based theory to interpret the dynamics of whole food webs. According to Robert Holt, modules are “as motifs with muscles.” Holt's modular theory focuses on the implications of the strength of the interactions on the dynamics and persistence of these units. In this book, the term “module” means all motifs that include interaction strength, whereas the term “motif” represents all possible subsystem connections, including the trivial one-node/species case to the *n*-node/species cases. Part 2 considers the dynamics of important ecological modules or motifs such as populations, consumer–resource interactions, food chains, and omnivory, while Part 3 uses the logic attained from this modular or motif-based theory in order to elucidate the dynamics of whole food webs. The book argues that ecologists must make a concerted effort to understand how coupling different modules ultimately modifies flux within each individual module.

*Laura Ruetsche*

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199535408
- eISBN:
- 9780191728525
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535408.003.0007
- Subject:
- Philosophy, Philosophy of Science

This chapter highlights other surprising aspects of QM∞: unlike theories of ordinary QM, theories of QM∞ can traffic in algebras none of whose countably additive states are pure. The chapter sketches ...
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This chapter highlights other surprising aspects of QM∞: unlike theories of ordinary QM, theories of QM∞ can traffic in algebras none of whose countably additive states are pure. The chapter sketches and provides simple illustrations of the mathematical basis of this circumstance: atomless von Neumann algebras. It also illustrates the uses to which atomless von Neumann algebras are put in QM∞.Less

This chapter highlights other surprising aspects of QM_{∞}: unlike theories of ordinary QM, theories of QM_{∞} can traffic in algebras none of whose countably additive states are pure. The chapter sketches and provides simple illustrations of the mathematical basis of this circumstance: atomless von Neumann algebras. It also illustrates the uses to which atomless von Neumann algebras are put in QM_{∞}.