Characterizing Species Interactions to Understand Press Perturbations: What Is the Community Matrix?

The community matrix is among ecology's most important mathematical abstractions, formally encapsulating the interconnected network of effects that species have on one another's populations. Despite its importance, the term “community matrix” has been applied to multiple types of matrices that have differing interpretations. This has hindered the application of theory for understanding community structure and perturbation responses. Here, we clarify the correspondence and distinctions among the Interaction matrix, the Alpha matrix, and the Jacobian matrix, terms that are frequently used interchangeably as well as synonymously with the term “community matrix.” We illustrate how these matrices correspond to different ways of characterizing interaction strengths, how they permit insights regarding different types of press perturbations, and how these are related by a simple scaling relationship. Connections to additional interaction strength characterizations encapsulated by the Beta matrix, the Gamma matrix...

[1]  D. L. Angelis,et al.  STABILITY AND CONNECTANCE IN FOOD WEB MODELS , 1975 .

[2]  Owen L. Petchey,et al.  Interaction strengths in food webs: issues and opportunities , 2004 .

[3]  Neo D. Martinez,et al.  Simple prediction of interaction strengths in complex food webs , 2009, Proceedings of the National Academy of Sciences.

[4]  S. Carpenter,et al.  Stability and Diversity of Ecosystems , 2007, Science.

[5]  J. Castilla,et al.  Challenges in the Quest for Keystones , 1996 .

[6]  Jérôme Casas,et al.  Erratum to: Sensitivity analysis of continuous-time models for ecological and evolutionary theories , 2015, Theoretical Ecology.

[7]  R. Solé,et al.  Perturbations and Indirect Effects in Complex Food Webs , 2005 .

[8]  R Pilette,et al.  The potential for community level evaluations based on loop analysis. , 1987, Bio Systems.

[9]  Ricard V Solé,et al.  Press perturbations and indirect effects in real food webs. , 2009, Ecology.

[10]  R. Paine.  A Note on Trophic Complexity and Community Stability , 1969, The American Naturalist.

[11]  Stefano Allesina,et al.  The stability–complexity relationship at age 40: a random matrix perspective , 2015, Population Ecology.

[12]  T. Miki,et al.  Intraguild predation promotes complex alternative states along a productivity gradient. , 2007, Theoretical population biology.

[13]  J. Wootton,et al.  The Nature and Consequences of Indirect Effects in Ecological Communities , 1994 .

[14]  R. Paine.  Road Maps of Interactions or Grist for Theoretical Development , 1988 .

[15]  S. Metcalf.  Qualitative Models to Complement Quantitative Ecosystem Models for the Analysis of Data-Limited Marine Ecosystems and Fisheries , 2010 .

[16]  G. Polis,et al.  TOWARD AN INTEGRATION OF LANDSCAPE AND FOOD WEB ECOLOGY : The Dynamics of Spatially Subsidized Food Webs , 2005 .

[17]  J. Timothy Wootton,et al.  THEORETICAL CONCEPTS AND EMPIRICAL APPROACHES TO MEASURING INTERACTION STRENGTH , 1998 .

[18]  R. Lewontin,et al.  The meaning of stability. , 2020, Brookhaven symposia in biology.

[19]  Ben Raymond,et al.  Comprehensive evaluation of model uncertainty in qualitative network analyses , 2012 .

[20]  J. Wootton,et al.  Using experimental indices to quantify the strength of species interactions , 2010 .

[21]  Mark Novak,et al.  Estimating nonlinear interaction strengths: an observation-based method for species-rich food webs. , 2008, Ecology.

[22]  L. Stone,et al.  Conditions for a Species to Gain Advantage from the Presence of Competitors , 1991 .

[23]  J. Lawton.  Are there general laws in ecology , 1999 .

[24]  Lewi Stone,et al.  Advantageous indirect interactions in systems of competition. , 2004, Journal of theoretical biology.

[25]  Karen C. Abbott,et al.  Balls, cups, and quasi‐potentials: quantifying stability in stochastic systems , 2015, Ecology.

[26]  S. Carpenter,et al.  ESTIMATING COMMUNITY STABILITY AND ECOLOGICAL INTERACTIONS FROM TIME‐SERIES DATA , 2003 .

[27]  J. Lawton,et al.  Number of trophic levels in ecological communities , 1977, Nature.

[28]  J. Estes,et al.  Stability, resilience, and phase shifts in rocky subtidal communities along the west coast of Vancouver Island, Canada , 2011 .

[29]  S. Carpenter,et al.  Resilience indicators: prospects and limitations for early warnings of regime shifts , 2015, Philosophical Transactions of the Royal Society B: Biological Sciences.

[30]  J. Lawton,et al.  On feeding on more than one trophic level , 1978, Nature.

[31]  Alex James,et al.  Constructing Random Matrices to Represent Real Ecosystems , 2015, The American Naturalist.

[32]  Habitat destruction, habitat restoration and eigenvector-eigenvalue relations. , 2003, Mathematical biosciences.

[33]  Mark Emmerson,et al.  Predicting community responses to perturbations in the face of imperfect knowledge and network complexity. , 2011, Ecology.

[34]  P. Yodzis,et al.  The stability of real ecosystems , 1981, Nature.

[35]  Owen L. Petchey,et al.  The ecological forecast horizon, and examples of its uses and determinants , 2015, bioRxiv.

[36]  Ian T. Carroll,et al.  Understanding and predicting ecological dynamics: are major surprises inevitable? , 2008, Ecology.

[37]  Annette Ostling,et al.  Sensitivity analysis of coexistence in ecological communities: theory and application. , 2014, Ecology letters.

[38]  A. Neutel,et al.  Energetics, Patterns of Interaction Strengths, and Stability in Real Ecosystems , 1995, Science.

[39]  Charles J. Puccia,et al.  Qualitative modeling in ecology: loop analysis, signed digraphs, and time averaging , 1991 .

[40]  Peter Yodzis.  The Dynamics of Highly Aggregated Models of Whole Communities , 1988 .

[41]  P. Abrams,et al.  Positive Indirect Effects Between Prey Species that Share Predators , 1996 .

[42]  Masashi Murakami,et al.  Timescale Hierarchy Determines the Indirect Effects of Fluctuating Subsidy Inputs on In Situ Resources , 2008, The American Naturalist.

[43]  G. Gauze.  The struggle for existence, by G. F. Gause. , 1934 .

[44]  Extension of Levins' Loop Analysis to transient and periodic disturbances , 1980 .

[45]  W. M. Post,et al.  Dynamics and comparative statics of mutualistic communities. , 1979, Journal of theoretical biology.

[46]  R M May,et al.  Niche overlap as a function of environmental variability. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[47]  T. P. Burns,et al.  Quantifying direct and indirect effects of perturbations using model ecosystems , 2014 .

[48]  S. Schreiber,et al.  Why intraspecific trait variation matters in community ecology. , 2011, Trends in ecology & evolution.

[49]  J. Reum,et al.  Evaluating community impacts of ocean acidification using qualitative network models , 2015 .

[50]  Peter A. Abrams,et al.  Describing and quantifying interspecific interactions: a commentary on recent approaches , 2001 .

[51]  R. Levins,et al.  Effects of density dependence, feedback and environmental sensitivity on correlations among predators, prey and plant resources : models and practical implications , 1996 .

[52]  R Levins,et al.  DISCUSSION PAPER: THE QUALITATIVE ANALYSIS OF PARTIALLY SPECIFIED SYSTEMS , 1974, Annals of the New York Academy of Sciences.

[53]  L. Lawlor.  Direct and indirect effects of n-species competition , 1979, Oecologia.

[54]  Stephen H. Levine,et al.  Competitive Interactions in Ecosystems , 1976, The American Naturalist.

[55]  Samuel J. Mason,et al.  Feedback Theory-Some Properties of Signal Flow Graphs , 1953, Proceedings of the IRE.

[56]  Rodrigo Ramos-Jiliberto,et al.  A conceptual framework for studying the strength of plant-animal mutualistic interactions. , 2015, Ecology letters.

[57]  M. Novak,et al.  Complexity Increases Predictability in Allometrically Constrained Food Webs , 2016, The American Naturalist.

[58]  D. O. Logofet.  Stronger-than-Lyapunov notions of matrix stability, or how "flowers" help solve problems in mathematical ecology , 2005 .

[59]  A. Neutel,et al.  Linking saturation, stability and sustainability in food webs with observed equilibrium structure , 2016, Theoretical Ecology.

[60]  T. Hallam,et al.  The community matrix in three species community models , 1982 .

[61]  P. Yodzis.  The Indeterminacy of Ecological Interactions as Perceived Through Perturbation Experiments , 1988 .

[62]  Don A. Driscoll,et al.  Metacommunities: Spatial Dynamics and Ecological Communities , 2006 .

[63]  R. Levins,et al.  On the variance of eigenvalues of the community matrix : Derivation and appraisal , 2000 .

[64]  H. Caswell,et al.  ALTERNATIVES TO RESILIENCE FOR MEASURING THE RESPONSES OF ECOLOGICAL SYSTEMS TO PERTURBATIONS , 1997 .

[65]  P. Yodzis,et al.  DIFFUSE EFFECTS IN FOOD WEBS , 2000 .

[66]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[67]  Mark Novak,et al.  Bayesian characterization of uncertainty in species interaction strengths , 2017, Oecologia.

[68]  Hisao Nakajima,et al.  Sensitivity and stability of flow networks , 1992 .

[69]  Maria-Josefina Hernandez,et al.  Disentangling nature, strength and stability issues in the characterization of population interactions. , 2009, Journal of theoretical biology.

[70]  Jeffrey M Dambacher,et al.  UNDERSTANDING AND PREDICTING EFFECTS OF MODIFIED INTERACTIONS THROUGH A QUALITATIVE ANALYSIS OF COMMUNITY STRUCTURE , 2007, The Quarterly Review of Biology.

[71]  Oswald J. Schmitz,et al.  PRESS PERTURBATIONS AND THE PREDICTABILITY OFECOLOGICAL INTERACTIONS IN A FOOD WEB , 1997 .

[72]  Lars-Anders Hansson,et al.  Press perturbation experiments and the indeterminacy of ecological interactions: effects of taxonomic resolution and experimental duration , 2001 .

[73]  Peter Yodzis.  Food Webs and Perturbation Experiments: Theory and Practice , 1996 .

[74]  H Nakajima,et al.  Indirect effects in ecological interaction networks. I. The chain rule approach. , 1995, Mathematical biosciences.

[75]  D. Haydon.  Pivotal Assumptions Determining the Relationship between Stability and Complexity: An Analytical Synthesis of the Stability-Complexity Debate , 1994, The American Naturalist.

[76]  Peter A Abrams,et al.  When does greater mortality increase population size? The long history and diverse mechanisms underlying the hydra effect. , 2009, Ecology letters.

[77]  B. Ebenman,et al.  Using sensitivity analysis to identify keystone species and keystone links in size-based food webs , 2011 .

[78]  R. Paine,et al.  Food-web analysis through field measurement of per capita interaction strength , 1992, Nature.

[79]  Peter J. Taylor,et al.  Consistent scaling and parameter choice for linear and Generalized Lotka-Volterra models used in community ecology , 1988 .

[80]  Thilo Gross,et al.  Generalized modeling of ecological population dynamics , 2010, Theoretical Ecology.

[81]  J. Vandermeer,et al.  Interspecific competition: a new approach to the classical theory , 1975, Science.

[82]  Jeffrey M. Dambacher,et al.  RELEVANCE OF COMMUNITY STRUCTURE IN ASSESSING INDETERMINACY OF ECOLOGICAL PREDICTIONS , 2002 .

[83]  Jeffrey M Dambacher,et al.  Qualitative mathematical models to support ecosystem-based management of Australia's Northern Prawn Fishery. , 2015, Ecological Applications.

[84]  M. Emmerson,et al.  Predator–prey body size, interaction strength and the stability of a real food web , 2004 .

[85]  J. Vandermeer.  The Community Matrix and the Number of Species in a Community , 1970, The American Naturalist.

[86]  Stefano Allesina,et al.  Googling Food Webs: Can an Eigenvector Measure Species' Importance for Coextinctions? , 2009, PLoS Comput. Biol..

[87]  M. Gilpin,et al.  Perturbation Experiments in Community Ecology: Theory and Practice , 1984 .

[88]  Carl Boettiger,et al.  Tipping points: From patterns to predictions , 2013, Nature.

[89]  M. Emmerson,et al.  MEASUREMENT OF INTERACTION STRENGTH IN NATURE , 2005 .

[90]  Stefano Allesina,et al.  Predicting global community properties from uncertain estimates of interaction strengths , 2015, Journal of The Royal Society Interface.

[91]  Indirect effects in ecological interaction networks. II. The conjugate variable approach. , 1995, Mathematical biosciences.

[92]  Johan van de Koppel,et al.  Reconciling complexity with stability in naturally assembling food webs , 2009, Nature.

[93]  V. Fairén,et al.  Lotka-Volterra representation of general nonlinear systems. , 1997, Mathematical biosciences.

[94]  Albert-László Barabási,et al.  Universal resilience patterns in complex networks , 2016, Nature.

[95]  Thilo Gross,et al.  How to predict community responses to perturbations in the face of imperfect knowledge and network complexity , 2013, Proceedings of the Royal Society B: Biological Sciences.

[96]  Shuichi Matsumura,et al.  Ancient DNA from the First European Farmers in 7500-Year-Old Neolithic Sites , 1975, Science.

[97]  Serhiy Morozov,et al.  A Distributed, Architecture-Centric Approach to Computing Accurate Recommendations from Very Large and Sparse Datasets , 2011 .

[98]  Rodrigo Ramos-Jiliberto,et al.  Consequences of adaptive behaviour for the structure and dynamics of food webs. , 2010, Ecology letters.