A MATHEMATICAL REVIEW OF RESILIENCE IN ECOLOGY

Rising interest in the resilience of ecological systems has spawned diverse interpretations of the term's precise meaning, particularly in the context of resilience quantification. The purpose of this paper is twofold. The first aim is to use the language of dynamical systems to organize and scrutinize existing resilience definitions within a unified framework. The second aim is to provide an introduction for mathematicians to the ecological concept of resilience, a potential area for expanded quantitative research. To frame the discussion of resilience in dynamical systems terms, a model consisting of ordinary differential equations is assumed to represent the ecological system. The question “resilience of what to what?” posed by Carpenter et al. [2001] informs two broad categories of definitions, based on resilience to state variable perturbations and to parameter changes, respectively. Definitions of resilience to state variable perturbations include measures of basin size (relevant to one-time perturbations) and basin steepness (relevant to repeated perturbations). Resilience to parameter changes has been quantified by viewing parameters as state variables but has also considered the reversibility of parameter shifts. Quantifying this reversibility and fully describing how recovery rates determine resilience to repeated state-space perturbations emerge as two opportunities for mathematics research.

[1]  S. Carpenter,et al.  Early-warning signals for critical transitions , 2009, Nature.

[2]  C. Conley.  The gradient structure of a flow: I , 1988, Ergodic Theory and Dynamical Systems.

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

[4]  Stephen R. Carpenter,et al.  Management of eutrophication for lakes subject to potentially irreversible change , 1999 .

[5]  S. Levin,et al.  Resilience, Robustness, and Marine Ecosystem-based Management , 2008 .

[6]  Stephen R. Carpenter,et al.  VARIABILITY OF LAKES ON THE LANDSCAPE: ROLES OF PHOSPHORUS, FOOD WEBS, AND DISSOLVED ORGANIC CARBON , 2003 .

[7]  S. Pimm.  The complexity and stability of ecosystems , 1984, Nature.

[8]  C. S. Holling,et al.  Resilience, Adaptability and Transformability in Social–ecological Systems , 2004 .

[9]  C. S. Holling,et al.  Regime Shifts, Resilience, and Biodiversity in Ecosystem Management , 2004 .

[10]  J. Anderies,et al.  From Metaphor to Measurement: Resilience of What to What? , 2001, Ecosystems.

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

[12]  C. Wissel.  A universal law of the characteristic return time near thresholds , 1984, Oecologia.

[13]  C. S. Holling.  Resilience and Stability of Ecological Systems , 1973 .

[14]  D. Haydon,et al.  Alternative stable states in ecology , 2003 .

[15]  Guillaume Deffuant,et al.  Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication , 2013 .

[16]  A. Weigelt,et al.  Grassland Resistance and Resilience after Drought Depends on Management Intensity and Species Richness , 2012, PloS one.