Among the analytic and conceptual frameworks in use by ecologists, many are inherently topological in nature. They are informed by either the mathematical or conceptual underpinnings of the branch of mathematics known as topology. Topological thinking manifests in ecology in terms of the mapping or parameterization of interaction networks or, alternatively, in terms of dynamic fields describing the continuous phase space of ecological systems.
Interaction networks range from exchanges of information within monospecific social systems through mutualistic relationships between pollinators and flowers. These networks provide an analytic framework with topological properties that are ostensibly meaningful in an ecological context. In certain cases, however, complex sets of interactions are coupled with continuous and sometimes abrupt changes in system state that are more appropriately described using dynamical fields. In these instances, the collective system state is better described by a qualitative phase space characterizing the continual transition of a dynamical system rather than the discrete structures associated with a network representation.
When should we view the interactions taking place in an ecological system in a network context versus a field oriented dynamical systems context? Should the scale of our question influence our choice in conceptualizing interactions? Are network and field models of interaction mutually exclusive?
Results/Conclusions
Representations of ecological systems are either explicitly dynamic or, alternatively, are temporal snapshots of otherwise dynamic systems. While we see both network and field topological perspectives supporting increased understanding of dynamic ecological interactions, we differentiate between discrete dynamics and phase dynamics.
Network representations are preferred for addressing discrete system dynamics. By discrete, we mean dynamics associated with exchanges between individual discrete elements in an assumed steady state system. Questions answered via a network representation are thus dependent on structures arising from connections between discrete elements in the system (whether or not individual elements and connections are modeled as incrementally changing over time).
Phase dynamics refers to collective sets of system interactions subject to smooth and or discontinuous responses in one or more driving system control parameters. Phase dynamics are better treated as manifold topologies.
Using topological methods as a modeling motif and sustainability as a values context, this presentation addresses network and manifold views of topology in relation to problems in human-dominated ecosystems. We differentiate between discrete versus continuous dynamism, explicate the role of topology in understanding interactions in ecology, and discuss the influence of scale and context in methodological decisions.
