Equilibria and Stability

  • Equilibria arise when a process (or several processes) rate of change is zero.

  • There can be more than one equilibrium. Multiple stable states (e.g., long-term patterns that are returned to following a perturbation of the system) are typical of biological systems. The system dynamics may drive the process to any of these depending on initial conditions and history (e.g., the order of any sequence of changes in the system may affect the outcomes).

  • Equilibria can be dynamic, so that a periodic pattern of system response may arise. This period pattern may be stable in that for some range of initial conditions, the system approaches this period pattern.

  • There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).

  • Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. These bifurcations of equilibria arise in many nonlinear systems typical in biology.

Structure

  • Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.

  • Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.

  • Geometry of the aggregation can affect the resulting formulation.

  • Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.

Interactions

  • There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.

  • Some general properties can be derived based upon these (e.g., two-



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