SparcS Center

Glossary

Alternative stable states

Contrasting states to which a system may converge under the same external conditions

Attractor

A state or dynamic regime to which a model converges given sufficient simulation time.

Basin boundary

Divisions between sets of initial conditions that lead to different attractors

Basin boundary collision

Event at which internally-driven oscillations in the state of the system tip it into an alternative basin of attraction

Basin of attraction

Set of initial conditions that lead to a particular attractor

Bifurcation

Threshold in parameters (or conditions) at which the qualitative behavior of a model changes

Catastrophe fold

‘Folded’ response-curve of the state of a system to a conditioning factor, implying that for given conditions the system has alternative stable states.

Catastrophic bifurcation

Bifurcation where an attractor disappears so that the system is forced to move to an alternative attractor

Catastrophic shift

Substantial shift in the state of a model induced by a tiny perturbation bringing it across a catastrophic bifurcation.

Chaos

More precisely referred to as deterministic chaos. Unpredictable fluctuations in models resulting from deterministic rules.

Competitive exclusion

Exclusion of a species by a stronger competitor

Contingency

Used here for tendency of individuals to copy the attitude of peers

Critical slowing down

The tendency that the recovery of disturbances slows down close to a bifurcation.

Equilibrium

Situation in which processes that affect the state of a system precisely balance out so that it does not change. An equilibrium is stable if the system returns to it upon a small perturbation, and unstable if the system moves away from the equilibrium point upon such a perturbation.

Feedback loop

Set of cause-effect relationships that form a closed loop

Fold bifurcation

Critical threshold where a stable point is touched by an unstable point that marks the border of the basin of attraction. This bifurcation thus marks the abrupt disappearance of an equilibrium.

Ghost

The remains (ghost) of an attractor that has become unstable but still slows down the behavior of the system in its vicinity.

Hysteresis

The tendency of a system to stay in the same state if conditions change. Specifically, in the context of alternative stable states the term refers to the fact that as conditions are changing, the system remains on the same attractor until a catastrophic bifurcation is reached where it jumps to the alternative attractor. Now, if the conditions are changed in the opposite direction the system stays on the new attractor until another catastrophic bifurcation makes it jump back to the original state. This completes a ‘hysteresis loop’. The range of conditions between the two bifurcation points is also referred to as the size or width of the hysteresis.

Limit cycle

Stable cyclic dynamics of a system generated by internal processes. Such a cycle is an attractor as starting from different states the system is pulled towards this limit behavior.

Mechanistic model

A model that explicitly accounts for the mechanisms that are thought to play a role. By contrast empirical models (such as regression models) describe observed relationships between variables.

Minimal (strategic) model

Model that focus on a minimal set of mechanisms needed to produce a certain behavior.

Multiple stable states

A situation in which a system has more than one stable state given the same external conditions.

Niche

The way in which a species ‘makes a living’ in an ecosystem. For instance the niche of a particular bird may be to forage on small flying insects, and a particular plant may be specialized in growing in acidic, wet, nutrient-poor soils.   

Nonlinear system

A system in which the dynamics are not linearly dependent on the state. This can lead to phenomena such as thresholds, multiple attractors, cycles and chaotic dynamics.

Periodic forcing

Influence of changes in conditions that oscillate with a fixed period, such as seasonal or diurnal variations in temperature and light.

Positive feedback

A chain of effects through which something has a positive effect on itself. For instance, climate warming in some regions may cause snowmelt. Since the uncovered dark vegetation absorbs more solar radiation than the snow this leads to further warming.

Quasiperiodic

An oscillating system in which the behavior is not precisely periodic due to changing amplitude and period. In contrast with chaotic systems quasiperiodic systems do not have the sensitivity to initial conditions that causes long term unpredictability.

Regime shift

A relatively sharp change from one regime to a contrasting one, where a regime is a dynamic ‘state’ of a system with its characteristic stochastic fluctuations and/or cycles.

Repellor

The opposite of an attractor. Thus: a state or dynamic regime from which a model moves away. Repellors mark the border between alternative basins of attraction.

Resilience

The magnitude of disturbance that a system can tolerate before it shifts into a different state.

A more liberal definition used for social-ecological systems is: The capacity of a system to absorb disturbance and re-organize while undergoing change so as to still retain essentially the same function, structure, identity and feedbacks.

Resistance

Force needed to cause a certain change in a system.

Runaway change

Change that is accelerating and self-propelling due to a positive feedback.

Saddle

A particular unstable equilibrium that attracts in some directions, but repels in others

Self-organized patterns

Patterns that emerge automatically from the interaction between many units.

Slow-fast cycle

A limit cycle with contrasting slow and fast phases that can be interpreted as internally generated periodic shifts between alternative stable states.

Spatial heterogeneity

Variation of factors in space so that sites differ.

Stable equilibrium

Attractor point.

Strange attractor

Attractor on which the system shows chaotic behavior.

Threshold

A point where the system is very sensitive to changing conditions.

Transient

Behavior of a dynamical system on its way to an attractor

Unstable equilibrium

Repellor point.