Alternative stable states
Contrasting states to which a system may converge under the same external conditions
A state or dynamic regime to which a model converges given sufficient simulation time.
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
Threshold in parameters (or conditions) at which the qualitative behavior of a model changes
‘Folded’ response-curve of the state of a system to a conditioning factor, implying that for given conditions the system has alternative stable states.
Bifurcation where an attractor disappears so that the system is forced to move to an alternative attractor
Substantial shift in the state of a model induced by a tiny perturbation bringing it across a catastrophic bifurcation.
More precisely referred to as deterministic chaos. Unpredictable fluctuations in models resulting from deterministic rules.
Exclusion of a species by a stronger competitor
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.
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.
Set of cause-effect relationships that form a closed loop
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.
The remains (ghost) of an attractor that has become unstable but still slows down the behavior of the system in its vicinity.
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.
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.
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.
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.
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.
Influence of changes in conditions that oscillate with a fixed period, such as seasonal or diurnal variations in temperature and light.
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.
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.
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.
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.
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.
Force needed to cause a certain change in a system.
Change that is accelerating and self-propelling due to a positive feedback.
A particular unstable equilibrium that attracts in some directions, but repels in others
Patterns that emerge automatically from the interaction between many units.
A limit cycle with contrasting slow and fast phases that can be interpreted as internally generated periodic shifts between alternative stable states.
Variation of factors in space so that sites differ.
Attractor on which the system shows chaotic behavior.
A point where the system is very sensitive to changing conditions.
Behavior of a dynamical system on its way to an attractor