Spatial and temporal Taylor's law in 1D chaotic maps

Area of Science:

• Ecology
• Dynamical Systems
• Statistical Physics

Background:

• Taylor\'s Law (TL) describes a power-law relationship between sample mean and variance.
• Understanding TL across spatial and temporal ensembles is crucial for ecological and dynamical system analysis.

Purpose of the Study:

• To investigate Taylor\'s Law (TL) in low-dimensional chaotic maps.
• To differentiate between spatial ensemble TL (STL) and temporal ensemble TL (TTL).
• To elucidate the origins of differences between STL and TTL.

Main Methods:

• Analysis of low-dimensional chaotic maps (logistic, tent, Hassell, Ricker).
• Distinguishing between spatial (independent sampling) and temporal (correlated dynamics) ensembles.
• Analytical derivations and examination of trajectory structures.

Main Results:

• STL is explained by distribution skewness.
• Differences between TTL and STL arise from temporal correlations.
• Bartlett\'s law (quadratic relationship) identified analytically for logistic and tent maps.
• Hassell model TTL linked to trajectory chunk structure; Ricker model TTL has a unique mechanism.

Conclusions:

• The study clarifies the distinct mechanisms underlying spatial and temporal Taylor\'s Law in chaotic systems.
• Temporal correlations and map-specific dynamics significantly influence TTL.
• Findings provide a deeper understanding of statistical properties in chaotic dynamics.