Figure 1
The concepts of scaling and memory-influenced site fidelity. (A) Lévy walk (LW) and Brownian motion (BM) may be characterized by the tail part of the distribution F(L j ) of inter-fix steps lengths L, where L j is a specific length range ( “bin size”). On a log-log scale β is expressing how “steeply” the frequency of larger steps fades, relative to any reference length L j , over the range of j where β is relatively constant. Since this ratio between two magnitudes of L j is independent on which absolute size of L j we choose for comparison (e.g., “meters” in both numerator and denominator cancels each other, leading to a dimensionless number), the movement is scale-free. For a scale-specific kind of movement like BM (dotted line), the distribution shows β increasing with increasing j with β>3. (B) Memory-influenced movement under the present model is conceptualized by three hypothetical goals along a spatial path. Long term goal (arrow towards target A’, to be reached within time ta’, decided at location A at time ta), medium term goal (B and B’) and short term goal (C and C’). The difference in time intervals for the three targets implies different process rates, and consequently an option to execute several goals at finer temporal scales for each goal at coarser scales. (C) The spatial scatters of two hypothetical sets of fixes illustrate a method to study scaling and site fidelity. Number of non-empty grid cells (I) will depend on both the grid resolution and the number of fixes (N) in the sample. The optimal resolution is found where the regression line may be interpolated close to (0,0); i.e., somewhere intermediate between the two examples. Linearity of the slope implies scale-free space use, and a magnitude intermediate between 1 and 0 (e.g., close to 0.5) indicates site fidelity and hence compliance with the non-Markovian framework.