Depending on the process premise for interpretation of step length distributions, patterns showing β > 3 may either verify BM under the Markov premise or MRW with ρ ≪ 1 under the non-Markovian model premise. Pane (A) illustrates this principle conceptually for three typical variants of the F(L) function under log-transformation of both axes. The coloured areas indicate how the influence from return steps may typically inflate F(L
) under three strengths of the site fidelity ratio ρ = tret/tobs. The dotted line for ρ > 1 indicates the expectation either from very large ρ (leading to pseudo-LW), or from MRW under limited memory horizon (approaching the Markov condition, and thus expressing LW compliant movement). Panes (B-D) shows the result from simulations of MRW with β = 2, verifying that the observed distribution of step lengths depends on the ratio ρ; the individual’s return interval tret relative to the interval for fix collection tobs. (B) ρ = 10 and narrow memory horizon: estimated β is close to the simulation condition’s true β = 2. Under condition of ρ = 10 and memory horizon increased to 10,000 increments (blue line) a LW-like pattern with β ≈ 2 is still apparent but becoming more truncated under reduced tret (ρ < =1; yellow, red and green lines). (C) Infinite memory: increased shape-shift effect from reducing ρ, with plots appearing as truncated LW or BM-like (β close to 3). (D) Adding physical constraint on step length at spatial scale of the black arrow (see Methods) shows a similar pattern as in (C).