PMC

(A) Albatross data and the model with destructive scenario. Green circles: accumulated distribution of flight lengths between successive detected prey of the model forager with perception radius r = 0.001 following a Lévy process with µ = 1.5 (from 20 simulations of 75 captures each). The foraging ground is represented by a square of area unity and contained 5000 prey. Continuous line: exponential fit. Red triangles: of the Bird Island albatross takeoff/landing data . Blue triangles: of the Crozet Islands albatross prey capture data , converted into flight durations assuming a constant flight velocity m/s. Inset: same curves represented in semi-log to better emphasize the non exponential nature of the observed albatross data versus the exponential form of the model forager detections. (B) Albatross data and model with non-destructive scenario. Violet circles: accumulated distribution for the model forager performing a Lévy process with µ = 2. Continuous line: exponential fit. Prey number: 3000; r = 0.0003. In A) and B), the lengths in the model with foraging arena of area unity are converted in hours (t) by using with the scaling factor v = 0.12. Inset: same curves represented in semi-log to better emphasize the non exponential nature of the observed albatross data versus the exponential form of the model forager detections. (C) Murre data and model with destructive scenario. Green circles: accumulated distribution of flight lengths between prey of the model forager with perception radius r = 0.001 following a Lévy process with µ = 1.5 (from 20 simulations of 75 captures each). Continuous line: exponential fit. Brown dots are the murre flight durations from . Inset: same data represented in semi-log in order to better emphasize the exponential nature of both the observed murres data and the model forager. (D) Murre data and model with non-destructive scenario. Violet circles: accumulated distribution for a forager performing a Lévy process with µ = 2. Continuous line: exponential fit. Prey number: 3000; r = 0.0003. Inset: same curves represented in semi-log. Similar close-to-exponential detections are obtained in all simulations with , in both destructive and non-destructive scenarii.

Z-score growth velocity is the exponential model growth velocity predicted by weight Z-score change according to Fenton curves during neonatal hospital stay according to gender, gestational age and birth weight Z-score (n = 3,954).

(a) Log relative volume as a function of time following treatment for individual tumors after 10 Gy carbon (blue) or 30 Gy X-rays (green) with a graphical representation of a piecewise exponential model (black dashed line). (b) Volume data described in (a) with a graphical representation of a simple exponential model (red dashed lined). Note that the piecewise model characterizes the period of post-treatment stasis and the rate of regrowth better than the simple exponential model. (c) Example of an individual sarcoma evaluated with the piecewise exponential model after 10 Gy carbon ion radiation therapy. (d) Example of an individual sarcoma evaluated with the piecewise exponential model after 30 Gy X-rays radiation therapy. For details of the piecewise exponential model, please refer to .

(A) Single-step state-space used for B–G. (B,E) When the model consists of a set of exponential discounters with γ drawn uniformly from (0,1), the measured discounting closely fits the hyperbolic function. (C,F) When the model consists of a single exponential discounter with , the measured discounting closely fits the function (exponential). (D,G) When the model consists of a single hyperbolic discounter, the measured discounting closely fits the function (hyperbolic). (H) Chained state-space used for I–N. (I) If values are distributed so each exponential discounter has its own value representation, the result is hyperbolic discounting over a chained state space. (J,M) A single exponential discounter behaves as in the single-step state space, because multiplying exponentials gives an exponential. (K,N) A single hyperbolic discounter now behaves as an exponential discounter with , because each step is discounted by , where . (L) Likewise, a set of exponential discounters with shared value representation behave as an exponential discounter with , for the same reason.