Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion Statistical Physics and Anomalous Dynamics of Foraging Rainer Klages Max Planck Institute for the Physics of Complex Systems, Dresden Queen Mary University of London, School of Mathematical Sciences MPIPKS Dresden, Advanced Study Group 22 July 2015 Statistical physics and anomalous dynamics of foraging Rainer Klages 1
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Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Statistical Physics andAnomalous Dynamics of Foraging
Rainer Klages
Max Planck Institute for the Physics of Complex Systems, Dresden
Queen Mary University of London, School of Mathematical Sciences
MPIPKS Dresden, Advanced Study Group22 July 2015
Statistical physics and anomalous dynamics of foraging Rainer Klages 1
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Advanced Study Group 2015 on foraging
Topic: Statistical physics and anomalous dynamics offoraging
Team: 1 convenor and 5 team members
Duration: 6 months from July 1st until December 31st,2015
Concept: bring together a team of experts working on thechosen topic, supported by a vivid visitors programme
Statistical physics and anomalous dynamics of foraging Rainer Klages 2
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Motivation
Main theme:
Can biologically relevant search strategies be identified bymathematical modeling?
Four parts of this talk:
1 review the Lévy flight hypothesis
2 biological data: analysis and interpretation
3 own research: cell migration and foraging bumblebees
4 ASG: the team and key topics
Statistical physics and anomalous dynamics of foraging Rainer Klages 3
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Lévy flight search patterns of wandering albatrosses
famous paper by Viswanathan et al., Nature 381, 413 (1996):
for albatrosses foraging inthe South Atlantic the flighttimes were recorded
the distribution of flight timeswas fitted with a Lévy flightmodel (power law ∼ t−µ)
Statistical physics and anomalous dynamics of foraging Rainer Klages 4
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Lévy flights in a nutshell
Lévy flights have well-defined mathematical properties :
a Markovian stochastic process (no memory)with probability distribution function of flight lengthsexhibiting power law tails, ρ(ℓ) ∼ ℓ−1−α , 0 < α < 2;it has infinite variance, < ℓ2 >= ∞,satisfies a generalized central limit theorem (Gnedenko,Kolmogorov, 1949) andis scale invariant
• for an outline see, e.g., Shlesinger at al., Nature 363, 31 (1993)• for more details: A.V.Chechkin et al., Introduction to the theory ofLévy flights in: R. Klages, G.Radons, I.M.Sokolov (Eds.), Anomaloustransport (Wiley-VCH, 2008)
nb: ∃ the more physical model of Lévy walks; Zaburdaev et al., RMP87, 483 (2015)
Statistical physics and anomalous dynamics of foraging Rainer Klages 5
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Optimizing the success of random searches
another paper by Viswanathan et al., Nature 401, 911 (1999):
question posed about “best statistical strategy to adapt inorder to search efficiently for randomly located objects”random walk model leads to Lévy flight hypothesis:Lévy flights provide an optimal search strategyfor sparse, randomly distributed, immobile,revisitable targets in unbounded domains
Brownian motion (left) vs. Lévy flights (right)Lévy flights also obtained for bumblebee and deer data
Statistical physics and anomalous dynamics of foraging Rainer Klages 6
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Revisiting Lévy flight search patterns
Edwards et al., Nature 449, 1044 (2007):
Viswanathan et al. results revisited by correcting old data(Buchanan, Nature 453, 714, 2008):
no Lévy flights: new, more extensive data suggests(gamma distributed) stochastic processbut claim that truncated Lévy flights fit yet new dataHumphries et al., PNAS 109, 7169 (2012) (and reply...)
Statistical physics and anomalous dynamics of foraging Rainer Klages 7
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Lévy or not Lévy?
Lévy paradigm : Look for power law tails in pdfs!
Sims et al., Nature 451, 1098 (2008): scaling laws ofmarine predator search behaviour; > 106 data points!
prey distributions also display Lévy-like patterns...
Statistical physics and anomalous dynamics of foraging Rainer Klages 8
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Lévy flights induced by the environment?
Humphries et al., Nature 465, 1066 (2010): environmentalcontext explains Lévy and Brownian movement patterns ofmarine predators; > 107 data points!; for blue shark:
blue: exponential; red: truncated power law
note: ∃ day-night cycle, cf. oscillations; suggests to fit withtwo different pdfs (not done)
Statistical physics and anomalous dynamics of foraging Rainer Klages 9
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Optimal searches: adaptive or emergent?
strictly speaking two different Lévy flight hypotheses:
two types: wildtype (NHE+) and NHE-deficient (NHE-)
Statistical physics and anomalous dynamics of foraging Rainer Klages 13
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Experimental results I: mean square displacement
• msd(t) :=< [x(t) − x(0)]2 >∼ tβ and time dependentexponent β(t) = d ln msd(t)/d ln t
1
10
100
1000
10000m
sd(t
) [µm
2 ]
<r2> NHE+
<r2> NHE-
data NHE+
data NHE-
FKK model NHE+
FKK model NHE-
1.0
1.5
2.0
1 10 100
β(t)
time [min]
b
cI II III
• different dynamics on different time scales with superdiffusionfor long times; not scale-free!(solid lines: (Bayes) fits from our model)
Statistical physics and anomalous dynamics of foraging Rainer Klages 14
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Experimental results II: position distribution function
• green lines: results forBrownian motion
• other solid lines: fits fromour model; parameter valuesas before
100
10-1
10-2
10-3
10-4
10 0-10
p(x,
t)
x [µm] 100 0-100
x [µm] 200 0-200
x [µm]
OUFKK
100
10-1
10-2
10-3
10-4
10 0-10
p(x,
t)
x [µm] 100 0-100
x [µm] 200 0-200
x [µm]
OUFKK
2 3 4 5 6 7 8 9
500 400 300 200 100 0
kurt
osis
Κ
time [min]
a
b
c
NHE+t = 1 min t = 120 min t = 480 min
NHE-t = 1 min t = 120 min t = 480 min
data NHE+
data NHE-
FKK model NHE+
FKK model NHE-
• non-Lévy distributions with different dynamics on differenttime scales
Statistical physics and anomalous dynamics of foraging Rainer Klages 15
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Generalized Lévy walks for migrating T cells
T.H. Harris et al., Nature 486, 545 (2012):
• T cell motility described by a generalized Lévy walk (Zumofen,Klafter, 1995); claim: more efficient than Brownian motion
• mean square displacement (for 3 different cell types) andposition distribution function:
• microscopic justification of the model?• pdf not Lévy: how does the result fit to the Lévy hypothesis?
Statistical physics and anomalous dynamics of foraging Rainer Klages 16
Introduction The Lévy flight hypothesis Lévy or not Lévy? Cells and bees Conclusion
Foraging bumblebees
tracking of bumblebee flights in thelab
foraging in an artificial carpet offlowers with or without spiders
note: no test of the Lévy hypothesis but work inspired by the‘paradigm’
main result of data analysis and stochastic modeling:no change in the velocity pdf under predation thread; onlychange in the velocity autocorrelation function