Scaling in Movement Trajectory Analysis

Scaling in Movement Trajectory Analysis

Lévy or not Lévy? Path Characterization Random Walk Models Scaling with Size and Environment Step Identification Area Restricted Search

Size-Grain Hypothesis Kaspari & Weiser 1999, Funct. Ecol.

Kaspari & Weiser 1999, Funct. Ecol.

Kaspari & Weiser 2007, Ecol. Entom.

Body Size & Temperature vs. Running Speed Temperature Hurlbert et al. 2008, Ecol. Entom.

Hurlbert et al. 2008, Ecol. Entom.

P Metabolic Power Mass Coefficient (state, taxon, temp.) Speed Force on the locom. appendages Metabolic Rate Hirt et al. 2017, Ecology

Hirt et al. 2017, Ecology

Open questions: Temperature a factor? Natural rugosity of the surface a factor? In nature these relationships not all that relevant? Useful to reduce to mean? Differences/mechanisms more interesting?

How to characterize paths? Messor wasmanni Frizzi 2018, Ins. Soc.

Straightness

How squiggly? Tortuosity Frizzi 2018, Ins. Soc.

Fractal Dimension Intensity of Habitat Use Frizzi 2018, Ins. Soc.

MSD (cm 2 ) How dispersive? Mean Squared Displacement 10000 = (x x 0 ) 2 1000 100 y = 0.0448x 2.4073 10 r.m.s. = x 2 x 0 2 (x x 0 ) 2 1 0.1 1 10 100 0.01 time (s)

Indices Summary Straightness Sinuosity Intensity of Habitat Use Fractal Dimension Mean Squared Displacement Root Mean Squared Displacement Only if goal is present Depends on sample size Biological significance? Reason to assume? Population Dispersivity Individual Depends on type of question, organism, environment. Hard to objectively pick a priori

Patterns Behavioral Mechanism Optimality

Area Restricted Search

How to detect ARS Moving Average Speed or Turning Angle t Robinson et al 2007, Deep-Sea Res

How to detect ARS Moving Average Speed or Turning Angle t

How to detect ARS Moving Average Speed or Turning Angle t

How to detect ARS First Passage Time First Passage Time t Newell 1962, J Math Mech

How to detect ARS First Passage Time First Passage Time t

How to detect ARS First Passage Time First Passage Time t

How to detect ARS First Passage Time First Passage Time r 3 r 2 r 1 t

How to detect ARS First Passage Time First Passage Time r 3 r 2 r 1 t Var(FPT) r

How to detect ARS Residence Time First Passage Time r 3 r 2 r 1 t Var(FPT) r Barraquand & Benhamou 2008, Ecology

How to detect ARS Fractal Landscape Tremblay 2007, J Exp Biol

How to detect ARS Partial Sums Sinuosity + Speed Knell & Codling 2012, Theor Ecol

How to detect ARS Hidden Markov Models E I Knell & Codling 2012, Theor Ecol

Hidden Markov Models Basari et al. 2014, J Exp Biol

Methods Identifying ARS Mean Average (of Turning angle/speed) First Passage Time/Residence Time Fractal Landscape Partial Sums Hidden Markov Models

Patterns Behavioral Mechanism Models! Optimality

Quick history 1920s 1970s 1990s/2000s Edwards et al. 2007 Nature

E (t+1) = E (t)+rule Expectation = location Step length Distribution Turning angle Distribution

Ballistic movement Step length Arbitrary/ f Turning angle 0 f Food -180 0 180 No boundaries Food sources: Randomly dispersed Scarce Non-replenishing Non-predictable

Spiral of Archimedes N Central Place Forager Food sources: Randomly dispersed Scarce Non-replenishing Non-predictable

Simple Random Walk Step length Turning angle Constant/normal distr. Uniform distr. -180 0 180 Central Place Forager (or not) Food sources: Randomly dispersed Abundant Quickly replenishing Non-predictable

Correlated Random Walk f Step length Turning angle Some distribution Some distr. about 0 (Cauchy, van Mises, Normal, ) Central Place Forager (or not) Food sources: Randomly dispersed Scarce Non-replenishing Non-predictable -180 0 180

Step length Turning angle Power Uniform -180 0 180 Lévy flight Food sources: Randomly dispersed Scarce Non-replenishing Non-predictable

Real data: what are steps and turns?

Usually equal time sampling interval Reynolds et al. 2007, Ecology

Corridor de Knegt et al. 2007, Beh Eco

Local Method Non-Local Method Reynolds et al. 2007, Ecology

Non-Local Method Reynolds et al. 2007, Ecology

2/3D 1D projection Humphries et al. 2013, Meth Ecol Evo

Alternative Approaches υ (T) t = at β Hunt et al. 2016, Roy Soc Open Sci

Alternative Approaches Hunt et al. 2016, Roy Soc Open Sci

Alternative Approaches

Turn ID Corridor Local Non-local 1D Projection Limited Arbitrary Potentially changing character of trajectory Indicators of Meaningful Steps Promising but hard to get

Lévy or not Lévy, that is the question! Patchily distributed food Mechanism pattern Hot debate

p l ~l μ James et al. 2011, J Royal Soc Interf

Shlesinger & Klafter 1986, On Growth and Form Paul Lévy

Papers about Lévy flights in animal movement Viswanathan et al. 1996, Nature Pyke 2015, Meth Ecol Evol

Ancient worms Human Hunter Gatherers Movement of Money Viruses T-cells Bacteria All sorts of animals: Spider monkeys Marine predators (reading) Bees Polar bears Vultures Sims et al. 2014, PNAS

CRW f Step length Turning angle 2 or more distr. 2 or more distr. Brownian Walk -180 0 180 Composite Correlated Random Walk

Truncated Lévy Walk CCRW Step length distribution Power law, P~l -µ ; 1.5<µ<2.5 Multiple Exponential (e.g. µ=1 + Turning angle distribution Uniform Peak around 0 (or uniform) Self-similarity Scale-free (until boundaries) Scale-dependent CRW f -180 0 180 L W f -180 0 180

Plank et al. 2013, Lec Notes in Math

Truncation CRWs always diffusive over long term but may be superdiff. in small scale. TLW also diffusivein large scale due to truncation. Step length Turning angle Power (a<l<b) Uniform -180 0 180 Truncated Lévy Walk

Optimality If food farther away from perception range than just outside, Lévy becomes worse. If memory, sampling, differential search horizon present, Lévy becomes worse.

How do Lévy Flights come about?

What is scaling? (1) Quantify how characteristics of the system change (or don t change!) with changes in the fundamental dimensions (2) Characterize scaling function (pattern!) (3) Why such a function? -> theory Working hypothesis.... Inmost cases a scaling function will point tothe operation of general laws/principles.

Truncated Ballistic Reynolds 2015, Phys Life Rev

Self Avoiding Walks Shlesinger 1983, J Chem Phys Sims et al. 2014, PNAS

Shlesinger 1983, J Chem Phys

Sims et al. 2014, PNAS

Simulation Reynolds 2007, EPL

SAWs observed in animals but no correlation to LWs. Guy 2008, Anim Beh Atkinson et al. 2002, Oikos

Lawler et al. 2000, arxiv Reynolds 2010, Am Nat Following CRW Boundaries CRW Boundary: D = 1.33 Lévy Walk: D = µ - 1

Nearest Unoccupied Space Lowest Bid Auction Radicci et al. 2012, PLoS One

Physical Streams Cho et al. 2018, biorxiv

Neuronal Firing Mazzoni et al. 2007, PLoS One

Multiplicative Noise Biró & Jakovác 2007, Phys Rev Let

Lévy Walks Lots of issues with identifying: Sampling frequency, step resampling, model fitting Actually, it is mentally taxing & not really adapted to nature naturally arises when following simple rules Not selected for, but against losing.

Problems for the Future Step identification: Objectively possible? Machine learning for motifs steps, characterization, comparison, underlying behavioral mechanism/algorithm Open questions: ontogeny, indiv/spp. Variation, Group movement, Barriers, Soil organisms Using methods to answer hypotheses (requires solid analysis)