aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Chapter 4
Search for Food of Birds, Fish and Insects
Rainer Klages
When you are out in a forest searching for mushrooms you wish to fill your basketwith these delicacies as quickly as possible. But how do you search efficiently forthem if you have no clue where they grow (Fig. 4.1)? The answer to this questionis not only relevant for finding mushrooms [1, 2]. It also helps to understand howwhite blood cells kill efficiently intruding pathogens [3], how monkeys search forfood in a tropical forest [4], and how to optimize the hunt for submarines [5].
Fig. 4.1
Illustration of a typical search problem [1,2]: A human searcher endeavours to find mush-rooms that are randomly distributed in a certain area. It would help to have an optimal searchstrategy that enables one to find as many mushrooms as possible by minimizing the search time.Rainer KlagesMax Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, D-01187 Dresden,Germany and Queen Mary University of London, School of Mathematical Sciences, Mile EndRoad, London E1 4NS, UK, e-mail: [email protected]
In society the problem to develop efficient search strategies belongs to the realmof operations research , the mathematical optimization of organizational problemsin order to aid human decision-making [6]. Examples are the search for landmines,castaways or victims of avalanches. Over the past two decades search research [5]attracted particular attention within the fields of ecology and biology. The new disci-pline of movement ecology [7,8] studies foraging strategies of biological organisms:Prominent examples are wandering albatrosses searching for food [9–11], marinepredators diving for prey [12, 13], and bees collecting nectar [14, 15]. Within thiscontext the
L´evy Flight Hypothesis (LFH) became especially popular: It predictsthat under certain mathematical conditions on the type of food sources long
L´evyflights [16] minimize the search time [9, 10, 17]. This implies that for a bumble-bee searching for rare flowers the flight lengths should be distributed according to apower law. Remarkably, the prediction by the LFH is completely different from theparadigm put forward by Karl Pearson more than a century ago [18], who proposedto model the movements of biological organisms by simple random walks as intro-duced in Chap. 2 of this book. His suggestion entails that the movement lengths aredistributed exponentially according to a Gaussian distribution, see Eq.(2.10) in thissection. L´evy and Gaussian processes represent fundamental but different classes ofdiffusive spreading. Both are justified by a rigorous mathematical underpinning.More than 60 years ago Gnedenko and Kolmogorov proved mathematically thatspecific types of power laws, called
L´evy stable distributions [19,20], obey a centrallimit theorem. Their result generalizes the conventional central limit theorem forGaussian distributions, which explains why Brownian motion is observed in a hugevariety of physical phenomena. But exponential tails decay faster than power laws,which implies that for L´evy-distributed flight lengths there is a larger probability toyield long flights than for flight lengths obeying Gaussian statistics. Consequently,L´evy flights should be better suited to detect sparsely, randomly distributed targetsthan Brownian motion, which in turn should outperform L´evy motion when thetargets are dense. This is the basic idea underlying the LFH. Empirical tests of it,however, are hotly debated [11, 21–24]: Not only are there problems with a soundstatistical analysis of experimental data sets when checking for power laws; theirbiological interpretation is also often unclear: For example, for monkeys living in atropical forest who feed on specific types of fruit it is not clear whether the observedL´evy flights of the monkeys are due to the distribution of the trees on which theirpreferred fruit grows, or whether the monkeys’ L´evy motion represents an evolu-tionary adapted optimal search strategy helping them to survive [4]. Theoreticallythe LFH was motivated by random walk models with L´evy-distributed step lengthsthat were solved in computer simulations [10]. A rigorous mathematical proof ofthe LFH remains elusive.This chapter introduces to the following fundamental question cross-linking thefields of ecology, biology, physics and mathematics:
Can search for food by biolog-ical organisms be understood by mathematical modeling? [8, 17, 20, 25] It consistsof three main parts: Section 4.2 reviews the LFH. Section 4.3 outlines the contro-versial discussion about its verification by including basics about the theory of L´evy
Search for Food of Birds, Fish and Insects 3 motion. Section 4.4 illustrates the need to go beyond the LFH by elaborating onbumblebee flights. We summarize our discussion in Sec. 4.5.
In 1996 Gandhimohan Viswanathan and collaborators published a pioneering articlein the journal
Nature [9]. For albatrosses foraging in the South Atlantic the flighttimes were recorded by putting sensors at their feet. The sensors got wet when thebirds were diving for food, see the inset of Fig. 4.2. The duration of a flight wasthus defined by the period of time when a sensor remained dry, terminated by a divefor catching food. The main part of Fig. 4.2 shows a histogram of the flight timeintervals of some albatrosses. The straight line represents a L´evy stable distributionproportional to ∼ t − µ with an exponent of µ =
2. By assuming that the albatrossesmove with an on average constant speed one can associate these flight times with arespective power law distribution of flight lengths. This suggests that the albatrosseswere searching for food by performing L´evy flights.For more than a decade albatrosses were considered to be the most prominentexample of an animal performing L´evy flights. This work triggered a large numberof related studies suggesting that many other animals like deer, bumblebees, spidermonkeys and fishes also perform L´evy motion [4, 10, 12, 13, 17]. t i
Fig. 4.2
Histogram where ‘scaled frequencies’ holds for the number of flight time intervals oflength t i (in hours) normalized by their respective bin widths. The data is for five albatrosses during19 foraging bouts (double-logarithmic scale). Blue open circles show the data from Ref. [9]. Thestraight line indicates a power law ∼ t − µ with exponent µ =
2. The red filled circles are adjustedflight durations using the same data set by eliminating times that the birds spent on an island [11].The histogram is reprinted by permission from
Macmillan Publishers Ltd: Nature Ref. [11], copy-right 2007 . The inset shows an albatross catching food; reprinted by permission from
MacmillanPublishers Ltd: Nature Ref. [5], copyright 2006 . Rainer Klages
In 1999 the group around Gandhimohan Viswanathan published another importantarticle in
Nature [10]. Here the approach was more theoretical by posing, and ad-dressing, the following general question:“What is the best statistical strategy to adapt in order to search efficiently forrandomly located objects?”To answer this question they introduced a special type of what is called a
L´evywalk [20] in two dimensions and studied it both by computer simulations and by an-alytical approximations. Their model consists of point targets randomly distributedin a plane and a (point) forager moving with constant speed. If the forager spotsa target within a pre-defined finite vision distance, it moves to the target directly.Otherwise the forager chooses a direction at random with a jump length ℓ randomlydrawn from a L´evy stable distribution ∼ ℓ − µ , ≤ µ ≤
3. While the forager is mov-ing it constantly looks out for targets within the given vision distance. If no target isdetected, the forager stops after the given distance and repeats the process.Although these rules look simple enough, there are some subtleties that exem-plify the problem of mathematically modeling a biological foraging problem:1. Here we have chosen what is called a cruise forager , i.e., a forager that sensestargets whenever it is moving. In contrast, a saltaltory forager would not sensea target while moving. It needs to land close to a target within a given radius ofperception in order to find it [26].2. For a cruise forager a jump is terminated when it hits a target, hence this modeldefines a truncated
L´evy walk [13].3. One has to decide whether a forager eliminates targets when it finds them or not,i.e., whether it performs destructive or non-destructive search [10]. As we willsee below, whether a monkey eats a fruit thus effectively eliminating it, at leastfor a long time, or whether a bee collects nectar from a flower that replenishesquickly defines mathematically different foraging problems.4. We have not yet said anything about the density of the targets .5. We have deliberately assumed that the targets are immobile , which may notalways be realistic for a biological foraging problem (e.g., marine predators[12, 13]).6. If we ask about the best strategy to search efficiently , how do we define optimal-ity ?These few points illustrate the difficulty to relate abtract mathematical randomwalk models to biological foraging reality. Interestingly, the motion generated bythese models often sensitively depends on right such details: In Ref. [10] foragingefficiency was defined as the ratio of the number of targets found divided by the totaldistance traveled by a forager, see Eq.(3) therein. Different definitions are possible,depend on the type of forager and may yield different results [26]. The foragingefficiency was then computed in Ref. [10] under variation of the exponent µ of theabove L´evy distribution generating the jump length. The results led to what wascoined the L´evy Flight Hypothesis (LFH), which we formulate as follows:
Search for Food of Birds, Fish and Insects 5
L´evy motion provides an optimal search strategy for sparse, randomly dis-tributed, immobile, revisitable targets in unbounded domains.
Intuitively this result can be understood as follows: Fig. 4.3 (left) displays a typ-ical trajectory of a Brownian walker. One can see that this dynamics is ‘more lo-calized’ while L´evy motion shown in Fig. 4.3 (right) depicts clusters interrupted bylong jumps. It thus makes sense that Brownian motion is better suited to find targetsthat are densely distributed while L´evy motion outperforms Brownian motion whentargets are sparse, since it avoids oversampling due to long jumps. The reason whythe targets need to be revisitable is that the exponent µ of the L´evy distribution de-pends on whether the search is destructive or not, cf. the third point on the list offoraging conditions above: For non-destructive foraging µ = µ = - - -
20 20 - - - - - - - - - - -
500 500 - Fig. 4.3
Brownian motion (left) vs. L´evy motion (right) in the plane, illustrated by typical trajec-tories.
Several years passed before the results by Viswanathan et al. were revisited in an-other
Nature article led by Andrew Edwards [11]: When analyzing new, larger andmore precise data for foraging albatrosses the old results of Ref. [9] could not berecovered, see Fig. 1 in Ref. [11]. This led the researchers to reconsider the old al-batross data. A correction of these data sets yielded the result shown in Fig. 4.2 asthe red filled circles: One can see that the L´evy stable law with an exponent of µ = Rainer Klages for the flight times is gone. Instead the data now seems to be fit best with a gammadistribution.What happened is explained in Ref. [21]: For all measurements the sensors wereput onto the feet of the albatrosses when the birds were sitting on an island, andat this point the measurement process was started. However, to this time the sen-sors were dry; and in Ref. [9] these times were interpreted as L´evy flights. Thesame applied to the end of a foraging bout when the birds were back on the island.Subtracting these erroneous time intervals from the data sets eliminated the L´evyflights.However, in Ref. [27] yet new albatross data was analyzed, and the old data fromRefs. [9, 11] was again reanalyzed: This time truncated power laws were used forthe analysis, and furthermore data sets for individual birds were tested instead ofpooling together the data for all birds. In this reference it was concluded that someindividual albatross indeed do perform L´evy flights while others do not.
The debate about the LFH created a surge of publications testing it both theoreti-cally and experimentally; see Refs. [8, 17, 20, 25] for reviews. But experimentallyit is difficult to verify the mathematical conditions on which the LFH formulatedin Sec. 4.2.2 is based. Often the LFH was thus interpreted in a much looser senseby ignoring any mathematical assumptions in terms of what one may call the
L´evyFlight Paradigm (LFP):Look for power laws in the probability distributions of step lengths of foraginganimals.We illustrate virtues and pitfalls related to the LFP by data from Ref. [13] onthe diving depths of free-ranging marine predators. Impressively, in this work over12 million movement displacements were recorded and analyzed for 14 differentspecies. As an example, Fig. 4.4 shows results for a blue shark: Plotted at the bottomare probability distributions of its diving depths, called move step length frequencydistribution, where a step length is defined as the distance moved by the shark perunit time. Included are fits to a truncated power law and to an exponential distribu-tion. Since here L´evy distributions were used whose longest step lengths were cutoff, the fits do not consist of straight lines but are bent off, in contrast to Fig. 4.2.The top of this figure depicts the corresponding time series from which the datawas extracted, split into five different sections. Each section is characterized by pro-foundly different average diving depths. These different sections correspond to theshark being in different regions of the ocean, i.e., either on-shelf or off-shelf. It wasargued that on-shelf, where the diving depth of the shark is very limited, the data canbe better fitted with an exponential distribution (sections f and h) while off-shelf thedata displays power-law behavior with an exponent close to two (sections g, i and j).Fig. 4.4 thus suggests a strong dependence of the foraging dynamics on the environ-ment in which it takes place, where the latter defines the food distribution. Related
Search for Food of Birds, Fish and Insects 7 switching behavior between power law-like L´evy and exponential Brownian motionsearch strategies was reported for microzooplankton, jellyfish and mussels.
Fig. 4.4
Top: time series of the diving depth of a blue shark. The red lines split the data intodifferent sections (a - e), where the shark dives deep or the diving depth is more constrained. Thesesections match to the shark being off-shelf or on the shelf, respectively. Bottom: double-logarithmicplots of the move step length frequency distribution (‘rank’) as a function of the step length, whichis the vertical distance moved by the shark per unit time, with the notation (f - j) correspondingto the primary data shown in sections (a - e). Black circles correspond to data, red lines to fitswith truncated power laws of exponent µ , blue lines to exponential fits. This figure is reprinted bypermission from Macmillan Publishers Ltd: Nature Ref. [13], copyright 2010.
The power law matching to the data in the off-shelf regions was interpreted insupport of the LFH. However, note the periodic oscillations displayed by the timeseries at the top of Fig. 4.4. Upon closer inspection they reveal a 24h day-nightcycle: During the night the shark hovers close to the surface of the sea while overthe day it dives for food. For the move step length distributions shown in Fig. 4.4the data was averaged over all these periodic oscillations. But the distributions insections g, i and j all show a ‘wiggle’ on a finer scale. This suggests to better fitthe data by a superposition of two different distributions [14] taking into accountthat day and night define two very different phases of motion, instead of using onlyone function by averaging over all times. Apart from this, one may argue that thisanalysis does not test for the original LFH put forward in Ref. [9]. But this requiresa bit more knowledge about the theory of L´evy motion; we will come back to thispoint in Sec. 4.3.5.
Rainer Klages
Our discussion in the previous sections suggests to distinguish between two different
LFHs:1. The first is the ‘conventional’ one that we formulated in Sec. 4.2.2, originallyput forward in Ref. [9]: It may now be further specified as the
L´evy Search Hy-pothesis (LSH), because it suggests that under certain conditions L´evy flightsrepresent an optimal search strategy . Here optimality needs to be defined rig-orously mathematically. This can be done in different ways given the specificbiological situation at hand that one wishes to model [26]. Typically optimal-ity within this context aims at minimizing the search time for finding targets.The interesting biological interpretation of the LSH is that it has been evolvedin biological organisms as an evolutionary adaptive strategy that maximizes thesuccess for survival. The LSH version of the LFH became most popular.2. In parallel there is a second type of LFH, which may be called the
L´evy En-vironmental Hypothesis (LEH): It suggests that L´evy flights emerge from theinteraction between a forager and a food source distribution. The latter may bescale-free thus directly inducing the L´evy flights. This is in sharp contrast tothe LSH, which suggests that under certain conditions a forager performs L´evyflights irrespective of the actual food source distribution. Emergence of novelpatterns and dynamics due to the interaction of the single parts of a complexsystem with each other, on the other hand, is at the heart of the theory of com-plex systems. The LEH is the hypothesis that to some extent was formulated inRef. [9], but it became more popular rather later on [4, 12, 13].Both the LSH and the LEH are bound together by what we called the L´evy FlightParadigm (LFP) in Sec. 4.3.2. The LFP extracts the formal essence from both thesedifferent hypotheses by proposing to look for power laws in the probability distri-butions of foraging dynamics by ignoring any conditions of validity of these twohypotheses. Consequently, in contrast to the LSH and LEH the mathematical, phys-ical and biological origin and meaning of power laws obtained by following the LFPis typically not clear. On the other hand, the LFP motivated to take a fresh look atforaging data sets by not only testing for exponential distributions. It widened thescope by emphasizing that one should also check for power laws in animal move-ment data.
Simple random walks as introduced in Section 2.1 represent examples of unimodal types of motion if the random step lengths are sampled from only one specific distri-bution. For example, choosing a Gaussian distribution we obtain Brownian motionwhile a L´evy-stable distribution produces L´evy flights. Combining two differenttypes of motion like Brownian and L´evy yields bimodal motion . A simple example
Search for Food of Birds, Fish and Insects 9 is shown in Fig. 4.5: Imagine you have lost your keys at home, but you have a vagueidea where to find them. Hence, you are running straightforwardly to the locationwhere you expect them to be. This may be modeled as a ballistic flight during whichyou quickly relocate, say, from the kitchen to the study room. However, when youarrive in your study room you should switch to a different type of motion, which issuitably adapted to locally search the environment. For this mode you may choose,e.g., Brownian motion. The resulting dynamics is called intermittent [25]: It con-sists of two different phases of motion mixed randomly, which in our example areballistic relocation events and local Brownian motion.
Fig. 4.5
Illustration of an intermittent search strategy: A human searcher looks for a target (key)by alternating between two different modes of motion. During fast, ballistic relocation phases thesearcher is not able to detect any target (non reactive). These phases are interrupted by slow phasesof Brownian motion during which a searcher is able to detect a target (reactive) [25].
This type of motion can be exploited to search efficiently in the following way:You may not bother to look for your keys while you are walking from the kitchen tothe study room. You are more interested to get from point A to point B as quicklyas possible, and while doing so your search mode is switched off. This is called a non reactive phase in Fig. 4.5. But as you expect the keys to be in your study room,while switching to Brownian motion therein you simultaneously switch on yourscanning abilities. This defines your local search mode called reactive in Fig. 4.5.Correspondingly, for aninmals one may imagine that during a fast relocation event,or flight, they are unable to detect any targets while their sensory mechanisms be-come active during slow local search. This is close to what was called a saltaltoryforager in Sec. 4.2.2, but this forager did not feature any local search dynamics.Intermittent search dynamics can be modeled by writing down a set of two cou-pled equations, one that generates ballistic flights and another one that yields Brow-nian motion. The coupling captures the switching between both modes. One fur-thermore needs to model that search is only performed during the Brownian mo-tion mode. By analyzing a respective ballistic-Brownian system of equations it wasfound that this dynamics yields a minimum of a suitably defined search time underparameter variation if a target is non-revisitable , i.e., it is destroyed once it is found.Note that for targets that are non-replenishing the L´evy walks of Ref. [10] did notyield any non-trivial optimization of the search time. Instead, they converged to pureballistic flights as being optimal. The LSH, in turn, only applies to revisitable, i.e.,replenishing targets. Hence intermittent motion poses no contradiction. A popular account of this result was given by Michael Shlesinger in his Nature article ‘How tohunt a submarine?’ [5].
We now briefly elaborate on the theory of L´evy motion. This section may be skippedby a reader who is not so interested in theoretical foundations. We recommendRef. [16] for an outline of this topic from a physics point of view and Chap. 5in Ref. [19] for a more mathematical introduction. We start from the simple randomwalk on the line introduced in Chap. 2 of this book, x n + = x n + ℓ n , (4.1)where x n is the position of a random walker at discrete time n ∈ N moving in onedimension, and ℓ n = x n + − x n defines the jump of length | ℓ n | between two positions.In Chap. 2 the special case of constant jump length | ℓ n | = ℓ was considered, wherethe sign of the jump was randomly determined by tossing a coin with, say, plusfor heads and minus for tails. The coin was furthermore supposed to be fair in thesense of yielding equal probabilities for heads and tails. This simple random walkcan be generalized by considering a bigger variety of jumps. Mathematically thisis modeled by drawing the random variable ℓ n from some more general probabilitydistribution than featuring only probability one half for each of two outcomes. Forexample, instead we could draw ℓ n at each time step n randomly from a uniformdistribution, where each jump between − L and L is equally possible given by theprobability density ρ ( ℓ n ) = / ( L ) , − L ≤ ℓ n ≤ L and zero otherwise. Alternatively,we could allow arbitrarily large jumps by drawing ℓ n from an unbounded Gaussiandistribution, see Eq.(2.10) in Chap. 2 (by replacing x therein with ℓ n and setting t constant). For both generalized random walks Eq. (4.1) would still reproduce in thelong time limit the fundamental diffusive properties Eq. (4) discussed in Chap. 2,i.e., the linear growth in time of the mean square displacement, and Eq. (2.10) inChap. 2, the Gaussian probability distribution for the position x n of a walker at timestep n . This follows mathematically from the conventional central limit theorem.We now further generalize the random walk Eq. (4.1) in a more non-trivial wayby randomly drawing ℓ n from a L´evy α -stable distribution [19], ρ ( ℓ n ) ∼ | ℓ n | − − α ( | ℓ n | ≫ ) , < α < , (4.2)characterized by power law tails in the limit of large | ℓ n | . This functional form isin sharp contrast to the exponential tails of Gaussian distributions and has impor-tant consequences, as it violates one of the assumptions on which the conventionalcentral limit theorem rests. However, for the range of exponents α stated above itcan be shown that these distributions obey a generalized central limit theorem : Theproof employs the fact that these distributions are stable , in the sense that a linearcombination of two random variables sampled independently from the same distri- Search for Food of Birds, Fish and Insects 11 bution reproduces the very same distribution, up to some scale factors [16]. This inturn implies that L´evy stable distributions are scale invariant and thus self-similar .Physically one speaks of ℓ n sampled independently and identically distributed fromEq. (4.2) as white L´evy noise . As by definition there are no correlations between therandom variables ℓ n the stochastic process generated by Eq. (4.1) is memoryless ,meaning at time step ( n + ) the particle has no memory where it came from at anyprevious time step n . In mathematics this is called a Markov processes , and L´evyflights belong to this important class of stochastic processes.What we presented here is only a very rough, mathematically rather impreciseoutline of how to define an α -stable L´evy process generating L´evy flights. Espe-cially, the function in Eq. (4.2) is not defined for small ℓ n , as the given power lawdiverges for ℓ n →
0. A rigorous definition of L´evy stable distributions is obtainedby using the characteristic function of this process, i.e., the Fourier transform ofits probability distribution, which is well-defined analytically. The full probabilitydistribution can then be generated from it [16, 19]. For α = h ℓ n i = Z ∞ − ∞ d ℓ n ρ ( ℓ n ) ℓ n = ∞ . (4.3)The above equation defines what is called the second moment of the probabilitydistribution ρ ( ℓ n ) . Higher moments are defined analogously by h ℓ k i , k ∈ N , andfor L´evy distributions they are also infinite. This means that in contrast to simplerandom walks generating Brownian motion, see again Chap. 2, L´evy motion doesnot have any characteristic length scale. However, since moments are rather easilyobtained from experimental data this poses a problem to L´evy flights as a viablephysical model to be validated by experiments.This problem can be solved by using the very related concept of L´evy walks [20]:These are random walks where again jumps are drawn randomly from the L´evystable distribution Eq. (4.2). But as a penalty for long jumps the walker spends atime t n proportional to the length of the jump to complete it, t n = v ℓ n , where theproportionality factor v , typically chosen as | v | = const . , defines the velocity of theL´evy walker. This implies that both jump lengths ℓ n and flight times t n are distributedaccording to the same power law. In contrast, for the L´evy flights introduced abovea walker makes a jump of length | ℓ n | during an integer time step of duration ∆ n = continuous time randomwalks [19, 28, 29], which further generalize ordinary random walks by allowing awalker to move by non-integer time steps. We do not discuss all the similaritiesand differences between L´evy walks and L´evy flights, see Ref. [20] for details, butinstead highlight only one important fact: While for L´evy flights the mean squaredisplacement h x i , see Eq.(1) in Chap. 2, does not exist, which follows from our discussion above, for L´evy walks it does. This is due to the finite velocities, whichtruncate the power law tails in the probability distributions for the positions of aL´evy walker. However, in contrast to Brownian motion where it grows linearly intime as shown in Chap. 2, see Eq.(2), for L´evy walks it grows faster than linear, h x i ∼ t γ ( t → ∞ ) , (4.4)with γ >
1. If γ = anomalous diffusion [19, 28]. The case γ > superdiffusion , since a particle diffuses faster than Brownian motion, cor-respondingly γ < subdiffusion . There is a wealth of different stochas-tic models exhibiting anomalous diffusion, and while superdiffusion appears to bemore common among foraging biological organisms than subdiffusion the wholespectrum of anomalous diffusion is found in a variety of different processes in thenatural sciences, and even in the human world [19, 28, 30].Often the difference between L´evy walks and flights is not quite appreciated inthe experimental literature, see, e.g., Fig. 4.4, where move step length frequencydistributions were plotted. By definition a move step length x per unit time corre-sponds to what we defined as a jump length ℓ n by Eq. (4.1) above, x = ℓ n . Hence, atruncated power law fit ∼ x − µ to the distributions plotted in Fig. 4.4 corresponds toa fit with a truncated form of the jump length distribution Eq. (4.2) with exponent µ = + α testing for truncated L´evy flights [20]. The truncation cures the problemof infinite moments exhibited by random walks based on ordinary L´evy flights men-tioned above. However, this analysis does not test the LFH put forward in Ref [10],which was derived from L´evy walks. But checking for L´evy walks requires an en-tirely different data analysis [3, 20]. The LFH and its variants illustrated the problem to which extent biologically rele-vant search strategies may be identified by mathematical modeling. What we thenformulated as the LFP in Sec. 4.3.2 motivated to generally look for power laws inthe probability distributions of step lengths of foraging animals. Inspired by the longdebate about the different functional forms of move step lengths probability distri-butions, and by further diluting the LFP, an even weaker guiding principle wouldbe to assume that the foraging dynamics of biological organisms can be understoodby analyzing such probability distributions alone. In the following we discuss anexperiment, and its theoretical analysis, which illustrate that one may miss crucialinformation by studying only probability distributions. In that respect, this last sec-tion provides a look beyond the LFH that focuses on such distributions.
Search for Food of Birds, Fish and Insects 13
In Refs. [31] Thomas Ings and Lars Chittka reported a laboratory experiment inwhich environmental foraging conditions were varied in a fully controlled manner.The question they addressed with this experiment was whether changes of environ-mental conditions, in this case exposing bumblebees to predation threat or not, ledto changes in their foraging dynamics. This question was answered by a statisticalanalysis of the bumblebee flights recorded in this experiment on both spatial andtemporal scales [14]. (a) (b) (c)
Fig. 4.6
Illustration of a laboratory experiment investigating the dynamics of bumblebees foragingunder predation risk: (a) Sketch of the cubic foraging arena together with part of the flight trajectoryof a single bumblebee. The bumblebees forage on a grid of artificial flowers on one side of the box.While being on the landing platforms, they have access to nectar. All flowers can be equipped withspider models and trapping mechanisms simulating predation attempts as shown in (b), (c) [14,31].
The experiment is sketched in Fig. 4.6: Bumblebees (
Bombus terrestris ) wereflying in a cubic arena of ≈ × the influence of previous experience with predation risk on the bumblebees’ flightdynamics; see Ref [31] for full details of the experimental setup and staging.It is important to observe that neither the LSH nor the LEH can be tested by thisexperiment, as the flight arena is too small: The bumblebees always sense the wallsand may adjust their flight behavior accordingly. However, there is a cross-link tothe LEH in that this experiment studies the interaction of a forager with the envi-ronment, and its consequences for the dynamics of the forager, in a very controlledway. The weaker guiding principle derived from the LFP that we discussed abovefurthermore suggests that the main information to understand the foraging dynamicsmay be contained in the probability distributions of flight step lengths only. On thisbasis one may naively expect to see different step lengths probability distributionsemerging by changing the environmental conditions, which here is the predationrisk. ρ ( v y ) v y [m/s] Fig. 4.7
Semi-logarithmic plot of the distribution of velocities v y parallel to the y -axis inFig. 4.6(a) (black crosses) for a single bumblebee in the spider-free stage 1. The different linesrepresent maximum likelihood fits with a Gaussian mixture (red line), exponential (blue dotted),power law (green dashed), and single Gaussian distribution (violet dotted) [14]. Figure 4.7 shows a typical probability distribution of the horizontal velocitiesparallel to the flower wall (cf. the y-direction in Fig. 4.6(a)) for a single bumblebee.This distribution is in analogy to the move step length frequency distributions of theshark shown in Fig. 4.4, which also represent velocity distributions if the depictedstep lengths are divided by the corresponding constant time intervals of their mea-surements as discussed in Sec. 4.3.5. The distribution of bumblebee flights per unittime is characterized by a peak at low velocities. Only a power law and a Gaussian
Search for Food of Birds, Fish and Insects 15 distribution can immediately be ruled out by visual inspection as matching func-tional forms. However, a mixture of two Gaussian distributions and an exponentialfunction appear to be equally possible. Maximum likelihood fits supplemented byrespective information criteria yielded the former as the most likely functional formmatching the data. This result can be understood biologically as representing twodifferent flight modes near a flower versus far away from it, which is confirmed byspatially separated data analysis [14]. That the bumblebee switches to a specific dis-tribution of lower velocities when approaching a flower reflects a spatially adaptedflight mode to accessing the food sources. As a result, here we encounter anotherversion of intermittent motion: In contrast to the temporal switching between dif-ferent flight modes discussed in Sec. 4.3.4 this one is due to switching in differentregions of space.Surprisingly, when extracting the velocity distributions of single bumblebees atthe three different stages of the experiment and comparing their best fits with eachother, qualitatively and quantitatively no differences could be found in these distri-butions between the spider-free stage and the stages where artificial spider modelswere present [14]. This means that the bumblebees fly with the very same statisticaldistribution of velocities irrespective of whether predators are present or not. Theanswer about possible changes in the bumblebee flights due to changes in the en-vironmental conditions is thus not given by analyzing the probability distributionsof move step lengths, as one may infer from our diluted LFP guiding principle. Wewill now see that it is provided by examining the correlations of horizontal veloci-ties v y ( t ) parallel to the wall for all bumblebee flights. They can be measured by the velocity autocorrelation functionv acy ( τ ) = (cid:10) ( v y ( t ) − µ )( v y ( t + τ ) − µ ) (cid:11) σ . (4.5)Here µ and σ denote the mean and the variance of the corresponding velocity dis-tribution of v y , respectively, and the angular brackets define an average over all bum-blebees and over time. This quantity is a special case of what is called a covariance in statistics. Note that velocity correlations are intimately related to the mean squaredisplacement introduced in Chap. 2 of this book: While the above equation definesvelocity correlations that are normalized by subtracting the mean and dividing bythe variance, unnormalized velocity correlations emerge straightforwardly from theright hand side of Eq. (2.1) in Chap. 2 by rewriting it as products of velocities. Thisyields the (Taylor-)Green-Kubo formula expressing the mean square displacementexactly in terms of velocity correlations [32]. Note that the velocity autocorrela-tion function is defined by an average over the product between the initial velocityat time τ = τ along a trajectory: By definition it ismaximal and normalized to one at τ =
0, because the initial velocity is maximallycorrelated with itself. It will decay to zero if on average all velocities at time τ arerandomly distributed with respect to the initial velocities. Physically this quantitythus measures the correlation decay in the dynamics over time τ by giving an indi-cation to which extent a dynamics loses memory. For example, for a simple random walk as defined in Chap. 2 and by Eq. (4.1) in our section the velocity correlationswould immediately jump to zero from τ = τ =
0, which reflects that these ran-dom walks are completely memory-free. This property was used in Chap. 2 to deriveEq. (2.2) from Eq. (2.1) by canceling all cross-correlation terms. (a) -0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 v y a c ( τ ) τ [s]-0.1-0.05 0 0.05 0.1 0.15 0 0.5 1 1.5 2 (b) -0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 v y a c ( τ ) τ [s] Fig. 4.8
Velocity autocorrelation function Eq. (4.5) for bumblebee velocities v y parallel to thewall at three different stages of the experiment shown in Fig. 4.6: (a) Experimental results forstage 1 without spiders (red), 2 under predation threat (green), and 3 under threat a day after thelast encounter with the spiders (blue). The data show the effect of the presence of spiders on thebumblebee flights. The inset presents the resampled autocorrelation for the spider-free stage in theregion where the correlation differs from the stages with spider models, which confirms that thepositive autocorrelations are not a numerical artifact. (b) Theoretical results for the same quantityobtained from numerically solving the Langevin equation (4.6) by switching off (red triangles,upper line) / on (green circles, lower line) a repulsive force modeling the interaction of a bumblebeewith a spider. These results qualitatively reproduce the experimental findings in (a). Figure 4.8(a) shows the bumblebee velocity autocorrelations defined by Eq. (4.5)for all three stages of the experiment. While for the spider-free stage the correlationsremain positive for rather long times, in the presence of spiders they quickly becomenegative. This means that the velocities are on average anti-parallel to each other, oranti-correlated. In terms of flights, when predators are not present the bumblebeesthus fly on average more often in the same direction for short times while in thepresence of predators on average they often reverse their flight directions for shortertimes. This result can be biologically understood as reflecting a more careful searchunder predation threat: When no predators are present, the bumblebees forage withmore or less direct flights from flower to flower. However, under threat the bum-blebees change their direction more often in their search for food sources, rejectingflowers with spiders. Mathematically this means that the distributions of velocitiesremain the same, irrespective of whether predators are present or not, while the topology , i.e., the shape of the bumblebee trajectories changes profoundly being onaverage more ‘curved’.In order to theoretically reproduce these changes we model the dynamics of v y by a Langevin equation [33]. It may be called Newton’s Law of stochastic physics,as it is based on Newton’s Second Law: F = m · a , where m is the mass of a tracerparticle in a fluid moving with acceleration a = d x / dt at position x ( t ) (for sake ofsimplicity we restrict ourselves to one dimension). To model the interaction of the Search for Food of Birds, Fish and Insects 17 tracer particle with the surrounding fluid, the force F on the left hand side is writtenas a sum of two different forces, F = F S + F b : a friction term F S = − η v = − η dx / dt with Stokes friction coefficient η , which models the damping by the surroundingfluid; and another term F b that mimicks the microscopic collisions of the tracerparticle with the surrounding fluid particles, which are supposed to be much smallerthan the tracer particle. The latter interaction is modeled by a stochastic force ξ ( t ) ofthe same type as we have described in Sec. 4.3.5 for which here one takes Gaussianwhite noise. Interestingly, the stochastic Langevin equation can be derived fromfirst principles starting from Newton’s microscopic equations of motion for the fulldeterministic dynamical system of a tracer particle interacting with a fluid consistingof many particles [32].At first view it may look strange to apply such an equation for modeling themotion of a biological organism. However, for a bumblebee the force terms maysimply be reinterpreted: While the friction term still models the loss of velocity dueto the surrounding air during a flight, the stochastic force term now mimicks boththe force actively exerted by the bumblebee to perform a flight and the randomnessof these flights due to the surrounding air, and to sudden changes of direction by thebumblebee itself. In addition, for our experiment we need to model the interactionwith predators by a third force term. This leads to Eq. (20) stated in Chap. 2, whichfor bumblebee velocities v y we rewrite as dv y ( t ) dt = − η v y ( t ) − dU ( y ( t )) dy + ξ ( t ) . (4.6)Here we have combined the mass m with the other terms on the right hand side. Theterm F i = − dU ( y ( t )) / dy with potential U mimics an interaction between bumblebeeand spider, which can be switched on or off depending on whether a spider is presentor not. Data analysis shows that this force is strongly repulsive [14]. Computing thevelocity autocorrelation function Eq. (4.5) by solving the above equation numer-ically for a suitable choice of a repulsive force qualitatively reproduces a changefrom positive to negative correlations when switching on the repulsive force, seeFig. 4.8(b).These results demonstrate that velocity correlations can contain crucial informa-tion for understanding foraging dynamics, here in the form of highly non-trivialcorrelation decay emerging from the interaction of a forager with predators. Thisexperiment could not test the LSH, as the mathematical assumptions on its validitywere not fulfilled. However, conceptually these results are in line with the idea un-derlying the LEH: Theoretically the interaction between forager and environmentwas modeled by a repulsive force, to be switched on in the presence of predators,which qualitatively reproduced the experimental results. Together with the spatiallyintermittent dynamics when approaching the food sources as discussed before, thesefindings illustrate a complex spatio-temporal adjustment of the bumblebees both tothe presence of food sources and predators. This is in sharp contrast to the scale-freedynamics singled out by the LFH. Of course, modeling bumblebee flights by a Langevin equation like Eq. (4.6)ignores many fine details. A more sophisticated model that reproduces bumble-bee flights far away from the flowers more appropriately has been constructed inRef. [15] based on the same data as discussed above.
The main theme of our chapter was the question posed to the end of the introduc-tion:
Can search for food by biological organisms be understood by mathematicalmodeling?
While about a century ago this question was answered by Karl Pearsonin terms of simple random walks yielding Brownian motion, about two decades agothe LFH gave a different answer by proposing L´evy motion to be optimal for for-aging success, under certain conditions. Discussing experimental results testing it,we arrived at a finer distinction between two different types of LFHs: The LSH cap-tured the essence of the original LFH by stating that under certain conditions L´evyflights represent an optimal search strategy for finding targets. In contrast the LEHstipulates that L´evy flights may emerge from the interaction between a forager andpossibly scale-free food source distributions. A weaker version of these differenthypotheses we coined the LFP, which suggests to look for power laws in the prob-ability distributions of move step lengths of foraging organisms. An even weakerguiding principle derived from it is to assume that the foraging dynamics of bio-logical organisms can generally be understood by analyzing step length probabilitydistributions alone. We thus have a hierarchy of different LFHs that have all beentested in the literature, in one way or the other.By elaborating on experimental results, exemplified by selected publications, weoutlined a number of problems when testing the different LFHs: miscommunicationbetween theorists and experimentalists leading to incorrect data analysis; the diffi-culties to mathematically model a specific foraging situation by giving proper creditto all relevant biological details; and problems with an adequate statistical data anal-ysis that really tests for the theory by which it was motivated. We highlighted thatthere are alternative stochastic processes, such as intermittent search strategies, thatmay outperform L´evy strategies under certain conditions, or at least lead to similarresults, such that it may be hard to clearly distinguish them from L´evy motion. Wealso discussed an experiment on foraging bumblebees, which showed that relevantinformation to understand a biological foraging process may not always be con-tained in the probability distributions that are at the heart of all versions of the LFH.These experimental results suggested that biological organisms may rather performa complex spatio-temporal adjustment to optimize their search for food sources,which results in different dynamics on different spatio-temporal scales. This is atvariance to L´evy motion, which by definition is scale-free.However, these results are well in line with another, more general approach tounderstand the movements of biological organisms, called the
Movement EcologyParadigm [7]: This theory aims at more properly embedding the movements of bi-
Search for Food of Birds, Fish and Insects 19
Fig. 4.9
Sketch of the
Movement Ecology Paradigm : It cross-links four other existing paradigmsrepresenting different scientific disciplines, which describe specific aspects of the movements ofbiological organisms. The aim is to mathematically model the dynamics emerging from the in-terplay between these different fields by an equation like Eq. (4.7); from [7], copyright (2008)National Academy of Sciences, U.S.A. ological organisms into their biological context as shown in Fig. 4.9. In this figure,the theory centered around the LFH is rather represented by the region labeled ‘ran-dom’, which focuses on analyzing movement paths only. However, movement pathsof organisms cannot properly be understood without embedding them into their bi-ological context: They are to quite some extent determined by the cognitive abilitiesof the organisms and their biomechanical abilities, see the respective two further re-gions in this diagram. Indeed, only on this basis the question about optimality maybe asked, cf. the fourth region in this diagram, which here is rather understood in abiological sense than as purely mathematical efficiency. Physicists and mathemati-cians are used to think of diffusive spreading, which underlies foraging, primarily interms of moving point particles; however, living biological organisms are not pointparticles but interact with the surrounding world in a very different manner. The aimof this approach is to model the interaction between the four core fields sketched inthis diagram by a state space approach . This requires to identify relevant variables,cf. the diagram, by establishing functional relationships between them in form of anequation u t + = F ( ΩΩΩ , ΦΦΦ , r t , w t , u t ) , (4.7)where u t is the location of an organism at time t . A simple, boiled-down exampleof such an equation is the Langevin equation Eq. (4.6) that we proposed to describe bumblebee flights under predation threat. Here du t + / dt = v y ( t ) and the potentialterm is related to the variable r t above while all the other variables are ignored. The discussion about the LFH is still very much ongoing. As an example we referto research on movements of mussels, where experimental measurements seemedto suggest that L´evy movement accelerates pattern formation [22]; however, see thediscussion that emerged about these findings as comments and replies to the abovepaper, which mirrors our discussion in the previous sections. A second example isthe debate about a recent review by Andy Reynolds [24], in which yet another newversion of a LFH was suggested; again, see all the respective comments and theauthors’ reply to them. While these two articles are in support of the LFH, we referto a recent review by Graham Pyke [23] as an example of a more critical appreciationof it.We conclude that one needs to be rather careful with following power law hy-potheses, or paradigms, for data analysis, here applied to the problem of under-standing the search for food by biological organisms. These laws are very attractivebecause of their simplicity, and because in certain physical situations they repre-sent underlying universalities. While they clearly have their justification in specificsettings, these are rather simplistic concepts that ignore many details of the biolog-ical situation at hand. This can cause problems when biological processes are morecomplex. What we have outlined represents not an entirely new scientific lesson;see, e.g., the discussion about power laws in self-organized criticality. On the otherhand, the LFH did pioneer a new way of thinking that goes beyond applying simpletraditional random walk schemes to understand biological foraging.Financial support of this research by the MPIPKS Dresden and the Office ofNaval Research Global is gratefully acknowledged.
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