The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion.

BACKGROUND
The Lévy flight foraging hypothesis predicts a transition from scale-free Lévy walk (LW) to scale-specific Brownian motion (BM) as an animal moves from resource-poor towards resource-rich environment. However, the LW-BM continuum implies a premise of memory-less search, which contradicts the cognitive capacity of vertebrates.


RESULTS
We describe methods to test if apparent support for LW-BM transitions may rather be a statistical artifact from movement under varying intensity of site fidelity. A higher frequency of returns to previously visited patches (stronger site fidelity) may erroneously be interpreted as a switch from LW towards BM. Simulations of scale-free, memory-enhanced space use illustrate how the ratio between return events and scale-free exploratory movement translates to varying strength of site fidelity. An expanded analysis of GPS data of 18 female red deer, Cervus elaphus, strengthens previous empirical support of memory-enhanced and scale-free space use in a northern forest ecosystem.


CONCLUSION
A statistical mechanical model architecture that describes foraging under environment-dependent variation of site fidelity may allow for higher realism of optimal search models and movement ecology in general, in particular for vertebrates with high cognitive capacity.


Markovian and non-Markovian memory implementation
Though many ecologists would argue for some kind of path analysis when analyzing GPS-data in detail, it is important to realize that GPS-data are typically collected with 1 hour intervals or so, i.e., they are not exactly tracking every mechanistic step along the movement path of an animal. Hence, a statistical-mechanical model framework is a feasible approach, whether the movement is under influence of long term memory or not.
Basically, two movement classes are available to simulate memory-influenced space use; Markovian-compliant and non-Markovian. In the Markovian case a stochastic element, frequently represented by correlated random walk, is combined with deterministic rules for memory-based directed returns towards previously visited locations [1]. This implies a mixture of intrinsic directional persistence of two kinds; a tendency to keep the direction of the foregoing step (correlated random walk) and a bias towards remembered locations. At the mechanistic level the strength of the bias depends on a moment-to-moment perception of the potential foraging gain from re-visiting a specific patch versus just "move along" in a correlated random walk manner. In this model, memory is implemented in a Markovian-style manner, since both directional persistence and memory-based bias are re-calculated momentby-moment by the actual movement algorithm. The calculation is based (a) on local conditions at the model animal's present location, and (b) on perceived gain from moving in the direction of the assumed most profitable patch at the present point in time. Hence, this model implements memory of past patch visits by expanding the individual's field of perception relative to its current location [2]. While the model is executed at a mechanistic scale, one could get a statistical-mechanical representation of the output by sub-sampling every n'th relocation (where n is sufficiently large). Properties of this coarser representation of space use (which would resemble GPS fixes; see above) could then be calculated to infer indirectly about the animal's behavior.
On the other hand, to implement memory in a non-Markovian manner one could postulate that an animal might have several goals running in parallel at different temporal resolutions. In this case, a Markovian structure will not suffice, owing to the intrinsic tension between execution of high-frequency short term goals under weaker or stronger constraint of medium-and longer term goals. It is in this case not straightforward to define a specific Markovian calculation for the next step's direction and movement speed, based on momentto-moment local conditions only (whether an element of memory-based bias towards a remembered location is superimposed on local conditions or not). Specifically, since a Markovian-structured memory model [1] includes the directional bias (the memory-dependent component) in the mechanistic step-by-step calculation of the next move, it is not possible for the model individual to execute a tactical left-directed step while the bias term of the equation is stronger than the other terms and thus determining a right-directed move towards a more distant goal. Since short term tactics and longer term strategy are superimposed in the movement algorithm that determines the next step's direction, conflicts between tactics and strategy is solved by the present "weights" of the respective terms in the movement function.
These weights are re-defined in a step-wise manner; i.e., Markovian, in accordance to the conditions at the given point in time and locality in space.
To avoid this issue where tactical and strategic considerations have to lead to a specific "averaged", or "democratic" goal (all aspects considered at the present point in time and space), one may choose to shift the focus on the system by temporal coarse-graining in the first place, rather than executing the model at the mechanistic scale and then sub-sample relocations from the simulation output (as described above for the Markovian model). This approach isbasicallythe outline for the Multi-scaled random walk (MRW) approach. In 4 this case one assumes that the simulated path is collected at a temporal resolution t that is large enough to ensure a statistical-mechanical level of system dynamics. In other words, t is assumed to be sufficiently large to embed mechanistic behavior at finer scales <<t. At the statistical-mechanical level, one may then postulate an alternative to the Markov-mechanistic moment-to-moment (sequential) calculation where tactics and strategic terms are simultaneously weighted against each other to produce a given result for the next move.
The MRW model postulates that tactics and strategy may run with separate goals at different temporal scales. In short, a short-term move to the right may be executed even if the longer term goal is to move to the right, given that the magnitude of such finer-scaled violations of the longer term strategy is not compromising the longer term goal. In this manner coarser-scale strategic dynamics represents a constraint on finer-scale tactics [3].
Since a movement algorithm involving both tactical and strategic terms by necessity has to be mechanistic and thus Markov-structured, the MRW framework alternatively describes movement at the coarser statistical-mechanical level to avoid the mechanistic "Markov-trap".
At this coarser level, movement tactics and strategy is expressed by a postulate about scalefree space use under influence of site fidelity in statistical terms, and a statistical-mechanical path algorithm that is based on this postulate. Hence, a given step calculation involves two stochastic terms; one for scale-free exploratory steps (mimicked by a LW function) and one for occasional return events to a previous location (the site fidelity aspect). The crucial point is that the MRW postulate leads to a set of predictions about space use patterns, which can be tested on real GPS data using non-scaling and/or non-memory models as null hypotheses.
A statistical-mechanical level implies an alternative set of observable quantities, which deviates qualitatively from what is observed at the mechanistic level [4]. As shown in the present results, these observables may be applied to test for memory influence on movement.
It is also possible to distinguish Markovian (e.g., ref. [1]) from non-Markovian memory execution (MRW), for example from differences in the expected fractal dimension of space use [2,3]. However, Markovian and non-Markovian approaches to implement memory share the basic principle that movement is considered a mixture of exploratory steps and affinity towards previously visited patches.

Simulation of MRW
MRW of length 10 7 steps (8 replicates of each condition) was simulated in a homogeneous environment as a set of successively independent step vectors with length L MRW =α(RND) -1/(β-1) with α=1 and β=2. RND is a random number between 0 and 1. Using a constant α implies a constant average step length for the simulations; i.e., movement speed (step length per unit time increment) is assumed constant on average. Inter-step direction was drawn uniformly from 0-2π radians, which implies simulation execution at a statistical mechanical meso-scale relative to the micro-scale where the actual behavioral algorithm is expressed (reflected by the fine-grained path of the animal). On average at every 10,000 th , 1,000 th , 100 th or 10 th time increment, representing respective t ret , the step was replaced by a directed return to a randomly chosen previous location in the series (representing infinite memory). These variants of boundary conditions describe a progression towards smaller ranging area for a given series length, along an environment gradient with assumed increasing resource abundance. The random choice of target reflects the implicit assumption that even in an objectively homogeneous environment there may be a subjective positive fitness value connected to site fidelity (auto-facilitation from patch familiarity [5]).
The infinite memory condition above was replicated by series where memory horizon for collection of historic locations was constrained to a trailing window of last 10,000 time increments and t ret varying as for the infinite memory conditions. Step length truncation was for all conditions above set to the maximum length within the largest arena for the simulations (LW-like condition), in practice implying no truncation from other factors than return steps. Thus, series were also produced with infinite memory but physical constraint on step lengths; i.e., mimicking a truncated power law. This "physical" truncation adds to the return step effect on observed step lengths when a series is sampled at scale t obs = 1,000 unit time increments. Truncation was effective in a range around L trunc =200,000 length units, by discarding long steps from the following rule: A replicate length L'' x for each successive step of length L' x (x=1, 2, 3, … , 10 4 ), was calculated from the formula L'' x = (L' x /2) + RND*(L' x ). If L'' x >L trunc , the actual step was discarded and replaced by a new step, and the test was repeated. Site fidelity was varied from variations of t ret as above, with return event every 10,000 th , 1,000 th , 100 th or 10 th time increment. The truncation condition reflects the situation where limitations from movement speed or environmental borders may terminate the largest displacements prematurely. If these extrinsic constraints are stronger (appearing at higher frequency) than the return step events, they will contribute additionally to limit the range of scale-free (power law) distribution of steps.
Additionally, one set of series was produced with return rate t ret =10 4 , but the target was chosen among the last 10 time increments to represent MRW mimicking memory-less and free-dispersing LW. This variant represents an interface towards classical LW. To illustrate additional transition towards a LW-like pattern when t ret is increased further, one series was run with t ret =100,000 and t obs =1,000.
Arithmetic binning (bin size L, 2L, 3L, …) was chosen for the 10-step time horizon, for better display of the "hockey stick" pattern in the extreme tail (see results). For other distributions, log-binning was applied [6].
The narrow memory horizon route towards LW may be interpreted as an adaption to an unpredictable environment (old information becomes swiftly outdated and is ignored).
One additional simulation was also run with memory-less correlated random walk (persistence 0.5 on a scale from 0 to 1) rather than MRW. The chosen temporal lag t obs =10 3 also for this series produces BM; i.e., directional persistence is vanishing at coarser observational scales [7].
Under all conditions the relocations (sample of fixes) were collected at t obs =10 3 , leading to N=10 5 collected steps pr series, and a ratio for the MRW conditions of ρ=t ret :t obs of 10:1, 1:1, 1:10 and 1:100 respectively. Each series except for the correlated random walk case was initiated by a memory-less LW of length 10,000 steps (1,000 steps for condition t ret =10) to represent an individual entering an unfamiliar area and then switching to memory map utilization. The first 1,000 collected steps from each series were discarded to minimize transient effects (i.e., N=9,000 for analysis).
Step length distributions were produced using standard histogram method (arithmetic binning) and power-law optimized method (geometric binning [6]). Slopes were estimated from least square regression after log transformation of both axes. Unit bin width for all series was set to 2600 length units, which represents ca 50% larger scale than median step length size for the data set with the least influence from memory. Eight replica of each of the 6 conditions were included in the analyses. MRW with extreme ratio tret/tobs=100:1 Figure A2 shows the log-transformed distribution of 9,000 step lengths (log-binning) from a simulation of MRW with t ret =1:100,000 and t obs =1:1,000 (implying a ratio ρ=t ret /t obs =100:1). At this magnitude of t ret /t obs , the influence from return steps is close to negligible even towards large binsand the pattern appears LW-like over the entire range of L.
Truncation was set to the neighborhood of 800,000 ≈ 2 20 length units. The slope -1.88 implies β=1.88, which is below the limit for transition towards stationary variance (the inset line for β=3 shows this transition slope, for visual comparison).

Study area
The study area is in Sogn og Fjordane county at the western part of southern Norway.
The vegetation is mostly in the boreonemoral zone (Abrahamsen et al. 1977). Forests are dominated by deciduous (mainly birch Betula sp. and alder Alnus incana) and pine forest (Pinus sylvestris). Norway spruce (Picea abies) has been planted in many areas [9]. The terrain is characterized by valleys and mountains from coastal to inland areas. Red deer is the most common cervid in the area. In addition livestock, in particular sheep, is common in some areas. More detail description of the habitat can be found elsewhere [9].

Red deer data
The GPS data comes from 18 female red deer caught by darting at winter feeding  [2]. Hence, space use was shown to be MRW-like rather than LW-like. Here we extend the analysis by applying the parallel shift method under the Lagrangian approach (step length distribution) and adjusting grid resolution for each individual to estimate the "spatial grain unit" parameter c under the Eulerian approach (incidence as a function of sample size of fixes).
The procedure for capture was approved by the Norwegian national ethical board for science ("Forsøksdyrutvalget", http://www.fdu.no). The deer were fitted with Televilt Basic "store-on-board" GPS (Global Positioning system) collars or Televilt Basic GPS collars with GSM (Global System for Mobile communications) option for transfer of data via cell phone network (Televilt TVP Positioning AB, Lindesberg, Sweden) [10]. The GPS collars were programmed to record hourly positions, and to release a drop-off mechanism after approximately 10 months (tracking period ranged from 6-12 months). The median location error for our GPS collars was 12 m [upper 95% CI=23 m [10]]. We removed points located further than 10 km/h from the preceding location (n=85; 0.024 % of all points), because they most likely represent large GPS errors and not true deer locations. We here use data from the summer period from June 1 st -September 16 th , except for four individuals where fix sampling was terminated earlier owing to abandoning of summer range and start of autumn migration [11]. Biologically this means the period when individuals are in their summer range and when most adult females give birth to a calf. Calving status of the marked red deer females was unknown, but most adult females ovulate and are likely accompanied by a calf [12]. The end of the period coincides with the onset of hunting season (from 10th September), fall migration [11], and rutting activities [13].

Statistical analyses of red deer data
To estimate super-diffusion and power law compliance from the Parallel shift method, each of the 18 series of fixes was re-sampled 1:10 (collecting every 10 th relocation; t obs =10 h), resulting in series lengths of ca 200-280 fixes pr individual. The distribution of step lengths F(L) for relocations of the pooled set of step lengths at this coarser t obs (averaged over 18 series) were then compared with the average distribution from 200-280 step lengths, uniformly sampled from each of the original series at t obs ≈1 h (the parallel shift method requires equal sample sizes due to non-stationary distributions under condition of LW or MRW; otherwise a correction term needs to be included). Bin width was set to 95 m, 13 representing 50% wider bin than the median step length for the pooled series of fixes. F(L) was also studied for each individual, both at t obs =1 h and t obs =10 h. Bin width was in this case calculated specifically for each individual, and set to 50% larger than the respective median step length.
Expectation for parallel shift is t 2obs , where t 2obs and t 1obs are observational lags at coarse and fine scales, respectively [14].