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A simple methodological framework was established for testing the BCR model using empirical datasets, consisting of the GPS data of 5 animals. For each of these 5 animals, the three parameters were accordingly tuned using a straightforward estimation procedure. This procedure uses the empirical datasets to infer the parameters’values (Fig. 1). We also used the datasets to assess the model’s reliability—or performance-. We also detail other analyzes that were carried out to ensure the robustness and consistency of the approach, including the deterministic nature of the 5 statistics and a sensitivity analysis. This analysis consists in evaluating the performance of the BCR using a sweep method that produce arbitrary values of the parameters instead of using data-driven estimations. All BCR simulations and the five statistics were performed using MATLAB Version 7.13.0.564 (R2011b).Figure 1Framework used for testing the BCR model performance, for one animal. Black lines detail the two operations processed from the GPS dataset. The 3 parameters are estimated from the GPS data and—using these parameters—1000 simulations of the BCR model are computed. No particular operations are associated with the dotted black lines, but they show how the BCR and the GPS dataset are evaluated and compared using the statistics.Full size imageDataThe locations of 5 GPS-collared red deer (Cervus elaphus) were gathered at La Petite Pierre National Hunting and Wildlife Reserve (NHWR), in north-eastern of France (48.8321 (Lat.) / 7.3514 (Lon.)). The reserve is an unfenced 2670 ha forest area characteristics by deciduous trees (mostly Fagus sylvatica) in the western part and by coniferous species (mostly Pinus sylvestris and Abies alba) in the eastern part in nature reserve surrounded by crops and pastures. It is located at a low elevation area of the Vosges mountain range, which rises up to 400 m a. s. l. The climate is continental with cool summers and mild winters (mean January and July temperatures of 1.4 and 19.6 (^{circ })C, respectively, data from Phalsbourg weather station, Meteo France, from 2004 to 2017). Three ungulate species are present and mainly managed through hunting in the NHWR: wild boar, red deer and roe deer. The present study focuses on female red deer for test model. A detailed overview of the landscape and surroundings is given in40. The GPS data had regular observation frequencies with high frequency sampling (Table 1). In the following text, we note (X_i = [X_i^{(1)}, X_i^{(2)}]) the locations of the individual with (X_i in {mathbb {R}}^{2}), (i=1,2,ldots ,n) and where (X_i^{(1)}), (X_i^{(2)}) represent the longitude and latitude respectively. We use (t_{i}) ((t_1=0)) as the time elapsed between two successive locations (X_{i-1}), (X_{i}) and$$begin{aligned} {overline{T}}= dfrac{1}{n} sum _{i=1}^{n} t_i end{aligned}$$
(1)
as the average sampling time. The trajectory of the animal, or ‘path’, was interpolated using linear interpolation between each pair of recorded observations (Fig. 2 and detailed in Supplementary Methods (Eq. 21) and associated Graphic 2). It approximates the animal travels in straight lines at constant velocity between each pair of locations41. The attractor (X_F) of one individual was estimated as the isobarycenter of all recorded locations:$$begin{aligned} X_F = left[ frac{1}{n}sum _{i=1}^{n} X_i^{(1)}, frac{1}{n}sum _{i=1}^{n} X_i^{(2)} right] end{aligned}$$
(2)
Table 1 Data summary. For each animal, the total number of observations n is given along with the period of collection (date and time), the sampling rate ({overline{T}}) (i.e. the average time between 2 observations) (in min.) and corresponding standard deviation, total distance (in kilometers), total recording time (in days) and average speed s (in (10^{-2}) m.s-1).Full size tableFigure 2Individual paths of the five red deer. Individual paths of the five red deer. The individual paths are plotted for the five red deer (left panel, a) along with the distribution of the relative turning angles (degrees) in polar plots (right panel, b). An angular value of 0 consists in a straight motion from the previous location, while a relative turning angle of 180 (^{circ }) c corresponds to a turn back.Full size image
BCR modelThe model aims at estimating the location at the next time step, given the actual location X at step i:$$begin{aligned} X_{i+1} = f left( X_i right) end{aligned}$$
(3)
such that the function (f(cdot )) is assumed to be representative of the behavior of the animal on sufficiently large time scales. We considered one individual of a given species with no interaction and simulated its movement in continuous space and discrete time in 2 dimensions. The BCR includes 3 parameters coupled with isotropic diffusion:

Diffusion: A random direction with uniform spatial distribution in a 2D plane,

Bias ((p_F)): An increased probability to go to a fixed point named attractor42. This attractor was estimated as the isobarycenter of all recorded locations, defined as (X_F) (Eq. 2). This yields a bias or advection parameter in the direction of (X_F). We use the term ’attraction’ for the bias component of the BCR and the term ’den’ for the attractor. In the data set we study, the den is equivalent to the deer’s bunk.

Correlated component ((p_I)): This parameter increases the probability to move forward, i.e. to perform one step in the direction of the previous step. This is equivalent to a short term bias in movement, when the animal has inertia. We refers to ’inertia’ for the correlated component,

Immobility ((p_s)): We included this as a specific parameter and the movement is stopped for one step. This takes into account the absence of movement between a pair of locations. It can be accredited to technological limitations with the satellite telemetry due to a weak GPS signal strength, possibly due to natural elements: such as when the animal was standing underneath a rock or due to dense clouds, dust particles, mountains or flying objects, such as airplanes). However, this can also be part of the behavior of the animals, during specific times: sleep cycles or foraging for instance. We use (d_{min }) to denote this distance cutoff and set (d_{min }=10)m which corresponds to the magnitude of the error typically found in GPS locations43. We also use (d_{min }) to encapsulate GPS error and peculiar ecological behavior, not associated with (p_I) or (p_F), that are beyond the scope of this study.

The effect of each parameter is detailed in Fig. 3. The typical model contains all three parameters: (p_I), (p_s) and (p_F) for describing animal motion while offering a trade-off between the number of parameters and the description of animal motion.Figure 3Simulated animal motions over arbitrary parameter values. Fifty motions of length (n_s=100) steps are simulated and originate from a common centroid (downward-pointing triangle) with increased levels of correlation ((p_I)), immobility ((p_s)) and bias ((p_F)). Both the location of the attractor (X_F) (black dot) and the log-normal parameters controlling the step size distribution are fixed ((mu =3), (sigma ^2 = 1)).Full size imageEstimation of the parametersThe estimation of the three parameters for each animal is based on the empirical datasets. We distinguished between the states, where one state is described by the pair (left{ X_{i-1}, X_iright} ) and the situations, where one situation is described by the past ((X_{i-1})), current ((X_i)) and future ((X_{i+1})) locations. Knowing both the state of the animal at a given time step i and its situation—the realization of movement at the next time step (i+1)—allowed for collecting the occurrences of inertia, immobilism and attraction. This could be done provided we account for the variability of the movement: the animal may not be heading exactly toward the den, or performing inertia with an exact angular value of (pi ). Thus we discretized the space around the animal in 8 quadrants at each time step i. For example, if the animal was heading straight forward with a margin of (pm pi /8) then it was considered in the situation of inertia. In other words, the state could fall in a situation of inertia with a margin of (pm pi /8). Such a discretization can be represented as a matrix, depending on the state of the animal, its location and the location of the den at each time step (see Supplementary methods, Eq. 19). In order to gather enough data samples per situation, we arbitrary used angular thresholds of (pi /8) as a convenient trade-off between data scarcity and precision loss. Using smaller threshold values (say (pi /10)) may result in too few samples per situations. Using larger threshold values such as (pi /4) may result in a loss of precision while capturing additional movement samples that may not correspond to the situation.We first needed to define in which state is the animal at each step i. A state is the 2-tuple containing the previous and actual observation ({X_{i-1}, X_i}). We wanted to distinguish between non-conflicting and conflicting states, where a non-conflicting state is when the animal is in one state only, while a conflicting state is when the animal is in two states at once. We defined two conflicting states:$$begin{aligned} {mathscr {H}}_{IF}:={i : widehat{ left| X_{i-1} X_i X_F right| } le pi /8} end{aligned}$$
(4)
when the animal was already heading toward the den (X_F), and:$$begin{aligned} {mathscr {H}}_{Is}:={i : dleft( X_{i-1}, X_iright) le d_{min }} end{aligned}$$
(5)
when the distance between two consecutive observations was too small ((le d_{min }) m.), describing an individual that was already immobile. Such that the subset of non-conflicting states is:$$begin{aligned} {mathscr {H}}:={1,cdots ,n} – {mathscr {H}}_{IF} – {mathscr {H}}_{Is} end{aligned}$$
(6)
We then needed to assess in which situation the animal was for each corresponding state. A situation is the 3-tuple (left{ X_{i-1}, X_i, X_{i+1}right} ). We defined three subsets of situations corresponding to a straight forward motion (I), no motion (s) and a motion toward the den (F):$$begin{aligned} I:= {i: pi – pi /8 < widehat{(X_{i-1} X_i X_{i+1})} le pi + pi /8} end{aligned}$$ (7) $$begin{aligned} s:= {i: dleft( X_i, X_{i+1} right) le d_{min }} end{aligned}$$ (8) $$begin{aligned} F:= {i: mid widehat{X_{i-1} X_t X_{i+1}} - widehat{X_{i-1} X_i X_F} mid le pi /8} end{aligned}$$ (9) With (d(cdot ,cdot )) the Euclidean distance between two locations. For the situations in s, we considered that the animal is not performing a motion if the Euclidean distance between two successive locations was (le d_{min })m.We counted the number of states falling in each situation, for states in ({mathscr {H}}) (Eq. 6). We defined (x_{1}), (x_{2}), (x_{3}) as the empirical proportion of cases corresponding to each situation:$$begin{aligned} {left{ begin{array}{ll} x_1 = dfrac{# I cap {mathscr {H}}}{# {mathscr {H}}} ; qquad x_1:=dfrac{1+p_I}{chi } \[16pt] x_2 =dfrac{# s cap {mathscr {H}}}{# {mathscr {H}}} ; qquad x_2:=dfrac{p_s}{chi }\[16pt] x_3 = dfrac{# F cap {mathscr {H}}}{# {mathscr {H}}} ; qquad x_3:=dfrac{1+p_F}{chi } end{array}right. } end{aligned}$$ (10) with (chi = 8+p_{I}+p_{s}+p_{F}). The values of (x_1), (x_2) and (x_3) were then gathered for each animal. We did not use immobile locations (i.e. distances separating two successive observations must be ( > d_{min }) m) for the estimations of (x_1) and (x_3). Solving Eq. (10) for (chi ) with respect to (x_1), (x_2), (x_3) yields:$$begin{aligned} chi = dfrac{6}{1-(x_{1} + x_{2} + x_{3})} end{aligned}$$
(11)
Plugging in Eq. 10:$$begin{aligned} {left{ begin{array}{ll} p_{I} = x_1 chi -1\ p_{s} = x_2 chi \ p_{F} = x_3 chi -1 end{array}right. } end{aligned}$$
(12)
Note that we assumed that (p_{IF} = p_I + p_F) in ({mathscr {H}}_{IF}) and (p_{Is} = p_I + p_s) in ({mathscr {H}}_{Is}) as a convenient arrangement and ignoring higher order conflicting cases. Investigating the step-size distribution in the 5 deers, we found a log-normal step size distribution (Supplementary Fig. S1). We then set a log-normal distribution (ln {mathscr {N}}(mu , sigma ^2)) for the step size distribution for the step size in the BCR.The same estimation procedure was used for configurations using a different number of parameters and quantity (chi ) is accordingly calculated depending on the number of parameters used. It is possible to obtain negative values using this inference method. A parameter with a negative values reflects a direction that is not favored by the animal. In such a case, one should rethink the design of the BCR by changing the parameters (see Supplementary methods, section “negative parameters”). In the subsequent sections, we only consider parameters with positives values.BCR dynamicsThe BCR dynamics for each animal are completely determined by the three parameters (p_F), (p_I), (p_s), taking values in ({mathbb {R}}^{+}), and the step-size distribution. If (p_F = p_I = p_s = 0), the BCR resumes to a typical two-dimensional random walk with a log-normal step size distribution (ln {mathscr {N}}(mu , sigma ^2)). The dynamics can be visualized in Fig. 3 for different values of each parameter. When simulating a step in the model, the motion in ({mathscr {H}}) is described by:$$begin{aligned} f left( X_i right) = {left{ begin{array}{ll} left{ X_i^{(1)} + d cos (alpha _1) ; X_i^{(2)} + d sin (alpha _1) right} &{} qquad text {if } x in [0,8[ \ left{ X_i^{(1)} + d cos (alpha _2) ; X_i^{(2)} + d sin (alpha _2) right} &{} qquad text {if } x in [8,8+p_I[ \ X_i &{} qquad text {if } x in [8+p_I, 8+p_I+p_s[ \ left{ X_i^{(1)} + d cos (alpha _3) ; X_i^{(2)} + d sin (alpha _3) right} &{} qquad text {else} end{array}right. } end{aligned}$$
(13)
with x, d, (alpha _1) random variables defined as (x sim {mathscr {U}} in [0, chi ]), (d sim ln {mathscr {N}}(mu , sigma )), (alpha _1 sim {mathscr {U}} in [0,2pi ]). Variables (alpha _2), (alpha _3) are related to the angular values (alpha _2 = {{,{mathrm{atan2}},}}(X_{i}^2 – X_{i-1}^2, X_{i}^1 – X_{i-1}^1)), (alpha _3 = {{,{mathrm{atan2}},}}(X_{F}^{(2)} – X_{i}^{(2)}, X_{F}^{(1)} – X_{i}^{(1)})) with ({{,{mathrm{atan2}},}}(y, x)) the four quadrant inverse tangent function (14):$$begin{aligned} {{,{mathrm{atan2}},}}(y, x) = {left{ begin{array}{ll} arctan left( {frac{y}{x}}right) &{} x > 0,\ arctan left( {frac{y}{x}}right) +pi &{} x< 0{text {, }}y ge 0,\ arctan left( {frac{y}{x}}right) -pi &{} x< 0{text {, }}y< 0,\ +{frac{pi }{2}} &{} x=0{text {, }}y > 0,\ -{frac{pi }{2}} &{} x=0{text {, }}y < 0,\ 0 &{} x=0{text {, }}y=0text {.} end{array}right. } end{aligned}$$ (14) The motion in ({mathscr {H}}_{Is}) is:$$begin{aligned} f left( X_i right) = {left{ begin{array}{ll} left{ X_i^{(1)} + d cos (alpha _1) ; X_i^{(2)} + d sin (alpha _1) right} &{} qquad text {if } x in [0,8[ \ X_i &{} qquad text {if } x in [8, 8+p_I+p_s[ \ left{ X_i^{(1)} + d cos (alpha _3) ; X_i^{(2)} + d sin (alpha _3) right} &{} qquad text {else} end{array}right. } end{aligned}$$ (15) The motion in ({mathscr {H}}_{IF}) is:$$begin{aligned} f left( X_t right) = {left{ begin{array}{ll} left{ X_i^{(1)} + d cos (alpha _1) ; X_i^{(2)} + d sin (alpha _1) right} &{} qquad text {if } x in [0,8[ \ X_t &{} qquad text {if } x in [8, 8+p_s[ \ left{ X_i^{(1)} + d cos (alpha _2) ; X_i^{(2)} + d sin (alpha _2) right} &{} qquad text {else} end{array}right. } end{aligned}$$ (16) Statistics for describing animal movementWe simulated (N=1000) BCR and used 5 statistics to assess the model reliability on spatial features including: (i) the distribution of relative turning angles which provides information about the movement of the animal, (ii) the home range which provides information about the spatial density of observations and (iii) observation counts using still and mobile transects, providing information on absolute observation abundance44. A detailed description of each statistic is provided in Supplementary Methods and Fig. S2. The reliability—or performance—was assessed in each animal and studied statistic using two error terms (e_1) and (e_2). Error (e_1) is the (L^1) norm to compare the differences between the statistic ({tilde{mathscr {S}}}) computed over a simulated path, and the statistic (smash {{mathscr {S}}}) computed over the data-set:$$begin{aligned} e_1 mathrel {mathop :}= sum {text {errors}} = sum _{k=1}^N |smash {{mathscr {S}}}- {tilde{mathscr {S}}}_k| end{aligned}$$ (17) With (k = 1,ldots , N) the number of simulations of the BCR. Error (e_1) is the sum of absolute differences in the given statistic, and is a natural way of measuring the distance between the statistics computed on the data set and the trajectories generated using the BCR. We also focused on the average relative error (e_2) as an indicator of the sensitivity:$$begin{aligned} e_2 mathrel {mathop :}= dfrac{1}{N} sum _{k=1}^N dfrac{{tilde{mathscr {S}}}_k}{smash {{mathscr {S}}}} end{aligned}$$ (18) Distribution of turning anglesFor each individual, the distribution of counter-clockwise relative turning angles (widehat{(X_{i-1} X_i X_{i+1})}) was gathered, provided (d(X_{i-1}, X_{i}) > d_{min }) and (d(X_{i}, X_{i+1}) > d_{min }). This means that we only kept the angles from observations that were separated by an Euclidean distance greater than (d_{min }).Home rangeWe used an adaptive kernel density estimator (matlab package kde2d—kernel density estimation version 1.3.0.0) as an estimator of the utilization distribution45 to represent the home range of the animal. The approach of Z.I. Botev provided an estimate of observation density using a bivariate (Gaussian) kernel with diagonal bandwidth matrix46. The density was estimated over a grid of (210 times 210) nodes and we computed the home range area (in m2) for various values: 100, 99, 95, 90, 80, …, 20, 10% of the estimated density. Similarly to the distribution of turning angles, we compared each value of the data’s home range against the simulated one.DilationDilation is generally used to account for the spatial attributes of an object such as to measure an area around the path or the volume of a brownian motion (see Wiener sausage47 and Gromov–Hausdorff distance). In our approach, we use dilation of both simulated and GPS paths for two reasons: to have a real—and comparable—number that accounts for how a trajectory has explored space and because it is natural tool from a census point of view (the dilated path corresponds to the area where the animal can be detected). Each simulated or real path was plotted in binary format in a window and dilated with a disk shape. The window size was set to a huge value in order to encapsulate the dilated path while preventing boundary effects, i.e. the convex envelope of the dilated area did not collide with any window border. We then estimated the surface covered by the dilated path for 100 different sizes of the disk, from disk size 1 to disk size 100. We compared each value of the data’s estimated surface against the simulated one.Immobile transectsWe used still transects that counted the number of times the animal was seen in their line of sight. We arbitrary set the line-of-sight value at 200 m. The number of sightings of each transect was gathered and ordered in decreasing order, thus breaking the spatial dependence. We then compared the bins of the resulting histogram in the data and in the simulated path.Mobile transectsFirst, the movement of the animal was linearly interpolated from the GPS data, meaning that between two recorded locations the individual followed a linear path. The speed of the animal between two locations was accordingly reconstructed using the recorded times (t_i) between each location. Second, we used mobile transects as the ecological sampling method, where each transect ‘count’ the intersection between its path and the animal’s one. The mobile transects followed a predefined path at a given constant speed as time increased. The area of vision of each transect was defined as a circle of a given radius. Each time the path of an individual collided with an area of vision, the count of the corresponding transect increased by 1. Two types of movements were used: linear and clockwise rotational transects. The initial locations of both types of transects are (X_1) and (X_F). Both the animal and mobile transects started to move at the same time. At each of the two locations (X_1), (X_F), 8 linear transects moved in the 8 cardinal directions, totalizing 16 transects. For the linear transects, every 10,000 time steps, we set (2 times 8) new transects starting at the same locations and following the same directions. Clockwise rotational transects were rotated around (X_1) and (X_F) using a 500 m radius. When we reached (t_n), we gathered the total count (i.e. the count of all transects). For the two types of transects, we gathered the total count for 6 different lines of sight: 50, 100, 200, 400, 500, 1000 m. and 4 speeds: s/4, s/2, s, 2 s with s the average speed of the animal. We then aggregated the overall count in each of the two types of transects, and compared the results from the data and the simulated path (Supplementary Methods and Fig. S2).Scale invarianceSeveral authors pointed out that the temporal resolution of the discretization is of importance: it should be relevant to the considered behavioral mechanisms5,48,49,50. Schlägel and Lewis focused on the quantification of movement models’ robustness under subsampled movement paths49. They found that increased subsampling leads to a strong deviation of the central parameter in resource selection models49,51. They underlined that important quantities derived from empirical data (e.g. parameters estimates, travel distance or sinuosity) can differ based on the temporal resolution of the data49,51. Moreover, Postlethwaite and Dennis highlighted the difficulty of comparing model results amongst tracking-datasets that vary substantially in temporal grain50). Each of the studied dataset has a relatively high sampling rate (roughly 10 m) and a period of study that is appropriate to the analysis of animal movement at the year scale (Table 1). In order to investigate such a possible effect on the BCR dynamic, we changed the sampling rate of the movement path to ensure that the three parameters (p_I), (p_s) and (p_F) are scale invariant. The movement path formed by the GPS observations (X_i) was subsampled (decimated) for each individual. We only kept every (k^{text {th}}) observation starting with the first one and (k in left[ 1,10right] ). For (k=1) the path corresponded to the original one. The time spent between each successive observation was also accordingly reconstructed in order to keep track of ({overline{T}}) in subsampled movement paths. The time between two locations (X_i) and (X_{i+k}) was reconstructed as:$$begin{aligned} t_{j}’ = sum _{i=j}^{i+(k-1)} t_i end{aligned}$$
(19)
with (j in left[ 1, 1+k, 1+2k, ldots , n-left( k-1 right) right] ). We did not change the value of (d_{min }) as we subsampled the movement path because we designed (p_s) for capturing GPS noise and movements that are associated with peculiar ecological behaviors that are beyond the scope of this study in terms of time and spatial scales (foraging for instance). We then compared the resulting parameters (p_I), (p_F) and (p_s) as the resampling rate k increased.FluctuationsWhereas the BCR is a stochastic process, the deterministic aspects of the 5 statistics were tested with an increasing number of steps (n_s). The statistic associated with each realization of the model (a simulated path) is a random variable. If the distribution of these random variables has low concentration (high variance) then it is not a convenient statistic as it cannot be used as a reference for assessing the model’s performance, even when averaging over multiples realizations. On the opposite, if the statistic is deterministic (no fluctuations) it can provide a reliable tool to assess the model’s performance. This was numerically tested over a range of increasing (n_s) values with (n_s = 10^4, 2times 10^4, ldots , 4 times 10^5). For each of those step values, a set of 100 BCR was simulated with parameters (p_I), (p_F) and (p_s) estimated from the first deer (see Table 2) and we studied the variance of the statistics.Sensitivity analysisIn order to assess whether the estimated parameters are optimal (i.e. providing the best possible performance) and to study parameter scarcity, we also evaluated the performance of the model using arbitrary weight values. We first started by evaluating how using one parameter instead of the three could alter the performance of the model. We then extended this sensitivity analyse by drawing arbitrary values for each parameter from a multi-dimensional square mesh, whose center corresponds to the estimated values of (p_I), (p_s), (p_F), estimated using GPS data (Fig. 1). We additionally used values that are distant from the estimated ones, up to (p_I=3), (p_F=3) and (p_s=5). We tested a total of 151 new configurations with these arbitrary values. For each configuration, we ran 150 simulations and evaluated them using the 5 statistics. The mean error of (|smash {{mathscr {S}}}- {tilde{mathscr {S}}}_k|) and its standard deviation are gathered and plotted for each arbitrary configuration. As a resume, we replicate the framework described in Fig. 1 but we inject arbitrary parameters instead of using data-driven parameterisations.ApplicationThe proposed model could be used to infer environmental and behavior information from the dataset. We chose to illustrate such an application by trying to detect anomalous voids (or holes) in the spatial territory of the individual using the GPS dataset and Monte-Carlo simulations of the model. Anomalous means that the observed void is not related to the randomness of the movement, but rather related to a geographical artifact. The parameters (p_I), (p_F), (p_s), (mu ) and (sigma ^2) of the BCR were accordingly estimated from the data of each individual, similarly to previous experiments (Fig. 1). A simple heuristic was used to find voids in empirical and simulated paths for each individual: we computed the alpha shape of all locations using a fixed alpha radius of 60 m. This allowed for determining the surface covered by all locations while preserving the voids. We then collected the area of each void provided they had an area of at least 100 m2. We focused on voids near the center of the alpha shape in order to avoid artificial voids, generated by the weak density of locations at the boundaries. We ran 10,000 iterations of the model for each animal and estimated the probability (p_{varnothing }) of finding voids of different sizes in the simulated paths. This probability was then compared to voids found in the GPS datasets and available environmental information was used to determine whether any geographical element(s) could explain the unexpected voids. More

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NEWS AND VIEWS
07 July 2021

How to buffer against an urban food shortage

There is widespread concern that the risk of food shocks — sudden disruptions to food supply — is increasing. It emerges that a city’s vulnerability to food shocks can be reduced by diversifying its supply chains.

Zia Mehrabi

0

Zia Mehrabi

Zia Mehrabi is at the Sustainability Innovation Lab at Colorado and at the Environmental Studies Program, University of Colorado Boulder, Boulder, Colorado 80303, USA.

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More than half of the world’s population lives in urban areas, a proportion that is set to increase1 to 68% by 2050. These urban residents depend on supply chains to produce, procure, prepare and deliver food, and they are exposed to potential supply-chain disruptions and food shortages from changes in human activity and natural processes. There is growing recognition that food-system resilience needs to be improved, but how best to buffer against urban food shortages remains an open question for both research and policy. Writing in Nature, Gomez et al.2 assess how the flow of agricultural products to a city depends on the diversity of the city’s trading partners. The authors apply ideas from engineering — such as those used when ensuring infrastructure is protected from flooding — to inform the design of food systems that can buffer cities against food shortfalls.For decades, scientists and industry have been warning governments and consumers about the risks of food shortages. Such shortages have a range of possible causes, including droughts and heatwaves, pest and disease outbreaks, financial downturns and trade policies3. More recently, the COVID-19 pandemic has led academics, and society more generally, to revisit the question of how fragile urban food supplies really are (Fig. 1).

Figure 1 | Empty supermarket shelves during the COVID-19 pandemic. Many city dwellers around the world experienced such scenes, which were largely driven by panic buying and changing consumer behaviour. Gomez et al.2 demonstrate that the resilience of food supply chains can be increased by boosting their diversity.Credit: Oli Scarff/AFP/Getty

There have been many proposed solutions to deal with the dangers of food shortages, from climate-resilient agricultural management practices to promoting local food systems and self-sufficiency4. One solution that is gaining attention is to increase the number and variety of agricultural products, farms and companies procuring and delivering food. Diverse food supply chains might buffer cities against food shortages — in the same way that, in finance, a varied portfolio of stocks limits investment risk and, in ecology, a diverse mixture of species maintains ecosystem functions.
Read the paper: Supply chain diversity buffers cities against food shocks
Gomez et al. used data on the origin and destination of different agricultural commodities for 284 cities and 45 non-city geographical areas in the United States. They identified domestic food systems for each city — that is, all of the geographical areas that supply crops, meat, live animals or animal feed to that city. The authors then determined how many cities faced different thresholds of abrupt food-supply disruptions, known as food shocks, using the percentage difference between the minimum and mean of supply amounts for each food sector over four years for each city. More specifically, they counted the number of cities in which the minimum was more than a particular percentage (ranging from 3% to 15%) smaller than the mean in any one of those four years.Next, Gomez and colleagues combined those data with simple indicators of geographical similarity — such as the physical distance and difference in climate between each city and the geographical areas in that city’s supply network. With this information in hand, the authors tested the idea that groups of cities with more-diverse supply chains are better able to buffer against food shocks than are groups whose supply chains are less diverse. Indeed, they found that cities importing food from suppliers that are more dissimilar from themselves are less likely to face shocks than are cities whose supply-chain partners are less diverse. Such supply-chain benefits would not be reaped from having solely local food systems.Gomez et al. then considered design concepts from engineering, where infrastructure systems should be planned to withstand shocks — such as extreme flooding — of a given frequency and magnitude. The authors undertook some bold extrapolations, in which they estimated the size of food shocks that would be faced by different US cities given their current supply-chain diversity. They found that a rare shock, such as one occurring once in 100 years, would cause a food-supply loss of about 22–32% across different cities.
Transforming the global food system
The other implicit finding from Gomez and colleagues’ model is that even moderate supply-chain diversity is effective at reducing the probability of extremely large shocks. The authors also applied their analysis to shocks happening in multiple food sectors simultaneously. They obtained similar results to those for single-sector shocks — with supply-chain diversity also providing a buffering effect for these even rarer occurrences.Gomez and colleagues’ work has major implications for the way in which resilient food systems should be built, but it also has a few caveats. First, the authors used only four years of data for each city, posing problems for characterizing the distribution of shocks at each city. This limited time series makes it difficult to define the baseline variation in food supply — that is, what is considered normal — for consumers and retailers alike. It also makes it hard to see to what extent diversified supply chains buffer against food shortages under normal conditions compared with years marked by extreme events, and whether the net benefits are large enough to trigger a change in food-procurement policies.Second, the food-flow data used by Gomez et al. do not represent actual flows for each year, but instead are simply annual production quantities proportionally distributed according to observed flows5 in 2012. Therefore, the authors’ analysis does not capture, or allow for, rerouting or other social responses at the onset of extreme events. Such social responses within and after shock years would result in changing food flows across the supply network.
Rural areas drive increases in global obesity
Third, Gomez and colleagues did not validate the predictive worth of their model beyond the four years considered, or outside the United States. This lack of verification is perhaps most limiting for applying the findings in practice — partly because the stability of food supply is itself dynamic, and will change with increasing volumes and types of food consumed, as well as with production technology. Although the observed phenomenon and general patterns might hold in other years and geographical regions, no data or analyses exist to validate whether the authors’ design suggestions will protect against future shocks to the degree claimed.Designing urban food systems to specification is not as easy as engineering a bridge or dam that won’t fail in 100 years. The major global concern with respect to urban food shortages and food security is for populations of middle- to low-income countries, particularly those that are dependent on imports6. Theoretically, supply-chain diversity will also have a buffering effect for these populations when the number of urban dwellers starts to drastically increase in the coming years, especially in Africa. However, such nations are probably not accurately described by the model presented. Moreover, they have different policy options and capacities for producing diverse supply chains compared with those possible in the United States. Nevertheless, Gomez and colleagues’ work provides a timely and refreshing reminder that building diverse supply chains offers a crucial mechanism for protecting urban dwellers from food shortages.

Nature 595, 175-176 (2021)
doi: https://doi.org/10.1038/d41586-021-01758-6

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Competing Interests
The author declares no competing interests.

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Dataset of food flow networksWe derive annual, intranational food flow networks for the USA using the Freight Analysis Framework version 4 (FAF4) database30. The derived networks are for different food sectors and include all metropolitan areas in the USA. The FAF4 database consists of annual commodity flows during 2012−2015 for 115 geographic areas in the USA and 43 different sectors. We focus on the following four food sectors in the FAF4 database: crops, live animals, animal feed and meat. The 115 geographic areas in the FAF4 database cover the entire contiguous USA, including 69 metropolitan statistical areas and 46 remainders of states (the remainder is the area of a state that is not part of a FAF4 metropolitan area).To obtain food flows for all metropolitan areas in the USA, we disaggregate the FAF4 database from 115 to 329 areas (Supplementary Fig. 4), out of which 284 are metropolitan or combined statistical areas (120 metropolitan and 164 combined statistical areas). The disaggregation is performed using different socioeconomic and agricultural-related variables as attractors of supply and demand. For each food sector, a flow with origin o and destination d in the FAF4 database is disaggregated to a metropolitan-level flow with origin o′ and destination d′ using a disaggregation variable a as the best attractor of supply or demand.The disaggregation is performed in two stages. In the first stage, the supply U of each FAF4 remainder of state is disaggregated to include all the metropolitan areas in that remainder of state as follows:$${U}_{{o}^{{prime} }d}^{c}=frac{{U}_{od}^{c}}{{a}_{o}}times {a}_{{o}^{{prime} }},$$
(3)
where ({U}_{{o}^{{prime} }d}^{c}) (in tons per year) is the disaggregated supply for food sector c in origin o′ that satisfies demand at the FAF4 destination d, ({U}_{od}^{c}) (in tons per year) is the FAF4 food flow for sector c between areas o and d, and ({a}_{{o}^{{prime} }}) and ao are the attractor variables for the new origin o′ and FAF4 origin o, respectively. In the second stage, ({U}_{{o}^{{prime} }d}^{c}) is further disaggregated into demand E using:$${E}_{{o}^{{prime} }{d}^{{prime} }}^{c}=frac{{U}_{{o}^{{prime} }d}^{c}}{{a}_{d}}times {a}_{{d}^{{prime} }},$$
(4)
where ({E}_{{o}^{{prime} }{d}^{{prime} }}^{c}) (in tons per year) is the demand at destination d′ for food sector c supplied by origin o′, while ({a}_{{d}^{{prime} }}) and ({a}_{d}) are the attractor variables at the disaggregated destination d′ and FAF4 destination d, respectively.The FAF4 database includes food flow data at both the state level (48 states) and metropolitan level (115 areas including 69 metropolitan areas). Prior to performing the disaggregation, we jointly use the FAF4 state data and the FAF4 metropolitan data to select the best performing attractor variables. That is, we first use equations (3) and (4) to disaggregate the FAF4 state-level data to the metropolitan-level for the metropolitan and remainder-of-state areas in FAF4. By comparing the performance of our disaggregated flow data against the empirical FAF4 metropolitan-level data, we select the best attractor variable for each food sector. The following attractor variables are considered: population47, employment48, wages48, number of establishments48 and cropland area49. These variables are selected on the basis of previous analysis and data availability50.To assess the performance of the attractor variables, we use the Pearson correlation coefficient between the empirical FAF4 flows and the disaggregated flows for the metropolitan areas and remainder-of-state areas in FAF4 (Extended Data Fig. 3). The performance is high with correlation values greater than 0.87 and an average of 0.95. Using the best-performing disaggregation variables, we build the food flow networks employed in this study. The nodes in the networks represent metropolitan and remainder-of-state areas, and the weighted links represent annual food flows during 2012−2015 for crops, live animals, feed and meat (Supplementary Fig. 5).The FAF4 metropolitan and remainder-of-state areas we used to select the attractor variables span a wide range of populations, cropland areas, and number of establishments, since these FAF4 areas include the largest cities in the USA and a broad range of medium-size cities. The values of the attractor variables used in the disaggregation are within the ranges implied by the FAF4 metropolitan data (Supplementary Fig. 6), indicating that the variables are reliable. The exception to this is population, which is only used to disaggregate meat demand. Population, however, has a high disaggregation performance with a correlation coefficient of 0.97 (Extended Data Fig. 3). In addition, the use of population to disaggregate meat demand is consistent with previous scaling results for metropolitan areas in the USA51 that have shown that metropolitan-level variables that are related to resource consumption scale approximately linearly with population.Food inflows supply chain diversityTo determine annual supply chain diversity, we extract the annual food buyer–supplier subgraph of each city and food sector from the food flow networks19. We refer to each of these subgraphs as a food system. The food buyer–supplier subgraph of a city i consists of all the supply chain interactions with its trading partners or neighbours j for a specific food sector. Our measure of supply chain diversity is based on the notion of functional distance52. We compute the functional distance d between i and any of its trading partners j by combining five different indicators: physical distance, climate correlation, urban classification, economic specialization and network modularity. The indicators are described below in the ‘Functional distance indicators’ section of the Methods. We also perform statistical analyses to evaluate the influence of the attractor variables on these indicators (Methods). The indicators represent stable characteristics of cities and therefore tend to remain fairly constant during our study period.The functional distance ({d}_{ij}^{r}) for an indicator r between any pair of connected nodes (i,j) is calculated as$${d}_{ij}^{r}={N}^{-1}|{r}_{k}-{r}_{i}|,$$
(5)
where the normalization constant N is determined as the maximum value of (|{r}_{k}-{r}_{i}|) between any node k in the network and i. In equation (5), ({d}_{ij}^{r}=0) for functionally similar nodes and ({d}_{ij}^{r}=1) for dissimilar nodes.For each city’s buyer−supplier subgraph and food sector, any pair of connected nodes has 5 different functional distance indicators associated with it. To combine these distance indicators into a single measure, we calculate the average functional distance indicator (langle {d}_{ij}^{r}rangle ) as the arithmetic average of the 5 functional distance indicators for any pair (i, j) of connected nodes. We use the discrete probability distribution of food inflows binned by (langle {d}_{ij}^{r}rangle ) categories, together with Shannon entropy53, to calculate the supply chain diversity ({D}_{i,c}^{t}) of node i and sector c at year t:$${D}_{i,c}^{t}=frac{-{sum }_{k=1}^{K}{Y}_{i,c}^{t}(k)mathrm{ln},{Y}_{i,c}^{t}(k)}{log ,K}.$$
(6)
For sector c and year t, ({Y}_{i,c}^{t}(k)) is the proportion of food inflows to node i within bin k. The k bin is obtained by binning all the (langle {d}_{ij}^{r}rangle ) values for node i into a total number of K bins.({D}_{i,c}^{t}) is sensitive to the total number of bins K. Thus, for each node in our food flow networks, we tested the sensitivity of ({D}_{i,c}^{t}) to the total number of bins K. For K = 15, D values stabilize (Supplementary Fig. 7); therefore, we used 15 bins when performing all calculations of functional diversity.Functional distance indicatorsThe average functional distance between a city and its trading partners is based on the following five indicators:

(1)

Physical distance indicator (PDI). The PDI is obtained by calculating the Euclidean distance from the centroid of each geographic area to the centroid of all other areas. The geometric centroids of all geographical areas are calculated using the GIS software ArcMap (https://desktop.arcgis.com/en/arcmap/).

(2)

Climate indicator (CI). To account for different climates in cities across the USA, the Palmer Drought Severity Index (PDSI) is used54. The monthly PDSI is obtained from the National Oceanic and Atmospheric Administration for the years 1895−2015 at the climate division geographic level. An area-weighted average is performed to aggregate the PDSI data to the metropolitan level. The CI is obtained by calculating the monthly correlation between an area and all other areas.

(3)

Urban classification indicator (UCI). To identify the urbanization level of a geographical area, the Urban-Rural Classification indicator of the National Center for Health Statistics is employed55. This indicator classifies counties using a scale from 1 to 6, where a value of 1 indicates the county is highly rural and a value of 6 means highly urban. The UCI is obtained at the metropolitan level using an area-weighted average of the county-level values.

(4)

Network modularity indicator (NMI). This indicator identifies geographical areas (network nodes) that belong to the same community. A community is a group of nodes whose strength interactions are stronger than with the rest of the network. To identify the network’s communities, we aggregate the flows from the four food sectors (crops, live animals, feed and meat) into a single-layer network. The communities are identified by maximizing the modularity measure of Newman56,57 using the greedy optimization algorithm of Blondel et al.58,59. Network nodes that lie in the same community are assigned a NMI of 1 and 0 otherwise.

(5)

Economic specialization indicator (ESI). Each geographical area is assigned a score based on its dominant economic supply sector. Supply is quantified using the FAF4 intranational commodity flows30. Areas with a dominant meat sector are assigned an ESI of 1, crops an ESI of 2, fruits and vegetables an ESI of 3, animal feed an ESI of 4, live animals an ESI of 5, milled grains an ESI of 6, and industrial products an ESI of 7.

Probabilities of food supply chain shockThe annual probability of food supply chain shock is calculated as the probability that food inflows to a city fall below a percentage of the average inflows for that city during 2012−20158. To compute this probability, we group all nodes from the 4 food flow networks (1,221 observations) into 6 diversity bins ordered from lowest to highest functional diversity D. The bin size is selected to obtain bins with similar number of observations, approximately 204 observations in each bin. For each city i and food sector c in a bin, we calculate the food supply chain shock ({S}_{i}^{c}) as$${S}_{i}^{c}=left[1-frac{min({I}_{i}^{c})}{langle {I}_{i}^{c}rangle }right]times 100,$$
(7)
where ({I}_{i}^{c}) is the time series of total food inflows to node i for sector c during 2012−2015, and ({rm{min }}({I}_{i}^{c})) and (langle {I}_{i}^{c}rangle ) are the minimum and average values of the time series ({I}_{i}^{c}), respectively.For each diversity bin b, we count the number of observations nb that meet the criteria ({S}_{i}^{c} > s) for (sin {3,4,5,ldots ,15}), with s being the shock intensity threshold. The probability of a food supply shock S being greater than s in bin b is calculated as:$${P}_{b}(S > s)=frac{{n}_{b}}{{N}_{b}},$$
(8)
where Nb is the total number of observations in bin b. Thus, for each shock intensity s, we obtain a set of probabilities of food supply chain shock,$$P(S > s)={P}_{b}(S > s),{rm{for}},b={1,ldots ,6}.$$
(9)
Furthermore, we adapt equations (8) and (9) to calculate the probability of a food supply chain shock S being greater than s, P′(S > s), under co-occurrence conditions. We define co-occurrence as any city that experiences a shock to 2 or more food sectors during 2012−2015. With this definition, P′(S > s) is calculated in a fashion similar to that described above. We bin the network’s nodes into 6 groups from lowest to highest diversity and determine the percentage of food supply chain shock with equation (7). Letting ({n}_{b}^{{prime} }) be the total number of cities for which ({S}_{i}^{c} > s,) holds for 2 or more food sectors, the probability of a food supply chain shock S being greater than the shock intensity s in bin b is now$${P}_{b}^{{prime} }(S > s)=frac{{n}_{b}^{{prime} }}{{N}_{b}^{{prime} }},$$
(10)
where ({N}_{b}^{{prime} }) is the total number of cities in bin b. Thus, under co-occurrence conditions, the new set of probabilities for each shock intensity s is$$P{prime} (S > s)={P}_{b}^{{prime} }(S > s),{rm{for}},b={1,ldots ,6}.$$
(11)
Statistical analysesWe use the disaggregated food flow data to calculate both the probability of food supply chain shock and supply chain diversity. Therefore, we perform two complementary analyses to test whether the attractor variables used in the disaggregation are causing a circularity issue in the empirical relationship between the probability of food supply chain shock and supply chain diversity. For the first analysis, we determine the Pearson correlation between the functional distance indicators (PDI, CI, UCI, NMI and ESI) and the attractor variables (Supplementary Fig. 8). We find that the attractor variables are weakly correlated with the functional distance indicators (Supplementary Table 1). For the second analysis, we determine the Pearson correlation between the food supply chain shock intensities and attractor variables for the 4 food sectors (Supplementary Fig. 9). The attractor variables are also weakly correlated with the food supply chain shock intensities (Supplementary Table 2). Thus, circularity is not unduly influencing the empirical relationship between the probability of shock and supply chain diversity.We also evaluate whether the empirical relationship between the probability of food supply chain shock and supply chain diversity is driven by the disaggregation of the original FAF4 data. For this, we recalculate the probability of shock and supply chain diversity using the FAF4 data. For the FAF4 data, the probability of shock also declines with rising supply chain diversity (Supplementary Fig. 10), similar to the reduction observed using the disaggregated food flow data (Fig. 1a), suggesting that the latter data are not driving the relationship.Furthermore, we test whether the relationship between the probability of food supply chain shock and supply chain diversity holds for different demand levels. To control for demand, we stratify all the data into low, medium and high demand levels using population or food inflows as proxies for demand. For both stratifications, the bounds are chosen so that each level has approximately the same number of data points. Using the stratified data, we recalculate the Pearson correlation between the probability of shock at 3%, 5%, 10% and 15% shock intensities and supply chain diversity for each level of population or food inflows (Supplementary Table 3). We find that the relationship between the probability of food supply chain shock and supply chain diversity holds for these different demand levels (Supplementary Table 3). Using the same exponential model in equation (1) to fit the relationship for the stratified data (Supplementary Fig. 11), we determine the exponential model parameters ks and D0,s for each demand level (Extended Data Table 6). These parameters fall within the 95% confidence interval of the parameters of the exponential model in the main text (Extended Data Table 1), indicating that the model is robust.We perform two different sensitivity analyses to assess the influence of the five distance indicators on the empirical relationship between the probability of food supply chain shock and supply chain diversity. The first analysis compares single-indicator diversity measures against the multi-indicator diversity measure calculated using all 5 indicators. Five different single-indicator diversity measures are compared, one measure for each of the 5 indicators: PDI, CI, UCI, NMI and ESI. For the second sensitivity analysis, we leave out one indicator at a time to calculate diversity using the 4 remaining indicators, which results in another 5 different diversity measures. The diversity measures for the sensitivity analyses are all calculated following the approach in the ‘Food inflows supply chain diversity’ section of the Methods. To perform the sensitivity analyses, we plot the empirical relationship between the probability of food supply chain shock and each diversity measure (Supplementary Figs. 12 and 13), and calculate the Pearson correlation of the data (Extended Data Table 7). The correlation coefficients are used to quantify the influence of the distance indicators on the relationship between the probability of food supply chain shock and supply chain diversity (Extended Data Table 7). The probabilities of food supply chain shock are calculated following the approach in the ‘Probabilities of food supply chain shock’ section of the Methods. We find that the five indicators have a varied influence on the relationship between the probability of food supply chain shock and supply chain diversity (Extended Data Table 7). The inclusion of all 5 indicators, however, in the supply chain diversity measure increases the Pearson correlation between the probability of food supply chain shock and supply chain diversity (Extended Data Table 7). More

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