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The authors have declared that no competing interests exist.

Conceived and designed the experiments: AK JJB JG. Performed the experiments: AK JG. Analyzed the data: AK VL RF SB SW JJB JG. Contributed reagents/materials/analysis tools: AK VL RF SB SW JJB JG. Wrote the paper: AK VL RF SB JG.

The goal of this study is to describe accurately how the directional information given by support inclinations affects the ant

The goal of the present study is to describe accurately the effect of support inclination on the ants

This study fits in a series of works devoted to the modeling of collective building processes in social insects

A methodology for modeling this kind of processes has been thoroughly reviewed in a previous article

Now, motivated by the need to progress towards an explanation of 3-dimensional structure construction (termites mounds, ants nests), we need to consider the major difference between motion on a horizontal plane and motion on a tilted surface developing in 3 dimensions, that is, the local inclination of the surface. In the building phase, the tilted and curved surfaces of the structure in progress are expected to modify the ants moving decisions and might thus have in turn an effect on the nest architecture itself

Numerous studies in insects show that the inclination of the support has indeed a strong effect on individual locomotion behavior. For instance, the speed of adult beetles decreased with an increase in the slope of the substrate as a reaction to the increased gravitational force vector opposing uphill movement

The first step of our analysis was to check that the trajectories of

To understand this global effect in terms of individual decision processes along the trajectory, we then proceed with the Boltzmann Walker framework. First, we check that this model is still relevant in the present context when ants move on the horizontal plane with no orientation field, and allows the quantitative correspondence between the individual parameters estimated from the trajectories and the population dispersal. We take this level plane condition as the reference case to test for inclination effects.

Then, we consider how precisely the inclination should affect the decision process. Organisms orient themselves to the effect of stimuli (such as heat, light, humidity, gravity etc.) in two ways. One is by a directed orientation reaction (taxis), in which the direction of motion of the organism is influenced by the stimulus. The other method of orientation is an undirected locomotory reaction (kinesis) in which the average speed or the average rate of turning of the organism, but not the direction in which it moves, are dependent on the stimulus

Finally, we discuss how this extended Boltzmann Walker model can be used in contexts of more natural landscapes with heterogeneous inclinations.

For experiments, a

We believe that this experimental data set benefits from being well controlled for factors affecting the ants' motion (temperature, humidity, stress), and uses high tracking precision to determine ants' positions. As we do not claim that the modeling framework we use below is exhaustive by nature, we made the whole data set available as supplementary information so as to offer the community an opportunity to analyze the ants' trajectories from a complementary point of view (e.g. with potential field approaches

Examples of ant trajectories on the inclines are illustrated in

Slopes are indicated by labels

In a second step, using the noisy tracked positions, we recovered a representation of the ants' trajectories compatible with the Boltzmann Walker description. For this, the time series of detected locations were converted into a series of straight segments separated by reorientation events. A full description of this segmentation procedure is given in the

A portion of a trajectory is shown in dots (same ant as in

Overall, we obtained 345 trajectories (69 per inclination value) containing from 3 to 2246 segments. The numbers of segments per trajectory for each inclination were (min–median–max)

From these trajectories, we derived individual time-averaged statistics such as the time needed to reach the border of the outer circle of radius

These quantities are computed over 69 trajectories for each inclination. The inclination has a major impact on the motion speed, which in turn induces longer residence times. However, since ants move straighter towards the upper or lower edges when the inclination is steeper, their total trajectory length within the disks is lowered.

Thus far, these time-averaged statistics confirm that inclination has a major effect on speed, but also that ants adapt locally their decision making about where to go, and/or how long to persist in the same direction, depending on how they are aligned with the steepest line. To give a full account of how the support inclination affects the ants' trajectories, we propose a behavioral model which accounts for this effect at the individual scale, as a stochastic decision process all along the trajectory. This model is developed by extending the standard BW model.

The classical Boltzmann Walker model is summarized in the

Being memory-less, the stochastic behavior of the Boltzmann Walker can furthermore be translated with no approximation into partial differential equations describing the time evolution of the probability density

In contrast, the effect of support inclination

Introducing the full dependencies of these parameters, the extended version requires:

a – When ants are aligned with the steepest line, they become slower

b – When ants are aligned with the steepest line, they increase their path lengths on average

c – When ants take new directions, they favor uphill or downhill directions

The first two predictions are a type of ortho-kinesis and klino-kinesis respectively, the third being a kind of taxis. Note that we assumed here that speed fluctuations (among and/or within individuals) are governed by a process uncorrelated with the reorientation and persistence decisions, and remain to be studied separately, if relevant. Hence speed, the mean free path and phase function are treated in this context as independent parameters. Accordingly, in the next part, the three predictions will be tested independently, and for each inclination

First, using the Mean Square Displacement, we checked that

The MSD (

Then, using the segmented series, we computed for each inclination the frequency distributions

The heading domain has been split into 16 sectors (each centered on the corresponding

For the null inclination, the distribution of the headings is flat (

Using the sector splitting of the parameters, we can now test each prediction in turn.

Regarding the average speed

We observe that the mean free path

The orientation domain has been split in 8 sectors. Segments were partitioned in sectors according to

The concentration of directional deviations, or heading persistence

The orientation domain has been split into 8 sectors. Segments were partitioned in sectors according to

We found a major effect of

As a final check that these observed effects of support inclination on the extended BW model features and parameters are fairly consistent with the observations at the population scale, we have generated numerically trajectories using the parameterized model (see

The exit headings were computed for each ant as the direction from the starting point to the point where they exited the disk of radius 0.2(left, N = 69 per

Overall, this analysis shows that the extended BW model parameters undergo two kinds of effects as the inclination increases, and the two predictions (b) and (c) should be considered. As for the prediction of klino-kinesis (b), ants moving on the steeper inclination actually appeared to increase their path lengths, on average, when they are aligned with the steepest line (

In this study, we have performed a detailed analysis of how

The present set-up was designed to isolate the effect of inclination on the ants' decision-making, so as to identify and quantify this effect. To this end, we managed to maintain the inclination the same all over the field, and keep everything else as constant as possible. In this homogeneous field, we consistently assumed that the influence of inclination on motion decisions was the same everywhere. It is noteworthy in this case that the lower speed on steeper inclines is more or less compensated for by straighter trajectories, so that the mean residence time in a definite area is only mildly affected by inclination. Further theoretical developments are required to derive the macroscopic equations corresponding to the extended BW model in the case of such a homogeneous directional field. Such a derivation of invariant characteristics (oriented diffusion, residence time, first-return statistics, statistics of visits…) is however expected to be challenging, especially considering the asymmetric shape of the phase functions for intermediate directions (

Moreover, the most interesting biological situations arise naturally for landscapes of varying inclinations. Since the characteristic shape of these variations (e.g. spatial frequencies spectrum) will probably be case-specific (dispersal within the nest, foraging in the external environment around the nest entrance, migration, etc), the functional consequences of the reaction to support inclination is expected to be highly context-dependent. In the context of building behavior, the next step will be to establish how the distributions of visits inside a given structure is affected by the preference for alignment with the steepest line (versus a uniform distribution predicted by pure diffusion).

The extended Boltzmann Walker model is a time-continuous description of the motion built upon the assumption of a memory-less process, so decision-making is considered instantaneous at the model time-scale, and only depends at any time on the information perceived at position

The predictions about how the extended BW model would shape the distribution of ants in a given landscape call for dedicated numerical studies, using Monte-Carlo simulations in complex geometries. There is no additional need for simulating the choice of a new direction since it remains a purely local decision at turning points. However, it would require specific algorithms (such as a null-collision algorithms

As for the speed variance (either for one individual across time, or among individuals), we have indicated that we focused on the geometrical aspects of the trajectories, considering the speed process as independent. As a matter of fact, this assumption is well supported a posteriori by the result that we did not find an effect of the heading on the average speed of ants, that is, the speed process does not seem to be affected by directional information. As it is known that speed can vary with temperature, replicating the same study with higher and lower temperatures would constitute a good test for the independence between the process governing speed, and the two processes governing trajectory geometry, which we have assumed here.

Considering macroscopic statistics, using average speed and neglecting speed distribution has proven to be a fair approximation in previous studies

The isotropic distribution of average speed appears as a surprising result since it would be expected, for instance, that ants progressing uphill should be slower than when moving downhill. For instance, Seidl et al. found lower speeds on steeper inclines in desert ants moving uphill, but indicate that desert ants progressing downhill displayed high velocities

Regarding the statistics of exit heading direction (

More generally, the coarseness of the substrate on which ants are moving should also be considered, as it can impact greatly on both speed and the sinuosity of trajectories

Finally, is this influence of ground inclination relevant for contexts other than ants' motion behavior? Understanding how animal movements are explicitly driven by environmental factors is a challenge for further advances in dispersal ecology

Three colonies of ants

Inclination | Colony | Day (YYYY-MM-DD) | Hour (HH:MM–HH:MM) |

D | 2012-05-29 | 14:15–15:00 | |

D | 2012-05-29 | 15:25–16:20 | |

D | 2012-05-29 | 16:25–17:30 | |

C | 2012-05-31 | 15:45–16:40 | |

C | 2012-05-31 | 17:15–18:05 | |

C | 2012-06-01 | 11:20–12:35 | |

C | 2012-06-01 | 16:55–18:45 | |

C | 2012-06-01 | 19:45–20:50 | |

C | 2012-06-04 | 11:30–12:40 | |

C | 2012-06-04 | 14:10–14:40 | |

A | 2012-06-15 | 14:00–15:45 | |

A | 2012-06-15 | 16:10–17:30 | |

A | 2012-06-15 | 17:50–18:40 | |

A | 2012-06-18 | 11:30–12:50 | |

A | 2012-06-18 | 14:50–15:30 | |

D | 2012-06-18 | 16:00–17:20 | |

D | 2012-06-18 | 17:40–18:50 |

The tracking program was written from scratch using the Core Image infrastructure of Mac OSX (Objective-C+GPU-based Image manipulation), starting from the CIColorTracking example (

The whole set of data is made available as supplementary information. The data are given as supplemental data files (zip archives) :

Each archive file contains a series of 69 files, one file per ant. Each file contains the data of a trajectory in a tab-delimited text format with 9 columns, corresponding in order to the inclination index, the colony index, the temperature, the humidity, the recording date, the individual index, the rank of the video frame, the corresponding time in second, and the x and y coordinates in meters. Each file starts with a header line labeling this information.

For the distribution of headings over the time shown in

We will detail in this section the algorithm we used to split the ants' trajectories into series of consecutive segments. Our algorithm is sourced from the field of time series data mining. This matter has received much attention over the last decade in relation with the increase of computer power and the explosion of data time series in a wide range of fields, from Life Sciences

The piecewise linear approximation in our context addresses the following problem: given a time series of locations in the plane, finding the best partitioning in linear segments. Such a process will thus aggregate consecutive points that belong to the same segment into one representation of this segment even if those points are not perfectly aligned. As an approximation, it can give a compact representation of the data, but compromises accuracy.

Hence the major concern for series segmentation is the balance between compactness and accuracy, i.e. the optimal number of segments

Given a time series T, produce the best representation using a fixed number

Given a time series T, produce the best representation such that the maximum error for any segment does not exceed some user-specified threshold (local error,

Given a time series T, produce the best representation such that the combined error of all segments is less than some user-specified threshold (total error

The problem of finding the best partitioning is combinatorially complex, and the data time series are up to approximately

As a first check for the algorithm consistency, we have tested its performance on an artificial set of data in a zero-noise situation. For this, we have generated an artificial trajectory following the Boltzmann Walker model on a large area (1.5 m), with parameters close to the ones found in ants in first approximation:

The next step was to estimate the sensitivity of the segmentation procedure to the accuracy of our estimation of tracking noise

For each couple of values, we generated 300 artificial trajectories as above, using parameters

The procedure is parameterized by a stopping criterion

Each set is associated with

We denote

We denote

We denote

We denote

1:

2:

3.

4:

5: Compute

6:

7:

8:

9: Compute

10:

11:

12:

13:

Since ants displayed varied speeds, and showed some periods of stopping from time to time, we computed the Mean Square Displacement as a function of the number of reorientation events rather than time, following

In order to conduct heading statistics analysis, we used circular statistics, taking heading distribution as the input. Linear statistical measures cannot be used because angles on a unit circle have modulus

The mean angle

In order to obtain of the undirected axis angle of the original sample, we must cancel the effect of doubling:

The final angular deviation value of our bimodal samples is:

The function to compute the P-value for the test of uniformity was adapted for R from

When the need is to simply generate trajectories from the standard BW model, parameterized by a mean free path

Input parameters: mean free path

Variables: position

1:

2:

3:

4:

5:

6:

7:

8:

To simulate the extended BW model, we need to further take into account the dependence of the parameters on the heading

These simulations were performed in the R environment

For instance, let's denote F(a) as the discretized cumulated function of heading deviations for a given sector and a given inclination, with

Denoting sector-based random sampling

Input parameters: inclination

Variables: position

1:

2:

3:

4:

5:

6:

7:

8:

Generating artificial data required sampling the angular deviation according to a probability density function governed by parameter

Input parameter:

1:

2:

3:

4:

5: return

In the

When particles such as photons are involved, velocity changes are triggered by local interactions with molecules or particles. As far as ants or other animals are concerned, the velocity changes

However, this random component of the path can be precisely specified as follows: the velocity change can occur at any time, it does not depend on how long the animal has been walking since the last velocity change event — this is a memory-less process. Let

Starting from the location of the last change, the probability that the next change does not occur before the ant has walked

If the rate is constant over space,

What happens at turning points under this model ? Let us denote

It can be shown that the statistics of space occupancy corresponding to this model are well approximated by a diffusion process (see below). Moreover, the corresponding diffusion coefficient that would govern the spreading rate of a population over time is strictly related to the parameters of the individual decision model following (in 2D):

With the BW model, a single walker is followed over time along its trajectory, making free paths and turning events. There is an alternative description focusing on what happens at a given position and a given direction over time. Let

In the same way,

Considering furthermore that the temporal variation of

Starting from a location

The corresponding displacement

The Mean Square Displacement is then a measure of the spatial spreading of the ant over time.

In cases when speed varies with time, it can be computed as a function of the number of reorientations events

For more formal developments, see for example

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We warmly thank Bernard Auduc, of the Ecole des Mines d'Albi, for his experimental assistance. We are grateful to Sepideh Bazazi and Stephen C. Pratt for their help in improving the manuscript.