Trail traffic flow prediction by contact frequency among individual ants

Abstract

Ants build a trail that leads to a new location when they move their colony. The trail’s traffic flows smoothly, regardless of the density on the trail. To the best of our knowledge, such a phenomenon has been reported only for ant species. The trail’s capacity is known as trail traffic flow. In this paper, we propose a probabilistic model of trail traffic flow, which overcomes some inadequacies of the kinetic model previously proposed in the literature. Our model answers a question unsolved by the previous model, namely, how many worker ants form such a density-independent trail. We focus on ants’ responses to mutual contacts that involve individuals in trail formation. We propose a model in which contact frequency predicts the number of worker ants that form a trail. We verify that our model’s estimates match the empirical data that ant experts reported in the literature. In modeling and evaluation, we discuss an intelligent ant species, the house-hunting ant Temnothorax albipennis, which is popular among the ant experts.

Appendices

Appendix A: Ant behaviors in modeling

The influence of density on contact frequency is an open question among ant experts (Gordon et al. 1993). Contact frequency does not increase in crowded colonies of either the European fire ant Myrmica rubra or the red imported fire ant Solenopsis invicta (Gordon et al. 1993). Contact frequency increases for small density and stays low for large density, regarding the hyper-social parasite ant Lasius fuliginosus (Gordon et al. 1993). We have not found any study that confirms the relationship between contact frequency and density regarding the house-hunting ant Temnothorax albipennis (see Pratt 2005a). Therefore, we have not assumed any relation between contact frequency and population density in this paper.

Ant experts use contact frequency per worker per minute to show how many times workers make physical contact with one another in their colonies (see Pratt 2005a, 2005b, 2008; Franks and Richardson 2006). The mutual contacts among workers are a probabilistic phenomenon. To simulate the probabilistic phenomenon each parameter should be defined as a probabilistic one. Therefore, the contact frequency that we define in this paper is the frequency of approach by workers toward other workers. These approaches end with either a mutual contact or a non-contact behavior, depending on the mutual distances.

The ants’ quorum sensing behavior has not been fully studied. Our model refers to the case of the house-hunting ant Temnothorax albipennis. A small number (20 to 30) of workers scout potential new locations. They assess the locations and determine a new location for moving their colonies by mutual contact (Pratt 2005a, 2005b, 2008; Franks and Richardson 2006). The contact frequency in the new location tells how many other workers they recruit from their current nests (Pratt 2005a, 2005b). Recruited workers are involved in trail formation. How the recruiting workers communicate with the recruited workers needs further research. However, it is reasonable to consider that the contact frequency is the only available way of communication between the recruiting workers and the recruited ones. In modeling, the recruiting workers approach the recruited ones at a certain contact frequency. We then assume that the recruited workers also approach other workers in the current nests at the same frequency.

The memory of the house-hunting ant Temnothorax albipennis is very short lasting (Pratt 2005a). The recruiting workers should exhaustively approach other workers in their current nests within one minute. Otherwise, their memory expires and wastes the time invested in nest search. We study by simulation whether such a small number of workers can approach all the other workers inside their nests.

In the simulation, we prepare 400 workers per colony, from which we randomly choose 20 recruiting workers. At a contact frequency of 20, the 20 recruiting workers successfully approach and make physical contact with all the other workers at least once within four chances of mutual contact. After the fourth chance of mutual contact (i.e., in one minute), the largest trail traffic flow emerges. This result supports our modeling based on the ants’ memory-lasting-time.

Appendix B: Formulation of the mutual distance distribution

We modify the parameters of the two types of functions of John et al. (2009). These functions describe the two different types of headway distance distributions for food-searching individuals. The modified functions can describe the mutual distance distributions for nest-locating individuals.

First, we add a constant value of $0.5 \times10 ^{-1} (= \frac{1}{ \sqrt{n}} = \frac{1}{ \sqrt{400}})$ to the log-normal function that is one of the two functions. This constant removes any unwanted bias in the sampling process of the mutual distances. Such unwanted bias should otherwise occur due to the large data size (see the simulation results in Appendix C). Furthermore, that constant normalizes the mutual distance distribution for any different colony worker population. The negative-exponential function does not need such a constant to be included in, because this function is robust against large data size.

Second, we scale down the location parameter value (see Appendix C) of the log-normal function by 1/10. For doing that, we refer to the study of the desert ant Cataglyphis niger (Lahav et al. 1999). Although the same behavior is observed in the house-hunting ant Temnothorax albipennis, no equivalent study has been proposed about this species. The desert ants Cataglyphis niger recognize one another from a short distance of 10–20 mm. The average body length of this species is 55–96 mm (Knaden and Wehner 2003). The average distance within which encountering individuals can recognize one another is 1.82 to 2.08 of the average body length. The location parameter value of $0.1~(= 1 \times\frac{1}{ 10})$, which we choose, gives the mutual distance distribution with the average of $0.911 \times10 ^{-1} \ (= e ^{0.1+ \ln( 0.5 \times10^{ -1} )+ \frac{1}{2}} )$. Scaling up this average by the aforementioned constant value gives 1.82 ($= \frac{0.911 \times10 ^{-1}}{0.5 \times10 ^{-1}}$), which is the average distance within which encountering individuals can recognize one another. The location parameter value enables the trail traffic flow to be gauged on the basis of the average mutual distance among individuals recognizing one another.

John et al. (2009) report on the house-hunting ant Temnothorax albipennis that the average headway distance is 3.11 for middle population density [0.2, 0.4], and 1.66 for large population density [0.4, 0.8], respectively. The aforementioned value of 1.82 falls between the two average headway distances. The location parameter value enables the trail traffic flow to be gauged in reference to the average headway distance in the middle population density and the average headway distance in the large population density.

Appendix C: Simulation of the mutual distance distribution

We sample the mutual distances within a range [−1.00,1.00]. This range is smaller than the average distance of 1.82 within which encountering individuals can recognize one another (see Lahav et al. 1999). We mentioned this average distance in Appendix B. This range is smaller than the minimum headway distance 1.66 in the large population density (see John et al. 2009). Individuals approach one another much closer than the average headway distance 1.66, and make physical contact with one another. It is rational to have the range smaller than 1.66 and 1.82.

For introducing the mutual distance distributions, we generalize the two functions of John et al. (2009) below:

$$\begin{aligned} X \sim& a \ln N \bigl( \mu, \sigma^{2} \bigr) ^{+/-}; a f _{X} ( X; \mu, \sigma ) ^{+/-} \\ =& \left \{ \begin{array}{l} \{ X \mid P ( 0\leq x _{k} \leq1 ) \} = \int_{0} ^{1} f ( x _{k} ) ^{+} {d x _{k}}, \\ \quad \mathrm{s.t.}\ f ( x _{k} ) ^{+} = \frac{a}{ x _{k} \sqrt{2\pi \sigma^{2}}} e ^{- \frac{ ( \ln x _{k} - \mu ) ^{2}}{2 \sigma^{2}}}, \int_{0} ^{+\infty} f ( x _{k} ) ^{+} {d x _{k}} =1,\quad x _{k} \geq0,\\ \{ X \mid P ( -1\leq x _{k} <0 ) \} = \int_{-1} ^{0} f ( x _{k} ) ^{-} {d x _{k}},\\ \quad \mathrm{s.t.}\ f ( x _{k} ) ^{-} = \frac{a}{- x _{k} \sqrt{2\pi \sigma^{2}}} \ e ^{- \frac{ ( \ln - x _{k} - \mu ) ^{2}}{2 \sigma^{2}}}, \int_{-\infty} ^{0} f ( x _{k} ) ^{-} {d x _{k}} =1,\quad \mathrm{otherwise}, \end{array} \right . \end{aligned}$$

(5)

where μ is a location parameter and σ is a scale parameter; and, $a = \frac{1}{\sqrt{n}} = 0.5 \times 10 ^{-1}\ (= \frac{1}{\sqrt{400}} )$.

$$ X\sim \exp \bigl( \lambda^{\pm} \bigr);\quad f \bigl( X; \lambda^{\pm} \bigr) = \left \{ \begin{array}{l@{\quad}l} \lambda^{+} e ^{- \lambda^+ x _{k}},& x _{k} > 0,\\ - \lambda^{-} e ^{\lambda^- - x _{k}},& \mathrm{otherwise}, \end{array} \right . $$

(6)

where λ ^± (λ ⁺>0,λ ⁻<0) is a positive or a negative reciprocal mean. Equations (5) and (6) provide two mutual distance distributions.

In order to evaluate the accuracy of the model’s estimate of trail traffic flow, we consider the mean of each mutual distance distribution. We choose values for the parameters of the two different equations (5) and (6) so that the mean of the log-normal distribution given by (5) is the same as the mean of the negative-exponential distribution given by (6). We choose the following values for the parameters: μ=0.1, σ=1, and λ ^±=±10.976.

These parameter values give a set of positive and negative mean values of .

When individuals stay away from one another, the mutual distances have positive values. Otherwise, individuals approach to bite or suck one another. In the latter case, the mutual distances have negative values, as we describe in Sect. 3.2. Therefore, not only positive but also negative mean values are given in our model.

For avoiding over- or underestimation in sampling the mutual distances, we choose the positive and the negative mean value that is at least nine times larger than the upper bound (0.01) or is smaller than the lower bound (−0.01) of (3), which denotes the mutual distances of physical contacts. The two mean values are defined far beyond the bounds of (3) so that we can avoid over- or underestimation in sampling the mutual distances from the distributions that (5) and (6) provide.

We use (5) and (6) for sampling the mutual distances. Figure 5 shows the parent population of mutual distances provided by the log-normal function (Fig. 5a) or by the negative-exponential function (Fig. 5b), and the sampling distribution of mutual distances provided by the log-normal function (Fig. 5c) or by the negative-exponential function (Fig. 5d). For sampling, we prepare a synthetic data set of random values of size 30000 for mutual distances. We obtain a colony collection of size 1000. This size is sufficient to conduct statistical tests on the sampling results. We randomly choose 300 sample mutual distances per colony. From each of the 1000 colonies we choose 300 samples.

Fig. 5

The parent population of mutual distances that the log-normal function provides (a) or that the negative-exponential function provides (b), and the sampling distribution of mutual distances that the log-normal function provides (c) or that the negative-exponential function provides (d): The size of a synthetic data set of random values is 30000. The size of colony collection is 1000. The sampling distribution that the log-normal function provides better fits the mutual distances without over- or underestimation than the sampling distribution that the negative-exponential function provides

The number of 300 samples is equivalent to a population density of 0.75 (=300/400). It is close to the upper bound of the large population density, i.e., 0.8, regarding the house-hunting ant Temnothorax albipennis (see John et al. 2008, 2009). As far as there is no over- or underestimation in sampling the mutual distances in the large population density, no over- or underestimation should occur in the small or the middle population density.

The two sampling distributions do not over-fit the Gaussian distribution (its kurtosis is 3.00). The sampling distribution that the log-normal function provides has a value of kurtosis of 3.194 (see Fig. 5c). The sampling distribution that the negative-exponential function provides has a value of kurtosis of 2.866 (see Fig. 5d). The two values of kurtosis do not correspond to the kurtosis of 3.00 of the Gaussian distribution. We need to consider other points for finding the differences between the two sampling distributions.

The log-normal function better provides a bias-free sampling distribution than the negative-exponential function. A two-sided t-test reports that the sampling distribution of mutual distances that the log-normal function provides has the mean that is the same as its parent population’s mean 0.158×10⁻³ (t=0.346, $\mathrm{d.f.} = 999$, p=0.729). We call the mean average mutual distance. Meanwhile, another two-sided t-test reports that the sampling distribution of mutual distances that the negative-exponential function provides has a different mean 0.847×10⁻³ from its parent population’s mean (t=1.444, $\mathrm{d.f.} = 999$, p=0.149). We hence confirm that the sampling distribution that the log-normal function provides better fits the mutual distances without over- or underestimation.

We gauge the number of samples of mutual distances that are classified as physical contacts to the total number of samples for the two sampling distributions. The mutual distances of physical contacts are given by (3) of Sect. 3.2. We call this number degree of convergence. The degree of convergence of the sampling distribution of mutual distances that the log-normal function provides is 75.6 %. The degree of convergence 75.6 % is stable during five runs of the simulation. Meanwhile, the degree of convergence of the sampling distribution of mutual distances that the negative-exponential function provides is 81.7 %. The degree of convergence 81.7 % often drops to 80.4 % during five runs of the simulation.

Over- or underestimation occurs when an excessive amount of samples are chosen in a certain range. In sampling, many samples should not exist within the mutual distances of physical contacts. We avoid such sampling bias, as far as a smaller average mutual distance accompanies a smaller degree of convergence. The sampling distribution of mutual distances that the log-normal function provides has the average mutual distance of 0.158×10⁻³ that is at least five times smaller than the one that the negative-exponential function provides. The log-normal function provides the degree of convergence 75.6 % that is smaller by 6.1 % than the one provided by the negative-exponential function. We then select the log-normal function for our model’s simulation in this paper.

Appendix D: Estimates and observations

We refer to the three field researches of ant experts on the house-hunting ant Temnothorax albipennis. The three field researches gauge the number of workers that are involved in the tasks for moving their colonies. Once workers get involved in the tasks, they always commit to the active role (Dornhaus et al. 2008). Meanwhile, once workers are not involved in the tasks, they always wait until other workers carry them on the back and bring them to a new nest location (Dornhaus et al. 2008). Only the workers that commit to the active role form trails. This characteristic behavior ensures that trail traffic flow is equal to the percentage of workers that are involved in the tasks for trail formation in colony worker population.

We gauge the average trail traffic flow of colonies as done by Pratt (2005b) who uses the average number of workers that are involved in the tasks for trail formation in colony worker population.