Exploring the onset of collective motion in self-organised trails of social organisms

TL;DR

This paper uses an agent-based model to analyze how social organisms develop collective trails, identifying the minimal group size needed for cooperative motion and how this scales with environment size.

Contribution

It introduces a quantitative analysis of the transition from diffusive to directed motion in social organisms, highlighting the minimal colony size for collective behavior.

Findings

Existence of a minimal colony size for collective trail formation

Identification of a crossover point from diffusive to directed motion

Scaling behavior of this transition with environment size

Abstract

We investigate the emergence of self-organised trails between two specific target areas in collective motion of social organisms by means of an agent-based model. We present numerical evidences that an increase in the efficiency of navigation in dependence of the colony size, exists. Moreover, the shift, from the diffusive to the directed motion can be quantitatively characterised, identifying and measuring a well defined crossover point. This point corresponds to the minimal number of individuals necessary for the onset of collective cooperation. Finally, by means of a finite-size scaling analysis, we describe its scaling behavior as a function of the environment size. This last result can be of particular interest for interpreting empirical observations or for the design of artificial swarms.

Figures

Tables

Table 1. Table 1: Parameters of the model.

Parameter	Description
P	colony size
L	linear size of the system
$\gamma$	pheromone effective deposition constant
$ϵ$	pheromone evaporation rate

Full text

Keywords: self-organisation; interacting agent based models; finite-size scaling; social insects.

1 Introduction

Collective motion of social organisms is an elegant example of an emergent phenomenon that can produce efficient behaviours based on a distributed cooperative cognition [1]. Among the different structures that this phenomenon can generate [2, 3], one of the more intriguing is the self-organisation of trails, both in ants colonies [5, 6], and in human communities [4]. Ant trail formation can be generated by means of the deposition of chemical pheromones that enables indirect communication among ants through an environmental marking procedure. By means of this mechanism, ants implement a reinforcement rule that allows for selection of the shortest path to connect food source to nest location. In this way, an adaptive behaviour based only on local information and interaction is achieved.

This astonishing behaviour has motivated important technological applications. One of the most notorious is a technique for general purpose optimisation [7]. A more recent one is the experimental implementation of a navigation strategy for swarms of robots challenged to find a path between two target areas in an unknown environment [8]. The solution of this practical problem opened new questions in relation to the scalability of this approach to larger groups and larger environment size. In other words, the characterisation of the behaviour of this strategy of navigation in dependence of the group size and its scaling in dependence of the environment size became a central topic of investigation. An earlier result, related to the problem of the dependence on the community size, was obtained collecting field data from real ant colonies [9]. This study shown a general increase in the number of ants walking to the feeder along the trail in relation to colony size and a very simplified mean-field model suggested that a minimum number of ants is required for an effective trail formation. Inspired by these previous results, the purpose of our work is to study and clearly characterise the nature of these specific aspects of trail formation by means of an accurate numerical analysis of the results produced by a microscopic model which directly generates the trails.

Continuum microscopic models which are able to describe the self-organisation of ants trail formation are well known in the literature [10, 11]. These models are related to the general class of active Brownian particle models. The motion of this random particles is determined by a field which is directly influenced by the movement of the particles themselves. This non-linear feedback, which operates between the particles and the generated field at microscopic level, results in the self-organisation of the trails at the macroscopic level. The use of this formalism allows the introduction of analytical approximations for achieving general results. For example, in [13], a mean-field approximation determined the line that separates the system phase exhibiting a pure diffusion from the one where spatial structures of a general type can emerge. Note that these studies can determine theoretically the crossover line, but not the resulting patterns. Otherwise, explicit solutions and simulations of the process are obtained by means of the discretisation and numerical solution of the continuous model. Another approach consider a discretised space and time where the motion of each individual is described by some transition rules. These agent based simulations [12, 13, 14] describe the same process of the continuum models, by means of these rules, which implement the movements and the deposition of the pheromone.

The use of agent based models is very important since they allow for the understanding of the role of fluctuations and noise, as well as the limitations and validity of the continuous and the mean field descriptions. Indeed, the intrinsic stochasticity produced at the individual level generates an internal noise which, in general, can cause impacting consequences [15, 16]. Moreover, as the central aim of our work is to understand what is the minimum number of ants required for trail formation to become effective, the description of the discrete nature of individuals is essential to characterise threshold and finite size effects. These effects can not be characterised by a continuum description where every small amount of the density of population is acceptable, even if unrealistically small [17].

In the following (Sec. 2), we introduce the details of the agent based model. Even if it is inspired by previous works [11, 13], in order to implement an in deep numerical analysis of the phenomenon, and not just some specific examples of trails formation, we consider a more simplified modelling approach. In fact, the model counts on a single pheromone and its dynamics depends on only two parameters. Trail formation is obtained based only on local information and interaction. In Sec. 3 we report the numerical analysis for the characterisation of the efficiency of navigation and for the quantitative description of the shift from the diffusive to the directed motion. We would like to stress that the aim of the work is not to identify a classical phase transition, but rather we are interested in the scaling behaviour for finite-size systems. A discussion of these important points can be found at the end of the paper.

2 The model

The ant colony is composed by a population of $P$ individuals. They can move on a regular 2-dimensional square lattice with $L\times L$ sites and periodic boundary conditions. We choose $L$ odd, with the origin of the coordinate in the centre of the lattice. The nest is located in (0,0) and the food source in (0,D), with $D=(L-1)/2$. The time unit $t$ is the time interval between two updatings of the positions of all the individuals of the colony.

In the initial state each ant is located at the nest. An individual at site $(x,y)$ can move only on the top site ($x,y+1$) and on the right ($x+1,y$) or the left one ($x-1,y$). Steps towards the bottom site are forbidden. In this way, individuals effectively walk along paths where no loops are allowed. This rule, in a simplified form, takes into account the ants persistence to keep the direction of motion [11, 18], reducing the probability of moving abruptly backward before reaching the specific goal. In fact, various pheromone-following ants can correct their walks by using environmental and even magnetic cues [19].

We have also implemented a model where ants diffuse in all the four lattice directions until they find the food source. In this case, results are equivalent, just a longer transition towards the quasi-stationary state is observed.

When an ant reaches the objective site, which represents the food source, the possible directions of motion change, with all the movements allowed, except the step towards the top site. Moreover, the ant starts to deposit, in the new visited sites, an amount of pheromone $\phi=\exp[-\gamma(t-t_{act})]$. Here $t_{act}$ is the time when the ant left the food, and $\gamma$ controls the critical time for an effective deposition. This means that ants can effectively mark their trajectories only for a limited time after they left the food. When an ant reaches another time the nest, $t_{act}=t$, making it effective in depositing pheromone once more. This actualisation of $t_{act}$ is only implemented when an ant reaches the food from the nest or the nest from the food. It follows that ants which get lost are ineffective in depositing the pheromone. This function models the tendency to deposit pheromone in association with specific stimuli and conditions [11, 20], in this case the relative nearness of the nest or of the food, instead of considering ants which deposit pheromone all the time and in all the regions they visit.

At each time step all the deposited pheromone evaporates with a rate equal to $\epsilon$: $\phi_{t+1}(x,y)=(1-\epsilon)\times\phi_{t}(x,y)$.

Sites with a higher level of pheromone are more prone to be visited. In fact, ants move to site ($x,y$) with probability $(1+\phi_{t}(x,y))/(S+3)$, where $S$ is the sum of all the pheromone presents in all the neighbour sites and the other therms ensure that if the pheromone is absent there is an equal probability to move in the three possible directions.

3 Results and discussion

We run different simulations aiming at exploring the onset of the cooperative motion which allows the emergence of trails. The passage from a system where a diffusive behaviour is present, towards a system where a short path is selected, can be easily monitored measuring the efficiency of navigation between the two target areas. This is achieved counting the number of ants which realise the trajectory from the nest to the food in a time unit, normalised over the total population ($E_{f}$). The result is comparable if we measure the number of ants which realise the trajectory from the food to the nest. In fact, after passing the transient time, there is a rough symmetry in realising the two tasks. In the case where no pheromone is deposited ($E_{0}$), the efficiency depends just on the value of $L$. This efficiency value describes a pure diffusive behaviour. In contrast, if deposition is present, the efficiency is strongly dependent on all the parameters: $\gamma$, $\epsilon$, $P$ and $L$. As it can be seen in Figure 1, their dynamics are quite simple: after a fast transient the system reaches a quasi-stationary state where the value of $E_{f}$ and $E_{0}$ are maintained around a plateau. A good parameter for the description of the state of the system is $E=E_{f}/E_{0}$, which measures the gain in transportation efficiency for systems with self-organised trails. Organised states, where trail-based foraging emerged, present $E$ values clearly greater than the unit.

The dependence of $E$ as a function of $\gamma$ and $\epsilon$ has a clear behaviour. In the top of Figure 2 the behaviour of $E$ is depicted for a fixed value of $\epsilon$. Changing the value of the pheromone effective deposition, $E$ grows from small values, when the deposition is really fable (small $\gamma$), towards a maximal value for an optimal $\gamma$. Then, it returns to smaller values, when the deposition remains active also for the lost ants, increasing the noise level in the reinforced paths.

In the bottom of figure 2, we can appreciate the behaviour of $E$ for a fixed value of $\gamma$. As for the previous case, a maximum value of $E$ exists in correspondence with an intermediate value of $\epsilon$. For larger value of evaporation the system, quite obviously, looses efficiency. Diminishing the value of $\epsilon$, the same behaviour is obtained.

Note that if the evaporation is absent, $E$ grows sensibly. This is followed by an impressive growth in the value of the variance of the ensemble average. In fact, for $\epsilon=0$, if the system selects an optimal path in the first period of the simulation, high levels of efficiency are registered, otherwise low levels are reached. In this state the system is not effectively maximising its efficiency through a collective mechanism of exploration and signalisation, and efficiency strongly depends on the random configurations determined by the first paths.

Fixing the value of $\epsilon$ and $\gamma$, we turn to the most important part of our analysis. Our goal is to describe the onset of the cooperative motion that allows for the emergence of trails. This corresponds to estimate the minimal number of ants necessary for reaching the organised state. For this reason, we must characterise the behaviour of the efficiency in dependence of the colony size. Then, we analyse its scaling behaviour as a function of the environment size.

Figure 3 shows the clear increase in the efficiency of navigation between the two target areas in relation to colony size $P$. The depicted function has a typical sigmoidal shape. For small colony size a low value of $E$ is present, which corresponds to a diffusive motion. Increasing the colony size, a plateau with a high value of $E$ is reached, which corresponds to the organised state. For higher values of $P$, $E$ starts to slowly decrease. This is probably due to the fact that very large populations increase noise and generate saturation effects. In general, our results are in accordance with the field study results presented in [9], which suggests that a minimum number of ants is needed for enabling these trails. To our knowledge, this is the first time this fact is clearly reported by means of an agent-based microscopic model that describes the emergence of trails between two target areas. For the different problem of unspecific pattern formation, a qualitative example was outlined in [23]. In the same figure we show the behaviour of the variance $\chi$, which can be obtained by evaluating: $\chi=\langle E^{2}\rangle-\langle E\rangle^{2}$, where $\langle\;\;\rangle$ stands for the average over different simulations taken at the steady states. The existence of a clear peak of the variance suggests the location of a crossover point. Indeed, the identification of the position of the maximum of this quantity has been demonstrated to be a very robust approach for performing finite-size scaling analyses for classical equilibrium and for unconventional out-of-equilibrium systems [21, 22].

For estimating the minimal number of ants necessary to reach the organised state ($P_{m}$) we also use another approach [24]. We note that the behaviour of $E$ is fairly analogous to that of the order parameter of some equilibrium system, as the region close to the point $P_{m}$ can be described by the approximation: $E\propto(P-P_{m})^{\alpha}$ (see the continuous line in Figure 3). We determine $\alpha$ and $P_{m}$ by plotting $\ln{E}$ as a function of $\ln([P-P_{m}]/P_{m})$ and using the value of $P_{m}$ for which the plot is the straightest [24]. Hence, we can give a systematic estimation of $P_{m}$. The existence of a $P_{m}$ is in accordance with the experimental results of [9], which found that a minimum number of ants is needed for the emergence of trails. More general relations between recruitment strategies and colony size [25] are in agreement with these findings. Indeed, species with small colony size predominantly use solitary foraging. For increasing sizes other alternative methods are used [26], and large colonies commonly use scent trails analogous to the ones we model [27].

Now we turn our attention to the description of the scaling of these results in dependence of the environment size. By changing the value of $L$ we rescale the entire system with the selection of $D=(L-1)/2$. In Figure 4, we can see the behaviour of $E$ and its variance $\chi$, as a function of $P$, for different $L$ values. As expected, for finite sizes, $P_{m}$ is $L$ dependent: $P_{m}(L)$. As before, we estimate the different values of $P_{m}(L)$ from the shift constant of the power law description near the crossover area. Moreover, for checking the consistency and robustness of this approach, we rescale the data points using the rescaled parameter $[P-P_{m}(L)]/P_{m}(L)$. Efficiency is rescaled using the relation $E/L^{0.75}$, which was found looking at the scaling of the maxima of $E$ for different L. As shown in detail in Figure 5, it is possible to obtain a reasonable collapse of all the curves. Data roughly collapse presenting a common rate of vanishing close to $(P-P_{m}(L))^{0.5}$, strongly supporting the validity of this method. In fact, this result hardly can be considered a mere coincidence and it further legitimates the use of the power-law approximation for determining $P_{m}$. Using this procedure, the shift from the diffusive to the directed motion occurs at a critical value that depends on the system size as: $P_{m}\propto L^{3.1\pm 0.2}$. Note that the same conclusion can be obtained estimating the crossover points using the $P$ values where $\chi$ reaches the maximum. In this case the scaling has the form: $P_{m}\propto L^{3.2\pm 0.2}$. This scaling indicates that, in the thermodynamic limit, the transition point goes to infinite. Strictly speaking, this means that the system does not display a classical phase transition, which is rigorously defined at the thermodynamic limit in which the number of constituents tends to infinity. However, it is also true that every finite system presents a well defined transition point. Even if this point is not a genuine critical point, it has a clear physical meaning as it is the value of the $P$ parameter that identifies the shift from the diffusive to the directed motion, corresponding to the minimal number of individuals necessary to reach the organised state. Other models, where the transition is only observed for finite size systems, disappearing in the thermodynamic limit, and which present system size scaling, are well known in the literature [16, 28].

In general, when we transfer tools of statistical physics to problems of social or biological sciences, the population size is always considerably smaller than the Avogadro number, and so, not so large to justify the thermodynamic limit and its results. In fact, we are interested in the behaviour of finite-size systems, where important phenomena can appear in dependence of the number of individuals [16]. In particular, in our case, we expect that the relevant scaling is limited to populations ranging from a dozen of agents, as for a swarm of mobile robots, reaching less than half a billion of individuals, as can happen in supercolony of some ant species [29].

This finite-size scaling results can be interpreted considering the density of the colony population ($P/L^{2}$): it is not sufficient to maintain a constant density to obtain the trail formation. This fact highlights the non-linearity of the phenomenon and it poses important constraints for real robot implementations. Moreover, our results can be explained in term of a simple geometric scaling. In fact, the minimal value of density, which generates a trail-based foraging, scales with the linear size of the environment, which is proportional to the size of the trail. Also the scaling of $E$ with $L^{0.75}$ points out how the dynamics is far from merely depending on individual density. Remembering that the definition of $E$ is already normalised for the colony size, we can suppose that larger environments, postponing the effects of noise and saturation, allow the active interaction of larger community of individuals, which are able to maximise more efficiently the navigation problem.

The reported behaviour of the finite scaling of the system is the most important result of our work. In fact, in analogy with classical models of phase transition, we can expect that the minimal number of individuals necessary to reach the organised state can depends on the details of the model. This would not be the case for the scaling laws, which could show some universal character, not depending on model details. In this perspective, we hope that our simplified results, perhaps refined using a model implementing more specific conditions, could be successfully used for interpreting empirical observations of the scaling behaviour of ant colonies or of artificial swarms built by cooperative mobile robots.