Factors affecting information transfer from knowledgeable to naive individuals in groups

Abstract

There is evidence that individuals in animal groups benefit from the presence of knowledgeable group members in different ways. Experiments and computer simulations have shown that a few individuals within a group can lead others, for a precise task and at a specific moment. As a group travels, different individuals possessing a particular knowledge may act as temporary leaders, so that the group will, as a whole, follow their behaviour. In this paper, we use a model to study different factors influencing group response to temporary leadership. The model is based on four individual behaviours. Three of those, attraction, repulsion, and alignment, are shared by all individuals. The last one, attraction toward the source of a stimulus, concerns only a fraction of the group members. We explore the influence of group size, proportion of stimulated individuals, number of influential neighbours, and intensity of the attraction to the source of the stimulus, on the proportion of the group reaching this source. Special attention is given to the simulation of large group size, close to those observed in nature. Groups of 100, 400 and 900 individuals are currently simulated, and up to 8,000 in one experiment. We show that more stimulated individuals and a larger group size both induce the arrival of a larger fraction of the group. The number of influential neighbours and the intensity of the stimulus have a non-linear influence on the proportion of the group arrival, displaying first a positive relationship and then, above a given threshold, a negative one. We conclude that an intermediate level of group cohesion provides optimal transfer information from knowledgeable to naive individuals.

Appendix: mathematical description of the model

Let us consider an individual i and one of its influential neighbours j at time step t. Let $\overrightarrow {u_i }$ and $\overrightarrow {u_j }$ be their direction vectors and $\overrightarrow {u_{ij} }$ the vector from i to j. If i is a stimulated individual, we also define the vector $\overrightarrow {u_{{\text{is}}} }$ from i to the source of the stimulus. If i is a naive individual we set $\overrightarrow {u_{is} } = \overrightarrow 0$.

Let w _al be the weight associated to the alignment behaviour of i towards j. Here we chose w _al as a constant equal to w. Let w _ij be the weight associated to the attraction, or repulsion, behaviour, of i towards j. Here we chose w _ij as the following quadratic function of the distance d _ij between i and j:

$${\text{If}}\,d_{ij} <D_{{\text{min}}} \left( {{\text{repulsion}}} \right),w_{ij} = - 2w\left( {1 - \frac{{d_{ij} }}{{D_{{\text{min}}} }}} \right)^2 .$$

$${\text{If}}\,d_{{ij}} > D_{{{\text{min}}}} {\left( {{\text{attraction}}} \right)},w_{{ij}} = 2w{\left( {1 - \frac{{D_{{{\text{max}}}} - d_{{ij}} }}{{D_{{{\text{max}}}} - D_{{{\text{min}}}} }}} \right)}^{2} .$$

Note that because i and j are neighbours, we have d _ij  < D _max. In all simulations we have used w = 0,5, D _min = 15 and D _max = 60 (Fig. 1). These functions have been shown to induce the formation of homogeneous groups within a large range of parameter values (Mirabet et al. 2007).

To update the direction vector of i at time step t + 1, we compute a vector that sums up the attraction, alignment and repulsion behaviours of i towards all its influential neighbours j, and the attraction towards the stimulus:

$$\overrightarrow {v_i } = \frac{1}{N}\sum\limits_{j = 1}^N {\left( {w_{{\text{al}}} \frac{{\overrightarrow {u_j } }}{{\left\| {\overrightarrow {u_j } } \right\|}} + w_{ij} \frac{{\overrightarrow {u_{ij} } }}{{\left\| {\overrightarrow {u_{ij} } } \right\|}}} \right) + w_s \frac{{\overrightarrow {u_{{\text{is}}} } }}{{\left\| {\overrightarrow {u_{{\text{is}}} } } \right\|}}} .$$

We have used different values for the number of influential neighbours N and for the stimulus intensity w _s (see Table 1).

Finally, we compute the angle θ by which the individual i has to turn at time step t + 1, i.e. the angle between vectors $\overrightarrow {u_i }$ and $\overrightarrow {v_i }$.

The model is our own creation, written in C language and running on Linux platform.