11 research outputs found
Ants in a Labyrinth: A Statistical Mechanics Approach to the Division of Labour
Division of labour (DoL) is a fundamental organisational principle in human
societies, within virtual and robotic swarms and at all levels of biological
organisation. DoL reaches a pinnacle in the insect societies where the most
widely used model is based on variation in response thresholds among
individuals, and the assumption that individuals and stimuli are well-mixed.
Here, we present a spatially explicit model of DoL. Our model is inspired by
Pierre de Gennes' 'Ant in a Labyrinth' which laid the foundations
of an entire new field in statistical mechanics. We demonstrate the emergence,
even in a simplified one-dimensional model, of a spatial patterning of
individuals and a right-skewed activity distribution, both of which are
characteristics of division of labour in animal societies. We then show using a
two-dimensional model that the work done by an individual within an activity
bout is a sigmoidal function of its response threshold. Furthermore, there is an
inverse relationship between the overall stimulus level and the skewness of the
activity distribution. Therefore, the difference in the amount of work done by
two individuals with different thresholds increases as the overall stimulus
level decreases. Indeed, spatial fluctuations of task stimuli are minimised at
these low stimulus levels. Hence, the more unequally labour is divided amongst
individuals, the greater the ability of the colony to maintain homeostasis.
Finally, we show that the non-random spatial distribution of individuals within
biological and social systems could be caused by indirect (stigmergic)
interactions, rather than direct agent-to-agent interactions. Our model links
the principle of DoL with principles in the statistical mechanics and provides
testable hypotheses for future experiments
The distance separating neighbouring ants depends upon the difference in their response-thresholds.
<p>a) Map of ant locations. Symbol sizes are proportional to the threshold
of the ant, so sensitive ants have small symbols. b) Mean distance
between an active ant and its nearest neighbour (NND), as a function of
the <i>difference</i> in sensitivity between the two
(<i>θ<sub>i</sub></i> -
<i>θ</i><sub>NN</sub>). Error bars are one standard
deviation. Drive, <i>δ</i>S = 10, N
simulations = 100, N time-steps per
simulation = 5×10<sup>4</sup>. The
horizontal line shows the expected NND under conditions of complete
spatial randomness (Expected NND = 2.56,
<i>σ</i> = 1.25, N
simulations = 2000).</p
Definition of parameters and response statistics for the two-dimensional model.
<p>Definition of parameters and response statistics for the
two-dimensional model.</p
The skewed activity distribution.
<p>Individual ant activity is measured on a per-ant basis, as the work done
per time-step. Main panel; an activity-rank plot. A rank of
1<sup>st</sup> indicates the ant was the most active, and a rank of
20<sup>th</sup> indicates the ant was the least active. Panel
insert: the same data as the main panel plotted as the survivorship of
the individual ant activity. The distribution is exponential-like. Model
parameters as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018416#pone-0018416-g002" target="_blank">Figure
2</a> legend. All realisations were run for 50000 time-steps
after reaching the steady-state.</p
Emergence of one-dimensional spatial division of labour.
<p>a) Black line: The development of the steady state (world
circumference = 500, N
ants = 20, stimulus
drive = 0.1 stimulus units per time-step). Red
line: The total work done per time-step. b) The positions of the ants in
the ring nest as a function of time. The ants measure their position
clockwise from a fixed but arbitrarily chosen point along the ring.
There is a transition from a random initial configuration, to one in
which ants are aggregated into a few clusters, with low threshold ants
shuttling between the clusters. The clusters are represented by the
straight lines.</p
The stimulus landscape percolates at a critical response-threshold.
<p>a) A stimulus landscape as it appears to the outside observer
(<i>δS</i> = 1,
, N
ants/<i>l</i><sup>2</sup> = 0.04).
The more stimulus a site contains, the darker the grey. b)
Threshold-dependent site-occupancy for the same landscape as seen by
a sensitive ant
(<i>θ<sub>i</sub></i> = 1,
<i>p</i> = 0.759). Sites with
S<<i>θ<sub>i</sub></i> are white. The
largest cluster on the lattice is coloured in red. The cluster
‘percolates’ across the lattice. c) Threshold-dependent
site-occupancy for an ant with
<i>θ<sub>i</sub></i> = 1.55,
here <i>p</i> = 0.594. The occupancy
is just above the critical occupancy
(<i>p<sub>c</sub> = </i>0.5927…),
where the mean cluster area displays a phase-transition. d)
Threshold-dependent site-occupancy for a less sensitive ant, where
and
<i>p</i> = 0.32. To this ant most
sites do not contain stimulus, clusters of occupied sites do not
span the lattice, and hence the landscape does not percolate.</p
Stimulus input and ant activity update rules for the two-dimensional model.
<p>Stimulus input and ant activity update rules for the
two-dimensional model.</p
The stages involved in an 'ant bout'.
<p>The position of the ant is indicated by the red square. Each
time-step every ant checks its local neighbourhood (the four blue
squares) for any stimulus that exceeds its individual response
threshold (S><i>θ<sub>i</sub></i>). Here, at
<i>t</i><sub>1</sub> some stimulus arrives in the
ant's West square, such that
S<sub>W</sub>><i>θ<sub>i</sub></i>, so the
ant moves onto it, instantaneously reducing the stimulus at that
site to its threshold level, <i>θ<sub>i</sub></i>. If
more than one neighbouring site has
S><i>θ<sub>i</sub></i>, the ant chooses
randomly between them (at <i>t<sub>n</sub></i>). At
<i>t<sub>n+1</sub></i> the ant has exhausted
the stimulus in its four adjacent squares, so it is trapped.</p
The scale-free structure of the stimulus landscape.
<p>Both panels depict the survivorship (the complement of the cumulative
distribution) function for: a) mean site occupancy,
<<i>p</i>> and b) The mean cluster size, ,
normalised by the maximum cluster possible,
<i>l</i>×<i>l</i>. Both
<<i>p</i>> and are ensemble-averages,
calculated by averaging across all individuals irrespective of
threshold. The different curves represent different fixed drives (â—‹;
δS = 1×10<sup>−2</sup>, ⧫;
δS = 1×10<sup>−1</sup>, •;
δS = 1×10<sup>0</sup>, ▪;
δS = 1×10<sup>1</sup>).</p
The mean stimulus per site as a function of the stimulus drive.
<p>The different symbol types represent different colony threshold
distributions (•; Uniform CTD, minimum = 0,
maximum = 10, ⧫; Gaussian CTD,
S.D. = 1.5, ⧫; Gaussian,
S.D. = 1.0, ⧫; Gaussian,
S.D. = 0.5, ○; Homogeneous CTD (all ants are
identical), θ = 5,
<i>l</i>×<i>l</i> = 60×60,
ant density = 0.04, N simulations per parameter
combination = 60). The error bars are standard
deviations. The thin dashed line has a slope of one. Insert: The
fluctuation amplitude (CV = S.D./) for the
stimulus held across all sites on the lattice as a function of the
drive. The horizontal line indicates the null expectation, that is, when
the amount of stimuli held in a site is Poisson distributed.</p