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Electronic Supplementary Material
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Excavated substrate modulates growth instability during nest building in ants
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Etienne Toffin*, Jonathan Kindekens, and Jean-Louis Deneubourg
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Service d’Écologie Sociale, CP231, Université libre de Bruxelles, Plaine Campus,
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Boulevard du Triomphe, 1050 Bruxelles, Belgium
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*Author for correspondence (etoffin@ulb.ac.be)
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Content
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Supplemental methods
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Table S1
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Figure S1
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(a) Substrate characterization
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The main protocols that characterize the material properties are made on a
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macroscopic level as they require several hundred grams of materials to be performed.
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The mechanical forces tested by these methods (i.e. overall cohesion, resistance to
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shear stresses, quantitative composition) are not those that govern the ant behaviours
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(Bonner 2006). On the other hand, measuring the pellet size was inaccurate because
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of their irregular shape (Cassill et al. 2002).
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Hence, the mean individual pellet dry weight (table S1) was estimated by
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counting the number of loaded ants (N) exiting a nest still under excavation and
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weighing the overall extracted (m) pellets after 1 hour. Each substrate mean density
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(d) was determined by weighing the material content of a digging setup of known
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volume five times. The mean number of pellets per cm2 (Npel) and the mean volume of
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a pellet (V) were thus calculated.
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
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(b) Simulations
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All sets of Monte Carlo simulation contained, according to the substrate, the same
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number of replicates as the experimental groups (Granular: N=24; Cohesive: N=25).
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Each replicate was defined by two values, its state (initial state is N1) and its area
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(initial area: A=0). A simulation step corresponded to an increase of 1 mm2 in nest
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area and can be summarized as below.
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At each simulation step, A was increased and its state was checked to determine
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which transition could occur according to the model (figure 3). The probability ki of
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entering into each of the three alternative states i (2, 3 or Stop) was computed from
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the following equation
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ki (A) 
i
 i (A c i )
1 e
(S1)
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Thus, a random number P[0; 1[ was generated and the event that occurred was
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chosen as:
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- state Stop if 0≤P<kS, as long as current state≠NS,
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- state 2 if kS≤P<kS+k2, as long as current state=N1 and state≠NS,
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- state 3 if kS ≤P<kS+k3, as long as current state=N2 and state≠NS,
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- if P>kS+k2+k3, the replicate stays in its current state.
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A simulation ended as soon as a replicate entered the Stop state.
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Every time a state change occurred, the corresponding value of nest area was
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stored. The distribution through time (i.e. the increase of nest area) of the replicates
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according to their state could thus be generated. These results, like the mean area of
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each state change (morphological transitions or stopping of activity) were used to
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assess the agreement between experiment and simulation results.
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(c) Determination of 3 and c3 values with simulations
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The non-linear least square fitting method did not function to fit the survival curves of
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experiments in state 2 (occurrence of second transition). The values of 3 and c3 were
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thus determined by means of Monte Carlo simulations.
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For each couple of parameter values 200 sets of simulations were executed.
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For each substrate, all the simulated values of A2 and AM were grouped to get a global
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survival curve of Stage 2. The agreement of simulation and experimental results was
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assessed by computing the sum of squares between the two survival curves. The
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resulting landscape indicated that the highest possible value of survival curve non-
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linearity (3=0.95) and a threshold value equal to that of the first transition (c3=c2)
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was the best combination to generate results in good agreement with the experimental
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ones.
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REFERENCES
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Bonner, J. T. 2006 Why Size Matters: From Bacteria to Blue Whales. Princeton
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University Press.
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Cassill, D. L., Tschinkel, W. R., & Vinson, S. B. 2002 Nest complexity, group size
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and brood rearing in the fire ant, Solenopsis invicta. Insect Soc 49: 158-163.
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(doi:10.1007/s00040-002-8296-9)
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Table S1 Microscopic characterisation of digging substrates
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Figure S1 Dynamics of excavation and shape transitions in cohesive substrate
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(experiment 18). (a) Dynamics of nest excavation, showing evolution of both nest
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area A (and its fit by equation (3.1), parameters value: α=1.33, β=15.29 h,
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AM=16.94 cm2) and rate of digging against time. The two morphological transitions
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are represented by vertical dashed lines. (b) Characterisation of first transition point
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with the relationship between P and A (ω1=0.40; ω2=1.31; A1=3.80 cm2). (c)
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Determination of second transition time using scatterplot of A vs AC (γ1=1.82;
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γ2=7.75; A2=8.31 cm2).