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Supplementary Notes for “Social complexity, diet, and brain evolution:
modeling the effects of colony size, worker size, brain size, and foraging
behavior on colony fitness in ants”
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Behavioral Ecology and Sociobiology.
Ofer Feinerman1 James F. A. Traniello2
1. Weizmann Institute of Science, Department of Physics of Complex Systems, Rehovot,
Israel, 7610001. Corresponding author, email: ofer.feinerman@weizmann.ac.il
2. Boston University, Department of Biology, 5 Cummington Mall, Boston, MA USA 02215.
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Supplementary note 1: Optimal colony structure
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Since τ is positive, equation (iii) demands that Nopt>F. Therefore, in an optimal colony,
the number of workers must be large enough to deplete all available food during the
available activity period.
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The mechanisms by which larger brains can reduce search time are all given by the single
function τ(V). Next, we make some reasonable assumptions about the structure of τ(V)
and examine the consequences on optimal colony characteristics. We assume that
individual worker search time, τ(V), is a smooth monotonically decreasing function
tending toward very long search times if brain size is very small (V →0) and to some
minimal search time value, τ∞ (constrained by factors other than cognitive abilities), for
very large brain sizes (V →∞).
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As noted above, optimal colony characteristics must lie on the line on the (N,V) plane
described by equation iii (Figure 2A). Fixing the value of F, and applying our
assumptions regarding τ(V), we find that this line has two asymptotes: when V goes to
infinity (vertical asymptote) then Nopt tends to F+ τ∞/Tactive, which for F >> τ∞ can be
approximated as Nopt≈F. The horizontal asymptote is derived for large values of N such
that, to satisfy equation (iii), τ(Vopt) must be large as well and this implies that brain
volume goes to 0 (horizontal asymptote). For the resource dominated branch, the
gradient (illustrated in Figure 2A) at any point, 𝛻enet,1 = (-F/N2, -cbrain ), always points in
the negative x and negative y directions. Therefore, a maximum of enet (Figure 2, white
diamond) must occur at the transition area between these two asymptotes. The location
of the transition can be approximated by the vertical asymptote Nopt ≈> F. We thus
expect that the optimal number of workers in a colony will exceed the number required to
collect all available food and will be roughly linearly related to it.
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The precise location of (Nopt,Vopt) on the transition line between the two regimes
(equation (iii) and Figure 2A) depends on the exact details of τ(V). Moving to the right
along this line implies a decrease in V and an increase in N. The formula for 𝛻enet,1 shows
that decreasing V by one unit has a positive effect on net energy intake (cbrain) whereas
increasing N by one worker has a negative effect (-F/N2). Since Nopt is proportional to F,
the cost incurred by the addition of a worker to the colony, F/Nopt2, can be approximated
by 1/F and becomes increasingly smaller with larger values of F. Therefore, for large
values of F (F >> τ∞), one can expect an energy-saving reduction in brain size at the
expense of an energetically inexpensive increase in the number of workers (which
becomes lower at larger values of F). When sufficient food is available, we therefore
expect optimal brain size to decrease with increasing food abundance.
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Optimal colony structure further depends on the parameter cbrain. Most of the results in
this work are presented for the case of cbrain=1, this value was chosen such that the daily
energy expenditures per ant cbrain∙V is (for values of V that are on the order of 1)on the
scale of the amount of energy that the ant can collect daily. Varying cbrain around this
value induces only small quantitative changes in (Nopt,Vopt) (Supplementary figure 1). On
the other hand, if we choose cbrain>>1 then the point depicting the optimal colony will
move on the horizontal asimptote (see Supplemetary figure 1) such that brain volumes
tend to zero. On the other hand, cbrain<<1, will yield an optimal colony on the
perpendicular asymptote such that the number of ants in the colony is minimized to
include about F ants all with very large brains.
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Supplementary Fig. 1. Dependencies of optimal colony structure and brain costs.
This figure is identical to the one presented in figure 2A of the main text. The only
difference is that optimal colonies (diamonds) are presented for different values of
cbrain (over a range of 100 fold). Note that all these maxima reside on the full line
depicting the boundary between resource and collection time-limited regimes.
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Supplementary note 2: Worker polymorphism
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Consider a colony with N workers, i = 1…N, each with a brain volume of Vi that incurs
an energetic cost of cbrain∙ Vi and corresponds to a mean search time specified by τ(Vi). At
the colony level, the total energetic cost is cbrain ∙ ∑Vi = cbrain ∙ Vtot, and collective search
time is obtained, as before, by taking the minimum over N exponentially distributed
random functions. This yields another exponentially distributed variable with a mean of
τcolony (V1, V2,…, VN) = 1/ ∑ [1/ τ(Vi)].
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We compare the net energy intakes of monomorphic and polymorphic colonies with an
equal number of ants N and the same total brain volume Vtot. For a monomorphic colony
the brain volume for each worker is Vtot/N, whereas in a polymorphic colony Vtot remains
the same but it is now unevenly distributed among workers according to their body size.
Net energy intake is e2net = Tactive- 1/ ∑ 1/ τ(Vi)-cbrainVtot/N for a polymorphic colony and
this reduces to e2net = Tactive- τ (V tot/N) –cbrainVtot/N for a monomorphic colony. These two
expressions differ only in the second additive term which should be minimized (as it
appears with a negative sign). This problem can be translated into finding the colony
structure that maximizes the rate of food finding defined as r(V1,V2,…,VN)=∑ 1/ τ(Vi)
under the constraint ∑Vi =Vtot.
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We again consider functions of the form τ(V)=1/Vα+ τ∞ and, for simplicity, use τ∞=0.
The food finding rate, r(V1,V2,…,VN) is symmetric in all its arguments (i.e. Vi and Vj can
be interchanged for every i and j) and for α<1, it is also concave. The maximum of a
function with these characteristics is obtained when all its arguments are equal: Vi=Vtot/N
for each i. This implies that, for α<1, a monomorphic colony in which the total brain
volume is evenly distributed among workers is energetically favorable. The case is
different when α>1 and r convex; here, the maximum of r lies on the corners of its
defining set. Namely, search time will be shortest if all the brain volume is held by a
single worker. If we assume that other physical constraints set a maximal brain volume
for a single ant Vi≤Vmax then the structure of an optimal colony will include n=Vtot/Vmax
workers that have a brain volume of Vmax and N-n ants have near zero brain volume. This
implies a differentiation into two worker subcastes: one consisting of n scouts with larger
brains and higher cognitive capacity to efficiently search for food, and a second subcaste
of N-n workers that are specialized on food retrieval.
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