  The response threshold model explains the emergence of division of labor
(i.e., task specialization) in an unstructured population by assuming that the
individuals have different propensities to work on different tasks. The
incentive to attend to a particular task increases when the task is left
unattended and decreases when individuals work on it. Here we derive mean-field
equations for the stimulus dynamics and show that they exhibit complex
attractors through period-doubling bifurcation cascades when the noise
disrupting the thresholds is small. In addition, we show how the fixed
threshold can be set to ensure specialization in both the transient and
equilibrium regimes of the stimulus dynamics. However, a complete explanation
of the emergence of division of labor requires that we address the question of
where the threshold variation comes from, starting from a homogeneous
population. We then study a structured population scenario, where the
population is divided into a large number of independent groups of equal size,
and the fitness of a group is proportional to the weighted mean work performed
on the tasks during a fixed period of time. Using a winner-take-all strategy to
model group competition and assuming an initial homogeneous metapopulation, we
find that a substantial fraction of workers specialize in each task, without
the need to penalize task switching.
