Two-category model of task allocation with application to ant societies

In many network models of interacting units such as cells or insects, the c oupling coefficients between units are independent of the state of the unit s. Here we analyze the temporal behavior of units that can switch between t wo 'category' states according to rules that involve category-dependent cou pling coefficients. The behaviors of the category populations resulting fro m the asynchronous random updating of units are first classified according to the signs of the coupling coefficients using numerical simulations. They range from isolated fixed points to lines of fixed points and stochastic a ttractors. These behaviors are then explained analytically using iterated f unction systems and birth-death jump processes. The main inspiration for ou r work comes from studies of non-hierarchical task allocation in, e.g., har vester ant colonies where temporal fluctuations in the numbers of ants enga ged in various tasks occur as circumstances require and depend on interacti ons between ants. We identify interaction types that produce quick recovery from perturbations to an asymptotic behavior whose characteristics are fun ction of the coupling coefficients between ants as well as between ants and their environment. We also compute analytically the probability density of the population numbers, and show that perturbations in our model decay twi ce as fast as in a model with random switching dynamics. A subset of the in teraction types between ants yields intrinsic stochastic asymptotic behavio rs which could account for some of the experimentally observed fluctuations . Such noisy trajectories are shown to be random walks with state-dependent biases in the 'category population' phase space. With an external stimulus , the parameters of the category-switching rules become time-dependent. Dep ending on the growth rate of the stimulus in comparison to its population-d ependent decay rate, the dynamics may qualitatively differ from the case wi thout stimulus. Our simple two-category model provides a framework for unde rstanding the rich variety of behaviors in network dynamics with state-depe ndent coupling coefficients, and especially in task allocation processes wi th many tasks. (C) 2001 Society for Mathematical Biology.