How do ensembles occupy space?

Abstract

To find an answer to the title question, an attractiveness function between agents and locations is introduced yielding a phenomenological but generic model for the search for optimal distributions of agents over space. Agents can be seen as, e.g., members of biological populations like colonies of bacteria, swarms, and so on. The global attractiveness between agents and locations is maximized causing (self-propelled) `motion' of agents and, eventually, distinct distributions of agents over space. At the same token spontaneous changes or `decisions' are realized via competitions between agents as well as between locations. Hence, the model's solutions can be considered a sequence of decisions of agents during their search for a proper location. Depending on initial conditions both optimal as well as suboptimal configurations can be reached. For the latter early decision-making are important for avoiding possible conflicts: if the proper moment is missed, then only a few agents can find an optimal solution. Indeed, there is a delicate interplay between the values of the attractiveness function and the constraints as can be expressed by distinct terms of a potential function containing different Lagrange parameters. The model should be viewed as a top-down approach as it describes the dynamics of order parameters, i.e. macroscopic variables that reflect affiliations between agents and locations. The dynamics, however, is modified via so-called cost functions that are interpreted in terms of affinity levels. This interpretation can be seen as an original step towards an understanding of the dynamics at the underlying microscopic level. When focusing on the agent, one may say that the dynamics of an order parameter shows the evolution of an agent's intrinsic `map' for solving the problem of space occupation. Importantly, the dynamics does not necessarily distinguish between evolving (or moving) agents and evolving (or moving) locations though agents are more likely to be actors than the locations. Put differently, an order parameter describes an internal map which is linked to the expectation of an agent to find a certain location. Owing to the dynamical representation, we can therefore follow up the change of these maps over time leading from uncertainty to certainty.