Looking for the right thing at the right place: Phase transition in an agent model with heterogeneous spatial resources

We introduce a simple model describing an agent traveling within a medium containing resources randomly distributed in space. The agent, initially located at some origin, performs a single step toward a target of his choice that minimizes an economy function. We use an economy function that depends on the weight of the target and its distance from the origin. The cost and length statistics of the displacements are analyzed. We find a critical point for a particular resource weight distribution, such that the average cost vanishes (or becomes very small in practice). This problem can describe an economical agent looking for a firm with the best price or largest product choice, a tourist in search of an interesting and nearby city to visit, or an animal foraging in a complex landscape. (c) 2005 Wiley Periodicals, Inc.