How animals divide space can have fundamental implications for the population dynamics of territorial species. It has recently been proposed that space can be divided if animals tend to avoid fight locations, rather than the winner of fights gaining access to exclusive resources, behaviour that generates exclusive territories in two-dimensional space. A game-theory model has shown that this avoidance behaviour can be adaptive, but the adaptiveness has not been investigated in a spatially realistic context. We present a model that investigates potential strategies for the acquisition of territories when two-dimensional space must be divided between individuals. We examine whether exclusive territories form when animals avo id all encounters with others, or only those encounters that have led to losing fights, under different fighting costs and population densities. Our model suggests that when fighting costs are high, and the population density is low, the most adaptive behaviour is to avoid fight locations, which generates well-defined, exclusive territories in a population that is able to resist invasion by more aggressive strategies. Low fighting costs and high population densities lead to the break-down of territoriality and the formation of large, overlapping home ranges. We also provide a novel reason as to why so-called paradoxical strategies do not exist in nature: if we define a paradoxical strategy as an exact mirror-image of a common-sense one, it must respond in the opposite way to a draw as well as to wins and losses. When this is the case, and draws are common (fight outcomes are often not clear-cut in nature), the common-sense strategy is more often adaptive than a paradoxical alternative. © Springer-Verlag 2004.