Geometry for mutualistic and selfish herds: the limited domain of danger.

We present a two-dimensional individual-based model of aggregation behaviour in animals by introducing the concept of a "limited domain of danger", which represents either a limited detection range or a limited attack range of predators. The limited domain of danger provides a suitable framework for the analysis of individual movement rules under real-life conditions because it takes into account the predator's prey detection and capture abilities. For the first time, a single geometrical construct can be used to analyse the predation risk of both peripheral and central individuals in a group. Furthermore, our model provides a conceptual framework that can be equally applied to aggregation behaviour and refuge use and thus presents a conceptual advance on current theory that treats these antipredator behaviours separately. An analysis of individual movement rules using limited domains of danger showed that the time minimization strategy outcompetes the nearest neighbour strategy proposed by Hamilton's (J. Theor. Biol. 31 (1971) 295) selfish herd model, whereas a random strategy confers no benefit and can even be disadvantageous. The superior performance of the time minimization strategy highlights the importance of taking biological constraints, such as an animal's orientation relative to its neighbours, into account when searching for efficient movement rules underlying the aggregation process.