A method of modelling time dependent data: swimming in guppies (Poecilia Reticulata) under threat of a predator

The use of statistics that assume independence between observations may not be an adequate tool in the analysis of how prey fish co-ordinate their swimming when inspecting piscine predators. This is mainly caused by the fact that continuity is an inevitable feature of locomotion. Thus, if one takes observations at close intervals, the assumption of independence will probably be violated, while observations taken too far apart may lose significant information of what goes on in between. In this paper we present one way of dealing with these problems and demonstrate what may come out when applied to empirical data of predator inspection in fish. The method models measurements taken at equispaced points in time as dynamic systems of simultaneous stochastic difference equations, and may well be applied in general to auto- and cross-correlated data. A full account of statistical procedures and how to estimate the order of difference presented in an appendix. Swimming trajectories, represented as successive distances to predators at intervals of 2 s, were obtained from paired male guppies inspecting piscine predators. Dynamic models of swimming were then estimated and used for quantifying causes of locomotion that relates to inertia of own swimming, to motion of co-inspectors, and to independent causes generated elsewhere. The derived models complied with second order systems in 9 out of the 10 fish that were investigated. Feed-back relations within co-inspectors varied between pairs, but none of the fish moved independently of its partner. All models may be interpreted in terms of interacting attracting/repelling 'forces' between each prey fish and the predator. The way swimming by a particular fish affected co-inspectors was further analysed. Mean properties of this influence revealed that even when partners lagged behind inspecting individuals, they mostly behaved so as to make the leading fish approach the predators even further. This finding suggests other behavioural strategies to be involved than those assumed previously, and one alternative is discussed.