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Many flying insects tend to have very jagged trajectories. For moves of a fruit fly looks like a random walk.

Is there any research on the properties of trajectories (e.g. their fractal dimension or mathematical models producing similar behaviour)?

Besides the path of a single insect, I am also interested in the interaction of a small number of insects (e.g. 2 or 3).

I am aware of the research in flocking and swarming (e.g. Tamas Vicsek, Anna Zafiris, Collective motion 2010) and of techniques of obtaining such trajectories (e.g. Danping Zou, Yan Qiu Chen, Acquiring 3D motion trajectories of large numbers of swarming animals or an open access pdf, 2009). However, I haven't found within them an answer to my particular question.

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I'm curious what is your purpose in consideration of insect movements in isolation of their function (avoiding predators, looking for prey, finding new habitat patch etc.) –  Marta Cz-C Jan 23 '12 at 19:28
    
@MartaCz-C: Abstraction sometimes helps (I am a physicists). And properties of the trajectory are very likely in a close relationship both to optimizing certain actions (avoiding predators, looking for prey, finding new habitat patch etc.) while being bounded by constrains (physiological, energy, ...). –  Piotr Migdal Jan 24 '12 at 18:17

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Quick search - Some articles that may interest you:

1) Random walk model of insect movements
Kareiva P. M., Shigesada N. (1983). Analyzing insect movement as a correlated random walk. Oecologia 56(2-3) 234-238

2) Artificial life model of flying insects and its comparison to real insects navigation strategies.
Dale K., Collett T. C. (2001). Using artificial evolution and selection to model insect navigation. Current Biology 11(17) 1305–1316

3) The last one is not about insects, but it's a trial to practical usage of the mathematical analisys of invertebrate movements.
Shimizu N. et al. (2002). Fractal analysis of Daphnia motion for acute toxicity bioassay. Environmental Toxicology 17(5) 441–448

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Dickinson (2005) has a good review of insect flight, including behavior, biomechanics, electrophysiology, and neural control with links to more of the primary literature. What follows is a general summary thereof.

The jagged trajectories you mention are called saccades in the insect flight literature. In Drosophila, saccades are ~90° turns accomplished in approximately 50 ms. Because flies can't move their eyes independently of their heads, saccades are though to allow a relatively constant visual field most of the time, interspersed with rapid reorientations during which the fly is "blind."

The actual turn begins with the generation of a torque, which starts the fly turning. Shortly thereafter, the halteres sense this change and a counter-torque is produces, which stops the turn.

Rather than random turns, Drosophila are turning away from visual expansion as they near objects. So as they move closer to an object, it expands in their visual field, and they turn away from it.

Schematic of fruit fly turning

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Thanks, a good and informative answer. However, I am interested not only in saccades but the whole patterns of movement (so also structures at different time scales, e.g. minutes). –  Piotr Migdal Jan 23 '12 at 13:21

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