Megachile_sideways_expMegachile willughbiella males are able to fly sideways in the direction perpendicular to the body axis. This behaviour is sometimes apparent as part of the chasing behaviour when the male is hovering with a sideways oscillation near a female.An example is given in the figure above. The oscillatory sideways movement is explained as an instability or near instability of the sideways control system. The model, with sideways velocity control included, has been applied to this case. The result is given in the figure below. Megachile_sideways_expmod

The parameters have been optimized for a reasonable fit. The model does indeed show oscillatory behaviour with the right frequency. This is more clearly visible in a plot of the vertical y-coördinate as a function of time.Megachile_sideways_expmod_oscil

Background information :

Matlab .m file :  Megachile_Boed_080615_1245NT.m with parameters : Kr=10, KrB=10, KdB=0, Kt=100, τv = 0.04 s,  τ = 0.07 s, τf = 0.04 s The reference distance rref is chosen 3.4

Comparison with 1D theory : The oscillation of the male is approximately along a straight line perpendicular to the body axis, with a constant distance to the female. Therefore the 1D model amy deliver useful results. The 1D theory will be applied with female velocity zero.

From experiment : Oscillatory part of the experiment from t=0 until t=1.5 s.  Oscillation 3 periods in 1.05 second. Period : 1.05/3=350 ms. Frequency 2*π/0.350= 18 rad/s.

The oscillation is approximately perpendicular to the body axis of the male. The 1D theory may be applied with female velocity zero. From the 1D theory given τf=0.04 :

τ=1/( τf2)=0.077.  (Model τ=0.07)

Kt=3.4*(1/ τf+1/ τ) = 129. (Model Kt=100)

Conclusion : The 1D theory is a useful approximation in this case.

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