*The first simple one dimensional* (1D) model

To obtain further insight in the male *Megachile* chasing behaviour a model will be developed. Models of insect chasing and pursuit behaviour are already available in the literature. Here a model for blowflies (Boeddeker and Egelhaaf, 2005) has been adapted for use with *Megachile*. The most striking difference between the blowfly and *Megachile* is the ability of male *Megachile* to fly backwards. The first step in the development of the model will be a model for pursuit along a straight line. The model will be directly applicable to the part of the male path just before contact with a stationary female.(Chasing with rebound) The geometry of the model is given here.

In the model for the blowfly the distance r determines the speed sp. sp is filtered by a low pass filter. This low pass filter is modelling the time necessary to compute the distance i.e. by determining the size of the female as seen by the male. The output sp_{filt} of this low pass filter is input to a low pass locomotion filter to obtain the translation velocity v_{M}. The position x_{M} will be obtained by integrating the velocity v_{M}. For Megachile the speed sp is determined by a simple proportional control expression :

sp = Kr.(r-rref), (r_{ref }is a reference value and r>0)

The complete model is a proportional control system. The model is given in block-schematic form in the figure below.

The model is a third order proportional control system. This control system will not be stable when the gain K_{r} is large enough. This may explain the oscillation of the male about a mean position. A chase will be explained by assuming that the male is suddenly diminishing its reference value. When a male collides with the female in most cases the female will not fly away. In these cases the male will touch the female for some time and fly back a few cm before touching the female again. This rebounding behaviour is included in the model by including a collision detector. When a collision is detected the velocity v_{M} is reset. With a positive reference value for the distance the male will rebound, as does the real insect. **An example will be found here.**

The results of a simulation using Matlab/Simulink are given here.

(K_{r }= 40 τ_{v }= 0.04 and τ = 0.15 s)

The sudden change of r_{ref} should be triggered in some way. A trigger may be generated in the model by the occurrence of a distance r larger than some preset value i.e. r_{ref}+1. The complete one dimensional model is given below.