Modelling the hovering and egg shooting of Bombylius

 A female Bombylius major is filmed hovering on sandy soil near a shoe. At regular intervals the female is decreasing the distance to the rim of the shoe in an oscillatory way until the abdomen is curled in the direction of the shoe. Probably an egg will be deposited near the shoe. The distance as a function of time is given in the figure below.

Bombylius_shoe_distance_new

Figure 1 : Distance as a function of time.

Apart from oscillations and noise the distance at first diminishes and after egg ejection is larger again. The oscillatory behaviour before egg shooting is clearly recognized. This is suggesting a model with control of distance and a value of the gain larger than the critical value for oscillation. The model is given in Fig 2.

Bombyliidae_model

Figure 2 : The model

In the model the physical distance r between female and hole is filtered by a first order filter as a model of the neural calculation of the distance. The estimated distance is restim . This estimated distance is compared with a reference rref, the result is input of a controller. The controller is a proportional controller with gain Kr. The result is input to a filter representing the mechanical inertia. The result is the physical velocity vF. The velocity is integrated to obtain the position xF. The total control system consists of two first order systems and an integrator in series, therefore the system will oscillate when the gain Kr is large enough. It is assumed that gain and reference distance are determined by the availability of an egg in the sand chamber of the abdomen. Therefore the parameter "egg availability" has been introduced with a value between 0 and 1. To simulate the diminishing of the distance just before an egg shot the reference rref will be assumed to depend on egg availability in the way given in Fig. 3

Bombylius_reference

Figure 3

The gain Kr of the controller will be assumed to be proportional to the egg availability.

An egg is shot when the egg availability is >0.95 and the estimated distance restim is larger than a threshold.  Also a feedforward pulse is added to model the fast backward jump after shooting the egg.

Result of the model for the experimental results of Fig. 1 are given in Fig 4.

Bombylius_exp_model

Figure 4 : Comparison between experiment and theory.

The model with the parameters is given in Bombylius_parameters The fit of the model with the experimental results is not exact. The oscillations are modelled in the right way. The overshoot after the egg shot is ocuuring in the experiment but not always for the same egg shot.

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