A BIFURCATION ANALYSIS OF A 3D BLOWFLY MODEL IN DISCRETE AND CONTINUOUS TIME
In this paper, we extend the two-dimensional discrete time model of the virtual fly, developed by Boddeker and Egelhaaf [2003], into three space dimensions, and introduce its continuous time analog. Like real blowflies, the virtual counterparts exhibit a dichotomous chasing behavior: depending on size, velocity and the course of the targets; they catch the targets or pursue them at constant distance. Here, we analyze this behavioral characteristic with respect to the course of the target, in particular, we choose trajectories, spiraling upwards. After setting up the three-dimensional model, we transform it into the local coordinates of the pursuer, using equivariance properties. Then bifurcation tools apply, and it turns out that depending on the gradient and the curvature of the spiraling trajectory, a fixed point in the transformed system can lose or gain stability. A stable fixed point corresponds in the original system to a trajectory, on which the virtual fly follows the target at a constant distance. In this way, we explain the dichotomous behavior through the occurrence of bifurcations.
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