Modeling of swarm formation as a consequence of autotaxis

The simple model of school and swarm formation is proposed within the frameworks of Eulerian space models (reaction-diffusion-advection system). Assuming that the schooling and the processes of birth-and-death act on different time scales, we have excluded the local kinetics of species (the reaction term) from the model. The spatial dynamics of animals is circumscribed by scalar field of density and vector field of velocity. The basis of animal aggregation in space is the ability of animals to move in certain direction, i.e. taxis. As an example of swarming strategy the behavior of midges is taken: we presume that individuals accelerate towards higher swarm density but change direction when the density exceeded some maximum. In other words, acceleration of movement is assumed to be proportional (with density-dependent coefficient of proportionality) to the gradient of species density. This statement poses the equation for species velocity. Thus, our model adds the differential equation for velocity of autotaxis to the standard advection-diffusion model. The linear analysis of 1D problem with zero-flux boundary conditions has showed that homogeneous nonzero equilibrium looses its stability when the movement rate of animals (coefficient of proportionality in velocity equation) overpasses some bifurcation value. The numerical experiments have confirmed analytical results, displaying stationary spatially heterogeneous solution (standing waves) for the detected supercritical value of the movement rate.