Bifurcations of travelling waves in population taxis models

A systematic analysis of the wave dynamical modes of a conceptual population system described by the 'growth-taxis-diffusion' and 'growth-autotaxis-crossdiffusion' models is carried out for the case of the increasing powers of the growth and taxis (autotaxis) polynomial functions. It is shown that a 'suitable' nonlinear taxis can generate non-monotonous waves, such as trains and impulses, which represent the exact solutions of the model system. parametric 'critical' points whose neighborhood displays the full spectrum of possible wave modes are identified and a mode systematization in the form of bifurcation diagrams is given. This enables standard criteria to be developed, by which the proximity of 'dangerous boundaries' may be judged. As possible applications, 'pulsing density spots' in forest insect populations as well as plankton communities and some other examples are discussed.