A penetrating analysis of the wave dynamic modes of a conceptual population system described by the 'reaction-taxis-diffusion' and 'reaction-autotaxis-cross-diffusion' polynomial models is carried out for the case of increasing degrees of the reaction and taxis (autotaxis) functions. It is shown that a 'suitable' nonlinear taxis can affect the wave front sets and generate nonmonotone waves, such as trains and pulses which represent the exact solutions of the model system. Parametric critical points whose neighborhood displays the full spectrum of possible model wave regimes are identified and a wave mode systematization in the form of bifurcation diagrams is give. This enables standard criteria of approach to 'dangerous boundaries' to be developed. As possible applications, 'pulsing density patches' in forest insect populations as well as plankton communities and some other examples are discussed.
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