  We provide a short review of existing models with multiple taxis performed by
(at least) one species and consider a new mathematical model for tumor invasion
featuring two mutually exclusive cell phenotypes (migrating and proliferating).
The migrating cells perform nonlinear diffusion and two types of taxis in
response to non-diffusing cues: away from proliferating cells and up the
gradient of surrounding tissue. Transitions between the two cell subpopulations
are influenced by subcellular (receptor binding) dynamics, thus conferring the
setting a multiscale character.
  We prove global existence of weak solutions to a simplified model version and
perform numerical simulations for the full setting under several phenotype
switching and motility scenarios. We also compare (via simulations) this model
with the corresponding haptotaxis-chemotaxis one featuring indirect
chemorepellent production and provide a discussion about possible model
extensions and mathematical challenges.
