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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 invasionfeaturing two mutually exclusive cell phenotypes (migrating and proliferating).The migrating cells perform nonlinear diffusion and two types of taxis inresponse to non-diffusing cues: away from proliferating cells and up thegradient of surrounding tissue. Transitions between the two cell subpopulationsare influenced by subcellular (receptor binding) dynamics, thus conferring thesetting a multiscale character.  We prove global existence of weak solutions to a simplified model version andperform numerical simulations for the full setting under several phenotypeswitching and motility scenarios. We also compare (via simulations) this modelwith the corresponding haptotaxis-chemotaxis one featuring indirectchemorepellent production and provide a discussion about possible modelextensions and mathematical challenges.

Modeling multiple taxis: tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence