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.

We propose a novel approach to modeling cell migration in an anisotropic environment with biochemical heterogeneity and interspecies interactions, using as a paradigm glioma invasion in b...

A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $$L^2$$L 2 -theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $$\ell ^\infty (L^2)$$ℓ ∞ ( L 2 ) - and $$\ell ^2(H^1_0)$$ℓ 2 ( H 0 1 ) -norms. For the time increment $$\Delta t$$Δ t , the mesh size h and a conforming finite element space of polynomial degree $$k \in {\mathbb {N}}$$k ∈ N , the convergence order is of $$O(\Delta t^2 + h^k)$$O ( Δ t 2 + h k ) in the $$\ell ^\infty (L^2)\cap \ell ^2(H^1_0)$$ℓ ∞ ( L 2 ) ∩ ℓ 2 ( H 0 1 ) -norm and of $$O(\Delta t^2 + h^{k+1})$$O ( Δ t 2 + h k + 1 ) in the $$\ell ^\infty (L^2)$$ℓ ∞ ( L 2 ) -norm if the duality argument can be employed. Error estimates of $$O(\Delta t^{3/2}+h^k)$$O ( Δ t 3 / 2 + h k ) in discrete versions of the $$L^\infty (H^1_0)$$L ∞ ( H 0 1 ) - and $$H^1(L^2)$$H 1 ( L 2 ) -norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.