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We analyse a novel mathematical model of malignant invasion which takes theform of a two-phase moving boundary problem describing the invasion of apopulation of malignant cells into a population of background tissue, such asskin. Cells in both populations undergo diffusive migration and logisticproliferation. The interface between the two populations moves according to atwo-phase Stefan condition. Unlike many reaction-diffusion models of malignantinvasion, the moving boundary model explicitly describes the motion of thesharp front between the cancer and surrounding tissues without needing tointroduce degenerate nonlinear diffusion. Numerical simulations suggest themodel gives rise to very interesting travelling wave solutions that move withspeed $c$, and the model supports both malignant invasion and malignantretreat, where the travelling wave can move in either the positive or negative$x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fishermodels where travelling waves move with a minimum wave speed $c \ge c^* > 0$,the moving boundary model leads to travelling wave solutions with $|c| <c^{**}$. We interpret these travelling wave solutions in the phase plane andshow that they are associated with several features of the classicalFisher-Kolmogorov phase plane that are often disregarded as being nonphysical.We show, numerically, that the phase plane analysis compares well with longtime solutions from the full partial differential equation model as well asproviding accurate perturbation approximations for the shape of the travellingwaves.

A sharp-front moving boundary model for malignant invasion