(In)stability of Travelling Waves in a Model of Haptotaxis

We examine the spectral stability of travelling waves of the haptotaxis model studied in [K. Harley et al., SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 366--396]. In the process we apply Liénard coordinates to the linearized stability problem and use a Riccati-transform/Grassmannian spectral shooting method à la [K. Harley et al., Math. Biosci., 266 (2015), pp. 36--51; V. Ledoux et al., SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 480--507; V. Ledoux, S. J. A. Malham, and V. Thümmler, Math. Comp., 79 (2010), pp. 1585--1619] in order to numerically compute the Evans function and point spectrum of a linearized operator associated with a travelling wave. We numerically show the instability of nonmonotone waves (type IV) and the stability of the monotone ones (types I--III) to perturbations in an appropriately weighted space.