Figure 4

Solution profiles in the case of stable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when regular waves develop behind the invading tumour front (similar profiles of the vessel density not presented). Panel A (B) depicts the behaviour at t = 2000 (t = 4000). Behind the invading front, which travels with constant shape and connects the tumour-free steady state with the unstable co-existence steady state, regular spatio-temporal oscillations develop. Since the invading front moves faster than the evolving regular wave train, a large portion of the domain is at the unstable steady state ((p, v) = (0.35, 0.3)). Parameter values: η₀ = 0.5, d _p = 0.6, d _s = 0.2, = 0.2163 (using (A-11) in the Appendix (additional file 1)), δ = 0.65, D _v = 1, s _β = 0.4 and σ _p = 0. For these parameter values the waves are stable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3000 with (p(0, 0), v(0, 0)) = (0.01, 1) and (p(x, 0), v(x, 0)) = (0, 1) for x: 0 <x ≤ 1.