Mathematical models of tumour angiogenesis

We first study a parabolic–ODE system modelling tumour growth proposed by Othmer and Stevens [Aggregation, blowup, and collapse: the ABCʹs of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (4) (1997) 1044–1081]. According to Levine and Sleeman [A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (3) (1997) 683–730], we reduced it to a hyperbolic equation and showed the existence of collapse in [A. Kubo, T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential Integral Equations 17 (2004) 721–736]. We also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously. Next we apply the above result to another model proposed by Anderson and Chaplain arising from tumour angiogenesis and show the existence of collapse. Further we investigate a contact point between these two models and a common property to them.