This is a system of equations in three dependent variables \(u\), \(v\) and \(w\), all functions of one time variable \(t\) and any number of spatial variables \(x_i\). The general form is \[u_t = \varepsilon_u \nabla^2 u - \nabla \cdot (\gamma u \nabla w) \] \[v_t = \varepsilon_v \nabla^2 v + \alpha u - \beta w \] \[w_t = -\eta v w \]

Use Neumann conditions \(\partial_n u=0\) and \(\partial_n v=0\) for \(u\) and \(v\).

For the constants, you may choose \[ \varepsilon_u = \varepsilon_v = 10^{-3}, \qquad \eta=10, \qquad \alpha=0.1, \quad \beta = 0. \]

In light of the intended application, what are appropriate initial values? The system is most interesting where there are 2 spatial dimensions, but it will suffice to study just one.

Reference: A.R.A. Anderson et.al. Mathematical Modelling og Tumor Invasion and Metastasis, Journal of Theoretical Medicine, Vol. 2, pp. 129-154.

Challenges: 2 space dimensions, application.