Chemotactic and diffusive migration on a nonuniformly growing domain: numerical algorithm development and applications

A numerical algorithm to simulate chemotactic and/or diffusive migration on a one-dimensional growing domain is developed. The domain growth can be spatially nonuniform and the growth-derived advection term must be discretised. The hyperbolic terms in the conservation equations associated with chemotactic migration and domain growth are accurately discretised using an explicit central scheme. Generality of the algorithm is maintained using an operator split technique to simulate diffusive migration implicitly. The resulting algorithm is applicable for any combination of diffusive and/or chemotactic migration on a growing domain with a general growth-induced velocity field. The accuracy of the algorithm is demonstrated by testing the results against some simple analytical solutions and in an inter-code comparison. The new algorithm demonstrates that the form of nonuniform growth plays a critical role in determining whether a population of migratory cells is able to overcome the domain growth and fully colonise the domain.