  A hybrid stochastic individual-based model of proliferating cells with
chemotaxis is presented. The model is expressed by a branching diffusion
process coupled to a partial differential equation describing concentration of
a chemotactic factor. It is shown that in the hydrodynamic limit when number of
cells goes to infinity the model converges to the solution of nonconservative
Patlak-Keller-Segel-type system. A nonlinear mean-field stochastic model is
defined and it is proven that the movement of descendants of a single cell in
the individual model converges to this mean-field process.
