In the framework of Finite Element discretizations we introduce a fully nonlinear Newtonlike
method and a linearized second order approach in time applied to certain partial differential
equations for chemotactic processes incorporating two entities, a chemical agent
and the reacting population of certain biological organisms/species. We investigate the
benet of a corresponding monolithic approach and the decoupled variant. In particular
we analyze accuracy, eciency and stability of the dierent methods and their dependencies
on certain parameters in order to identify a well suited Finite Element solver for
chemotaxis problems