  We present a theoretical analysis of the temperature-magnetic
field-concentration phase diagram of the multiferroic Mn$_{1-x}$M$_{x}$WO$_{4}$
(M=Fe, Zn, Mg), which exhibits three ordered phases, with collinear and
non-collinear incommensurate and with a commensurate magnetic order. The middle
phase is also ferroelectric. The analysis uses a semi-phenomenological Landau
theory, based on a Heisenberg Hamiltonian with a single-ion anisotropy. With a
small number of adjustable parameters, the Landau theory gives an excellent fit
to all three transition lines, as well as the magnetic and the ferroelectric
order parameters. The fit of the magnetic and ferroelectric order parameters is
further improved by including the effect of fluctuations near the transitions.
We demonstrate the highly frustrated nature of these materials and suggest a
simple explanation for the dramatic effects of doping with different magnetic
ions at the Mn sites. The model enables an examination of different sets of
exchange couplings that were proposed by a number of groups. Small
discrepancies are probably a consequence of small errors in the experimental
magnetic parameters. In addition, using the Ginzburg criterion we estimate the
temperature range in which fluctuations of the order parameters become
important.
