We present the first successful attempt in modelling simultaneously the H$_{\alpha}$ and H$_{\beta}$ high resolution profiles and the H$_{\alpha}$ to Ly$_{\alpha}$ flux ratio for a dMe star (AU Mic, dM2.5e). We show that a very high transition region pressure and a thin transition region are simultaneously required to reproduce our observations. We give evidence that lower pressure model chromospheres can alos reproduce the Balmer line profiles, but give Lyman surface fluxes overestimated by more than an order of magnitude. Our model also reproduces the Balmer decrements, the Ly$_{\alpha}$ full width half maximum (upper limit) and the Balmer jump which is weak. Discrepancies in the profiles between observations and our model point to possible systematic upward flows in active stellar chromospheres. We obtain the Paschen lines in emission and Paschen decrements as large as 41 for Pa$_{\beta}$. For our model, we obtain a transition region column mass of 1.26 10$^{-3}$ g cm$^{-2}$ (log(M) $\simeq$ -2.9) which should be a lower limit (with the present modelling approximations). This large pressure chromosphere yields too large equivalent widths by a factor of 3.3. We believe that this points to the inhomogeneous character of the star's chromosphere. in this context, the plage filling factor is about 30\%. This disagrees with the concept of homogeneous atmospheres back-heated by large coronal X-ray radiation and suggests that additional non-thermal heating is taking place in active regions. The electron density at the top of the chromosphere is in the 10&^{12\}$-5 10$^{12}$cm$^{-3}$range, in agreement with previous estimates based on the Balmer decrements. The pressure is 6.3 Pa, of the same order as the current estimates for coronal pressures ($\sim$10 Pa). Discordant results for the HI Lyman and the HeI lines for our model atmosphere would imply conflicting filling factors for these lines which indicates that, like the Sun, stellar transition regions are quite inhomogeneous. Formation of all Hydrogen series is collisionally controlled and takes place in a rather thin layer at the top of the chromosphere (Balmer, Paschen, and Brackett) or in the transition region (Lyman). The temperature break acts as a disjunction in the domains of formation of the Lyman and other series. High electron density and collisional control is a necessary condition to drive the Balmer lines into emission in M dwarf atmospheres. Because of particularly low electron temperature and density in the middle chromosphere down to the photoshpere, the Balmer, Paschen and Brackett source functions are photoionisation controlled in this region. We compare our model to other dMe, dKe and solar model chromospheres. It shares common physical properties with solar flare models, thus implying a large and continuous chromospheric heating rate. The temperature break at 8200 K is very close to values obtained for the quiet Sun plages, flares and other stellar models. This emphasizes that this temperature is constrained by the plasma ability to radiate in the Hydrogen lines, over a wide range of the plasma and stellar parameters. The total radiative cooling in the Hydrogen lines and continua is about 1.22 10$^8$erg cm$^{-2}$s$^{-1}\$, of which 66\% arise from the Balmer series and continuum. The radiative cooling is dominated by the Lyman series and the Balmer series respectively in the transition region and the chromosphere. Cooling is dominated by the Paschen lines in the lower chromosphere. The short penetration of the Lyman radiation field produces a net backwarming in the upper chromosphere, while other series penetrate deeper and backwarm the lower chromosphere and the temperature minimum.
Key words: Stars: Late-type dwarfs - Stars: Chromospheric modelling - Stars: Activity - Hydrogen lines - Chromospheric and coronal heating