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Recent observations support the view that the universe is described by a FLRWmodel with $\Omega_m^0 \approx 0.3$, $\Omega_{\Lambda}^0 \approx 0.7$, and $w\leq -1/3$ at the present epoch. There are several theoretical suggestions forthe cosmological $\Lambda$ component and for the particular form of the energytransfer between this dark energy and matter. This gives a strong motive for asystematic study of general properties of two-fluid FLRW models. We consider acombination of one perfect fluid, which is quintessence with negative pressure($p_Q = w\epsilon_Q$), and another perfect fluid, which is a mixture ofradiation and/or matter components with positive pressure ($p = \beta\epsilon_m$), which define the associated one-fluid model ($p = \gamma\epsilon$). We introduce a useful classification which contains 4 classes ofmodels defined by the presence or absence of energy transfer and by thestationarity ($w = const.$ and $\beta = const.$) or/and non stationarity ($w$or $\beta$ time dependent) of the equations of state. It is shown that, forgiven $w$ and $\beta$, the energy transfer defines $\gamma$ and, therefore, thetotal gravitating mass and dynamics of the model. We study important examplesof two-fluid FLRW models within the new classification. The behaviour of theenergy content, gravitating mass, pressure, and the energy transfer are givenas functions of the scale factor. We point out three characteristic scales,$a_E$, $a_{\cal P}$ and $a_{\cal M}$, which separate periods of time in whichquintessence energy, pressure and gravitating mass dominate. Each sequence ofthe scales defines one of 6 evolution types

Two-fluid matter-quintessence FLRW models: Energy transfer and the equation of state of the universe