A two-fields model that self-consistently describes the coupled, spectral dynamics of flute mode large-scale flows turbulence is presented. This model has a characteristic form of a 'predator-prey' system, in which the populations of flute mode quanta (prey), growing via linear instability, generate large-scale flows (predators) through Reynolds stress. Concurrently, the mean flow growth regulates the prey population. To understand the long term nonlinear evolution of this one-prey two-predator system, a low-dimensional prototype of the model was constructed, assuming that the dynamics of such a complex system can be described within a phenomenological zero-dimensional approach. It is shown in the frame of this model that the dynamic outcome of interactions between the three system components may lead, depending on the system parameters, to their coexistence in the form of oscillatory solutions corresponding to quasiperiodic bursting of turbulence intensity level. These solutions are consistent with the time dependent behaviour of flute mode turbulence recently observed in numerical simulations.