We propose a variation of the Guiol-Machado-Schinazi (GMS) model of evolution of species. In our version, as in the GMS model, at each birth, the new species in the system is labeled with a random fitness, but in our variation, to each extinction event is associated a random threshold and all species with fitness below the threshold are removed from the system. We present necessary and suficient criteria for the recurrence and transience of the empty configuration of species; we show the existence of a long time limit distribution of species in the system, and present necessary and suficient criteria for the finiteness of the number of species in that distribution. There is a remarkable symmetry between both sets of criteria. We also highlight fundamental differences between ours and the GMS model, putting them in different universality classes.