Studies on generalized Yule models

We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the $OS$ property, while for the growth of species we use nonlinear time-fractional pure birth processes. Further, in two specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time $t$. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.