The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $M/M/1$ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The approximate solutions corresponding to different values of the birth and death rates are also compared.