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A study on the evolution of a community population by cumulative and fractional calculus approaches

Nowadays, in our globalized world,the local and intercountry movements of population have been increased. This situation makes it important for host countries to do right predictions for the future population of their native people as well as immigrant people. The knowledge of the attained number of accumulated population is necessary for future planning, concerning to education,health, job, housing, safety requirements, etc. In this work, for updating historically well known formulas of population dynamics of a community are revisited in the framework of compound growth and fractional calculus to get more realistic relations. Within this context, for a time t, the population evolution of a society which owns two different components is calculated. Concomitant relations have been developed to provide a comparison between the native population and the immigrant population that come into existence where at each time interval a colonial population is joined. Eventually at time t, the case where the native population becomes equal to immigrant population is discussed. Moreover, the differential equation of a probability function for the equilibrium state of accumulative population is obtained. It is seen that the equilibrium solution can be written as a series in terms of Bernoulli numbers. In the calculation of population dynamics, Mittag- Leffler(M-L) function replaces the exponential function.