A Stochastic Model for Population Evolution with Genocide Incidence
Prof. Dr. Roelly Sylvie2
1 Institut d’Enseignement Superieur de Ruhengeri
2 Institut f ̈ur Mathematik der Universit ̈at Potsdam
The probability generating function (PGF) for a birth- death-immigration-emigration (BDIE) Process with genocide incidence as a partial catastrophe is developed. In this paper, a BDIE process is considered under the influence of genocide incidence which, when it occurs, reduces the population size to a non-zero state. Upon utilizing the generating function, the population mean and its variance for BDIE processes with genocide can be found quite easily.
Key Words: BDIE process, PGF, Genocide, Partial Catastrophe
The stochastic process on birth-death type processes have been developed mostly Yule, Feller, Kandal and Getz among others. Recently, Granita  modeled linear growth birth and death processes with immigration and emigration using the stochastic differential equation. Getz , modeled birth-death processes with positive and negative controls using the probability generating function. The construction of the transient probabilities for a simple birth-death-immigration process under the influence of total catastrophe was constructed by Randall  and in his paper a total catastrophe that wipes out the total population to size zero sate was taken into consideration. Di Crescenzo  also worked on birth-death process subject to catastrophes using the Laplace transform of its probability density function to obtain the mean and variance. Moreover, Michael  modeled the immigration-emigration with catastrophe and found the steady-state solution using the classic recursive methods. However, the generating function model of birth-death-immigration-emigration process with genocide incidence as a partial catastrophe that depends on population size has not been discussed in the previous works. In this paper, we shall consider a genocide which occurs at a constant rate and when it occurs, reduces the population to a non- zero state. The purpose of this paper is to analyze the BDIE processes by introducing a genocide parameter depending on the size of the population like birth and death because genocide is considered as a forced death. This led to the development of differential difference equation and obtain the generating function for BDIE with genocide incidence by using the probability generating function method which will further lead to the determination of the mean and variance of BDIE with genocide incidence.