A Probabilistic Approach in Modeling the Growth of Cell Population

Abstract

A number of contraction mappings have been drawn by several researchers to generalize Banach type contraction. Some Cauchy sequences are applied to show the uniqueness of the fixed points obtained on the basis of certain fixed point theorems on these mappings. The aforementioned technique, using these theorems, is extensively used in finding the solution of numerous mathematical quandaries. M. Rotenberg, in his approach to model the growth of cell population under certain boundary conditions, utilized fixed point theorem in general Banach space. He suggested a mathematical model, represented by the partial differential equation shown in equation (1), by considering the population density of cells as a function of two parameters (degree of maturity, ? and maturation velocity, v) and time.
In Rotenberg mathematical model, application of fixed point method is found consistent by considering the quandary as a non linear boundary value problem because, each cell is assorted by its degree of maturity and velocity of maturation which are fixed and coupled biological boundaries as ?=0 and v=c (a positive value). Further, due to concentration and other density depending effects, the transition rates under a nutrient environment may fluctuate, so that, it must be a function of population density. This is again the clear depiction of the problem as non linear which is most suitable for modeling by using the said procedure and hence, applicable for employing a fixed point theorem.
Our approach in this paper is to further investigate the result by applying some more contraction mapping with associated fixed point theorem in probabilistic metric space.