Dynamics of cell growth: Exponential growth and division after a minimum cell size

In this paper, we consider a mathematical model for cell division using a Pantograph-type nonlocal partial differential equation, accompanied by relevant initial and boundary conditions. This formulation results in a nonlocal singular eigenvalue problem. We explore the possible eigenvalues that may lead to nontrivial solutions. We then consider cells that divide once they achieve a minimum size. Our model incorporates asymmetric cell division and exponential growth. We show that, unlike the constant growth rate case, a probability density function eigenvalue can be determined explicitly. Additionally, we demonstrate that a stochastic growth rate produces eigenfunctions expressed as an infinite series of modified Bessel functions. We extend our findings to encompass a wider range of dispersion and growth rates. The implications of this work are significant for understanding the dynamics of cell populations in biological systems. The work has potential applications in cancer research and developmental biology, where cell growth and division play critical roles.