Numerical analysis of a cell dwarfism model

The population of cells is described by a density function u(x, t), t represents time and x the measure of the cell-size. Functions μ, b and ν represents the mortality, division and migration processes which take place within the population and φ is the initial state of the population density. We assume that the environment is unlimited and all possible nonlinear mechanisms are ignored. In this model, cells grow exponentially, x′(t) = x(t), as in a petri dish experiment, and die with death rate μ(x) depending on cellular size. The cell division is considered into two equal cells [1]. Note that the exponential growth introduces the unavailability of a boundary condition at size x = 0, thus cell renewal is introduced through the division b(x) and immigration ν(x) rates. We want to point out that a proper combination of growth, division and mortality rates would introduce a natural maximum cell size [3], otherwise we could fix it as one (normalized) and we would consider that larger cells may only grow and die. The usual CPBM, as developed in [1], assumes the existence of a minimal cell size a > 0 for cellular division to take place which generates a minimal cellular size a/2. However, the model we study allows cell of any size in the interval (0, 1] to divide. Therefore, the minimal cellular size is a = 0. Although the idea of a cell with size zero is biologically unrealistic, we use it as the limiting value to describe an abnormality in the cellular division process: the production of unfunctional ”dwarf” cells. These kind of cells are