Time-delayed model of the unbiased movement of tetrahymena pyriformis (Q663114)

It is known that in an average eukaryotic ciliate Tetrahymena Pyroformis, up to one third of cells remain in a ``rest state'', that is, do not move or react to chemical compounds. Therefore, a delay in cell reactions to environmental changes is observed. To reflect this peculiarity in a mathematical model, the authors introduce a convolution of present and past states of the system with an appropriate density function \(s( t) \) in Fick's equation for the motion of cells: \[ \frac{\partial u\left( t,x\right) }{\partial t}=\int_{-\infty}^{t} D\frac{\partial^{2}}{\partial x^{2}}u\left( \tau,x\right) s\left(t-\tau\right) d\tau.\tag{1} \] Approximating (1) with the help of semi-discretization, they consider a discretized version of (1) in space which, unfortunately, cannot be solved analytically. This leads to the analysis of two special cases where the delay \(s( t) \) is assumed to be either the exponential or the gamma function. The main results of the paper are collected in Section 3. Theorem 3.1 guarantees existence of a unique solution to a discretized system. Positivity of solutions for all times is ensured in Theorem 3.3 for a special case which has an important application to a model of the capillary assay considered later in Section 4.