I'm new here so I hope this post is appropriate. I recently read in a bioengineering textbook about an approach to model cell proliferation and differentiation. They proposed the following partial differential equation:
$$ \frac{\partial X}{\partial t} + \delta \frac{\partial X}{\partial \lambda} = \big( \mu - \alpha \big) X $$
$X$ is number of cells.
$t$ is time.
$\lambda$ is the differentiation state, and can only have values between 0 and 1, which I will discuss below.
$\delta$, $\mu$, and $\alpha$ are differentiation rate, proliferation rate, and death rate, respectively, and have units of inverse time. I'll assume they are all constant for the time being.
I would be fine in solving this partial differential equation except for my interpretation of $\lambda$. In the population X, there are two kinds of cells, which I will call A and B. A transforms into B. Their sum is X. $$A + B = X$$ If $\lambda = 0$, then X is entirely composed of A. If $\lambda = 1$, then X is entirely composed of B.
That is all my textbook tells me, but I would like to understand this further.
From my understanding, $\lambda$ could therefore be defined as: $$\lambda = \frac{B}{X} = \frac{B}{A+B} $$
At first glance to me, $X$ should be a function of $t$ and $\lambda$, but if I'm saying that $\lambda$ is a function of A and B, does that mean that $X$ is only a function of time $t$?
Another question I have is about the implications of $\delta$. Does $\delta$ simply imply a transformation, or transformation and increase in numbers, based solely on the differentiation equation above.
I hope this question is not too biological for this forum. Thank you.