Is it possible to determine the initial population given the final one?

Question

It is possible to determine the initial population of, say, bacteria given a measure at a later time?

Let's say I use the logistic growth model

$$\dot N (t) = rN(t)\left(1 - \frac{N(t)}{K}\right)$$

and let's assume I know $K$. Let's imagine I have a mouse and I inject it with a dose of Shiga-toxin producing Escherichia coli (STEC). After 24 hours, the mouse dies, and I measure STEC with qPCR, obtaining a value of 20 million cells.

Can I calculate how many bacteria were in the initial shot, $N_0$?

And how many parameters do I need to know? For instance, let's say the mouse simply died some times after the injection. Can I know $N_0$ even if the absence of the time parameter?

How do I calculate $K$?

Is the logistic equation valid for live models? I understand it is good for chemostats, but is it valid for a mouse. Of course, a mouse can also reach saturation, since if $K$ is reached, the mouse dies, as in the example. Or is it a better description for bacterial growth in animals?

Thank you

Answer 1

For the logistic equation, $$\frac{\mathrm{d}N(t)}{\mathrm{d}t} = rN(t)\left(\frac{K - N(t)}{K}\right)$$ the solution that allows you to calculate population population size at any time is: $$N(t) = \frac{N(0)Ke^{rt}}{K - N(0) + N(0)e^{rt}}.$$ Looking at this equation algebraically with a set of knowns—$K$, $t$, and $N(t)$—and a set of unknowns—$r$ and $N(0)$—we see with this one equation that we have two unknowns, which are the intrinsic rate of growth and initial population size. So, the short answer is there is insufficient information to know the initial population size without additionally knowing the intrinsic rate of growth.

Hopefully this makes intuitive sense as well. If we know what the population size is, we just don't know how fast it arrived at that value. The intrinsic rate of growth could be small with a large initial population size on one extreme of the intrinsic rate of growth could be large with a small initial population size.

Calculating $K$ is problematic since $K$ is a maximum value. Better would be to use the original Verhuslt logistic equation, where the parameter $\alpha$ represents the decrease in population growth rate as a function of individuals added to the population: $$\frac{\mathrm{d}N(t)}{\mathrm{d}t} = rN(t) - \alpha N(t)^2.$$ Note here that is the same if equation if $K = r/\alpha$. In the $K$ equation, $K$ is conflated as both a maximum and an equilibrium, which is never the case in real populations or any extension of the logistic model (e.g., including harvest or species interactions).

There are many single-species models of growth. The equation is valid, depending on what's done with it. The logistic equation is a generalized model that is elegant in the sense that is simple and has relative high qualitative explanatory value. But if you are looking to forecast population dynamics or look at bacterial populations among populations of mice, then there are many different and preferred approaches that are more mechanistic, predictive, and realistic. HTH.