Proportion population dynamics

Question

We are told a population at any given time $t$ obeys the equation:

$$p(t) = 1 - \exp(-t).$$

Further, we are told that this total population actually consists of five segments. Say one of the segments at time $T > t$ is $0.35$. Can we determine the dynamics of this individual segment for all $t < T$?

My intuition says that this is not possible, since, roughly speaking,

$$p(t) = 1- \exp(-t) = p_1(t) + p_2(t) + p_3(t) + p_4(t) + p_5(t),$$ and there is an infinite number of ways to satisfy this equation, but, I'm not sure.

Answer 1

Mathematically, you are absolutely correct and cannot say much more than $0≤p_1(t)≤p(t)$.

Biologically, your scenario suggests that the size of your compound population is determined by some niche that limits its size to 1. In order to make statements about your population $p_1$ you need to make the following assumptions:

• Subpopulation 1 follows the same population law as the compound population.
• Subpopulation 1 has an exclusive sub-niche.
• Subpopulation 1 does not interact with the other subpopulations (e.g., by mingling or predator–prey relationships).

If all of this holds, you have:

$$p_1(t)= \frac{p_1(T) p(t)}{p(T)} = \left(1-\exp(-t)\right) \frac{0.35}{p(T)}$$