Loss of Fertility Per Unit of Time

Question

Consider a study examining fertility of a mammal.

Suppose there is a probability space $(\Omega,F,P)$.

Consider a random variable $t:\Omega\rightarrow [0,T]$ for some $T>0$, and suppose it represents a moment at which a mammal loses its fertility capacity. $T$ is the moment of its death.

Let $r:[0,T]\rightarrow\mathbb{R}_+$ be a function that represents how many babies the mammal can have. For example, $r(3)=12$ means at year 3, it has the capacity to produce 12 offspring if it lives until $T$.

I want to define a function that behaves like a probability density function that represents the loss of fertility per unit of time at time $t$.

Consider a definition:

$$f(t)=\frac{r(t)}{\int_0^T r(x)dx}.$$

It is a ratio between the number of babies the mammal can have at time $t$ (if it lives until death) and the cumulative sum of the babies it can produce over the entire lifetime. In other words, it compares the potential number of offspring it can have standing at time $t$ (reduced fertility in mid-life) and standing at time $0$ (i.e. assuming most fertile at the beginning of its life cycle).

My Questions:

1. Does this definition translate to the rate at which the fertility is lost (e.g. the loss of babies per unit of time)?

2. What would the CDF mean? Consider $F(t)=\int_0^tf(y)dy=\int^t_0\frac{r(y)}{\int_0^T r(x)dx}dy=\frac{\int_0^t r(y)dy}{\int_0^T r(x)dx}.$

I thought about this in a discrete case, and I am having hard time understanding what the intuition would be. For example, I considered a simple example where the mammal produces 10 babies at $t=0$ and 5 at $t=1$ and dies at $t=2$, so zero baby. According to the CDF definition, $F(1)=\frac{5}{15}=.33$. This number doesn't seem to relate to the fertility analysis, because the mammal is able to produce 10 offspring, so when it is at $t=1$, the loss of fertility is 50% and not 33%. I made $f(t)$ to behave like a density function, but normalizing with the area under the curve doesn't seem to provide any intuition. For example, why would my analysis be interested in the cumulative sum of babies the mammal can produce over the entire lifetime, because this would be as if I am double-counting its fertility capacity at each period.