Subtraction is perhaps best viewed as removing objects from a set. For instance, if I have \$10, and I give away \$5, I have \$5. I can visualize removing \$5 from the original \$10. But, for instance, in the following expression: $$ \triangle population_{t}=births_{t}-deaths_{t} $$
the change in population from $t-1$ to $t$, is essentially the difference between new births and deaths. How is subtraction best viewed in this case? We aren't “removing” anyone from the set of births, right?
Yes, $a-b$ can, in a sense, be thought of as removing the number $b$ from the number $a$.
In your example, you can repeatedly match each death with a birth, saying they cancel out. Once you are done removing a birth for every death, you're left with $births - deaths$ number of births.
There's a difference between subtracting sets and subtracting numbers. There is no contradiction in subtracting the sizes of two disjoint sets.
For natural integers $\{0,1,2,\ldots\}$, subtraction is defined as removal indeed.
But in that case you can't define $3-5$. You can't remove $5$ from $3$.
But what you can do is to define $3-5$ as the operation of adding $3$ then removing $5$ to something. For instance $1000 + 3 - 5 = 998$.
But this operation is equivalent to adding $2$ and then removing $4$, because $1000+2-4 = 998$ also. So $3-5 = 2 -4$. It is also the equivalent to adding nothing and removing $2$. So $3-5 = 0-2$, which is pure removal, which we denote by the shorthand $-2$ (forgetting the leading $0$).
In general, we say that $a-b$ is the same as $-d$ if $a+d = b$.
This is how mathematicians construct the relative integers $\{\ldots,-2,-1,0,1,2,\ldots\}$. This is explained in the wikipedia page about the integers, where you can find good sources such as Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. p. 83
I think it is pretty clear how this applies to your example. A variation of population is, by design, a relative integer (it can be negative, population can decrease). It corresponds to adding a certain number of people (the births) and removing another (the deaths). Hence your equation.
If someone is born they are alive and part of the population if they die they are no longer part of that population. Births and deaths are independent of each other.
Think of it like you're debit card. You get paid, your balance increase, you spend money your balance decrease, this doesn't mean you spent the money you just got paid.
Or you can think of places that check capacity, person A enters a building you add 1. Then persons B, C, D exit you subtract 3. Doesn't mean the same people that came in are leaving.
Subtraction can also be viewed as extra quantity in a set.
For instance, if you do $a-b=c$, then there are extra $c$'s in $a$ than $b$.
Try to understand it by this: $$10-7=3$$ Then, we can say that $10$ has extra $3$'s than $7$. You can best understand it by using the unitary subtraction.
So,
$$10\equiv1+1+1+1+1+1+1+1+1+1$$ $$7\equiv1+1+1+1+1+1+1$$ Then, $$10-7\equiv1+1+1+1+1+1+1+1+1+1-(1+1+1+1+1+1+1)=1+1+1=3$$
So, we say that $10$ is $3 \ units$ greater than $7$ OR $10$ has extra $3$'s than $7$ has.
So, in case of your example regarding population difference, you can interpret it as extra births / deaths in given time period.
For me personally, I think that visualizing shifts of populations, be them human or ecological, is easiest when I think about adding births and "minus births" (deaths) together. Imagine a cluster of $1$'s representing the newborns and a cluster of $-1$'s representing people just deceased. Smashing them together, you get the change in population. As time proceeds, you're generating a slew of $1$'s and $-1$'s each time a birth or death occurs, respectively, and the $\Delta \text{Population}$ value you get is the result of combining the two processes. This is equivalent to subtracting the deaths from births, but I find, for visualization, what I have just described works nicely.
