Population Sequence Proof

Question

For 10th Grade Math class we are modeling the population growth of dogs. For this task I also have to prove the models to show why and how they work. The formula attached below models the number of new puppies born to the mature parents per anum.

$U= m(1+0.5m)^{n-1}$

$U$ is the $n$-th term, and represents the number of puppies born in year $n$, $m$ is the number of puppies born to each female per annum $n$ is the year

can someone please show me how to prove this? this is a rather simple geometric sequence so it shouldn't be too hard but Ive never done anything like this so any help would be appreciated!

Answer 1

The word "prove" is never truly appropriate for mathematical models. Mathematics is logical deduction from initial premises, called axioms. It deals with a strictly conceptual world, that in and of itself have no attachment at all to reality. Modeling is attempting to tie this mathematical world to the real world. This is done by making assumptions about how the real world behaves and using these assumptions as axioms for your mathematical theory. Those axioms are not subject to proof. They cannot be validated mathematically. Only by accumulating experimental evidence can you gain any sort of confidence in their accuracy. And experimental evidence is never certain - there is always the possibility that factors you've overlooked could still affect the outcome.

Most mathematical models do not even claim to be be solid representations of reality. They only hold - imperfectly - in limited circumstances. That is certainly the case here. Your formula makes the following assumptions:

1. You start with a single litter of puppies (obviously, we are ignoring the issues with inbreeding).
2. Exactly half of the original litter, and of all puppies born each year, are female.
3. Every female gives birth to exactly 1 litter a year, including the females born in the previous year.
4. All litters are the same size, $m$.
5. No dogs die, ever.

Now, these assumptions can be tweeked to be more realistic, while still giving rise to the same equation. For instance, you can say that $m$ is the rate at which new dogs are born minus the rate at which they die.

To see how the equation arises, consider how the population changes in a year: Let $P$ be the population at the beginning of the year. Half of that population ($\frac 12 P$) is female and gives birth to $m$ new puppies each, adding $m(\frac 12 P)$ new dogs to the population. So at the end of year, there are $$P + m(\frac 12 P) = P\left(1 + \frac 12 m\right)$$dogs total.

In year $n=1$, there are $P = U(1) = m$ dogs. In year $n = 2$, that is multiplied by $\left(1 + \frac 12 m\right)$, so $$U(2) = m\left(1 + \frac 12 m\right)$$ In year $n = 3$, you start with $P = U(2)$ dogs and end with $$U(3) = U(2)\left(1 + \frac 12 m\right)$$ and so on.