# population growth?

i've heard that if in a population each pair has 4 children, then after 4 generations it's a 1600% increase..

I suppose after 1 generation it is 200%, and each successive generation is 200%, and they multiply 200%^4 But, if I try it out with figures like this, I never get 1600%

suppose a population of 10

g  g1 g2 g3  g4
10 20 40 80  160


So each generation is double the previous, and from the start of g, population of 10, each of 4 generations g1-g4 have 4 children.

10+20+40+80+160=310
(310-10)/10 = 3000%

Not 1600%

Howcome i'm not getting 1600%?

-
The value $g_n$ in your notation denotes the entire population at generation $n$. It is not necessary to add in previous generations (at least not very many) since a generation is assumed to have survived their parents. In your figures it is easy to see that $\frac{160}{10}\cdot 100$ is $1600$%. –  Joshua Shane Liberman Mar 31 '11 at 0:11
@Joshua Shane Liberman gn in my notation is not the entire population at generation n. 'cos each generation has 4 children.. so by the time of g1, the entire population is g1+g=30. –  barlop Mar 31 '11 at 0:54
I suppose 1600% is how many times bigger the last generation is than the first. Not % increase. And not comparing population sizes. –  barlop Mar 31 '11 at 0:56
Maybe I am not making myself clear. If $g_n$ were your notation for the entire population during time period $n$ (and we know that no members of generation $n-1$ are still alive and no members of the next generation have been born) then we do see the kind of progression you were asking about, which would explain why your answer is different. It is to a different question! An example of this kind of population is the cicada of genus Magicicada, the periodical cicada with the 13- or 17-year life cycle, although a female cicada often lays hundreds of eggs instead of four. –  Joshua Shane Liberman Mar 31 '11 at 3:19
Thanks Joshua. Well, the 2 posted answers are both excellent. Arturo's and Douglas's with his all important follow ups. When I consider follow-ups and I do and must, then i'm stumped! –  barlop Apr 1 '11 at 21:24

## 2 Answers

First, you need to be careful with the terminology: if you have a quantity $X$, and you go to quantity $Y\gt X$, then the percentage of increase is $$\frac{100(Y-X)}{X}.$$ For example, if you start with $X=10$, and double to $Y=20$, then the percentage of increase is 100%; however, $Y$ is "200%" of the original. And therein probably lies the problem.

A "1600%" increase would mean that if you start with 10, then the difference between the final total and 10 is $160$, so the final total will be $170$ (the increase plus the original); so the final total will be 17 times the original. On the other hand, if we mean that the total population at the end will be 1600% the original population, then the percentage of increase is "only" 1500%.

Now, for your problem: if you have a population of $P$, and we assume everyone in the population breeds, then you get $4$ offspring for every $2$ people, that means that in addition to the original $P$, you will have $4\left(\frac{P}{2}\right) = 2P$ offspring in the next generation.

If we assume that each generation "dies" after the offspring is born and grows to maturity (so each generation has only the number of individuals that were born in the previous generation), then you have the recurrence $$P = P_0,\qquad P_{n+1}=2P_n$$ each generation doubles in size from the previous one. You start with $10$, then the next generation has $20$, then $40$, then $80$; if you mean to go to the "next" one after that, it's $160$.

That's an increase of $150$, or 1500%. However, $160$ is of course 16 times the original population, and is 1600% of the original population, probably what was meant.

If the original population remains alive (and continues to breed) throughout, then we have the recurrence $$P = P_0,\qquad P_{n+1} = P_n + 2P_n = 3P_n$$ so each generation is three times as big as the previous one.

If we start with $10$, after the next generation is born we have $30$, then after the next generation is born we have $90$, and after the next generation is born, $270$. that's a total increase of $260$, and a growth of 2600%. If we mean the generation after that, then you would have $810$ individuals, for a total increase of 8000%, and a final population which is 8100% the original one.

-

First, I think the "$1600\%$ increase" is meant to be $16$ times the original, not $17$ times the original. For small percentages, people are usually careful not to call multiplying by $1.05$ a $105\%$ increase, but for larger increases they sometimes are off by $100\%$ when that's not large compared with the increase.

Second, I think they assume that after having $4$ children, the pair dies, or that you are comparing the size of the generations not the total number of individuals.

-
Somebody told me it was 1600%, was he wrong? if not, then how can I get 1600%? –  barlop Mar 31 '11 at 0:12
Yes, the "$1600\%$ increase" was wrong. –  Douglas Zare Mar 31 '11 at 0:14
What is the % increase then? 3000%? And then if g,g1,g2 die by the time g5 is reached, is that ((40+80+160)-10)/10=2700% ? –  barlop Mar 31 '11 at 0:49
The size of the generation has increased $1500\%$ from $g$ to $g4$. If $g$, $g1$, and $g2$ die while $g3$, $g4$, and $g5$ are alive, then you are correct that the increase is $2700\%$. –  Douglas Zare Mar 31 '11 at 4:25