population growth forumula

Me and my father got into a discussion about the enslavement of Jews in Egypt. We did some reading on the topic, and we found that the first Jews in Egypt were a single family, of about 70 people. We also found out that the Jews were in Egypt for 430 years, before the exodus. The source from which we got our information from, said that at the time of the exodus, the Jews population numbered 3 million. My father says this is impossible. My question is: how can i get an exponential formula that would approximate this growth. So starting population: 70, ending population: 3 million, time spent: 430 years. Any help would be appreciated.

Using the Classical Malthusian Model for population growth which uses the population of each previous generation (Pt) and a growth rate (r) to derive the next generations population (Pt+1). The equation takes the form:

Pt+1 = r * Pt

Assuming couples had babies around age 16 and on average had 3 babies per couple (r=1.5) we get the following data:

Generation Population Year
1 70 16
2 105 32
3 158 48
4 236 64
5 354 80
6 532 96
7 797 112
8 1,196 128
9 1,794 144
10 2,691 160
11 4,037 176
12 6,055 192
13 9,082 208
14 13,623 224
15 20,435 240
16 30,653 256
17 45,979 272
18 68,968 288
19 103,452 304
20 155,179 320
21 232,768 336
22 349,152 352
23 523,728 368
24 785,592 384
25 1,178,388 400
26 1,767,582 416
27 2,651,373 432

So I would say that yes it is possible to grow that fast. Over 27 generations with a 1.5 growth rate the population of 70 would reach 2.6 Million.

I have attached the excel file with the data.

The popular Population Growth Formula is:
N(t) = (No)*e^(kt)
Where
t = time
No = "N sub zero" = population at t=0 = initial population
N(t) = "N of t" = "N as a function of t" = population at any time t.
e = the natural number, 2.718.....
k = any real number

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We can make a simpler version of that:
B = A*e^(kt)
where
A = initial population, at time t=0.
B = final or new population, at time t.
e, k, and t, are the same as those in the original version.

In your question,
A = 70 persons
B = 3,000,000 persons
t = 430 years

e is always known, of course.

So only k is missing.

3,000,000 = (70)e^(k*430)
e^(430k) = 3,000,000 / 70
e^(430k) = 42,857.14286
Take the natural logarithm of both sides,
(430k)ln(e) = ln(42,857.14286)
430k = 10.6656276
k = 10.6656276 / 430
k = 0.024803785 ----------***

So, an exponential formula for your question is:

B = 70 *e^(0.02480378*t)

where
t is in years.
B is in number of persons.

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Samples:

At t = 0 year,
B = 70 *e^(0.024803785*0)
B = 70 *e^0
B = 70 *1
B = 70 persons.

At t = 10 years,
B = 70 *e^(0.024803785*10)
B = 70 *e^(0.24803785)
B = 70 *1.281508436
B = 89.7 or 90 persons.

At t = 100 years,
B = 70 *e^(0.024803785*100)
B = 70 *e^(2.4803785)
B = 70 *11.94578504
B = 836.2 or 836 persons.

At t = 200 years,
B = 70 *e^(0.024803785*200)
B = 9,989 persons.

At t = 300 years,
B = 70 *e^(0.024803785*300)
B = 119,323 persons.

At t = 400 years,
B = 70 *e^(0.02480378 *400)
B = 1,425,466 persons.

At t = 430 years,
B = 70 *e^(0.024803785*430)
B = 3,000,000 persons.

Ticbol's calculations seem to be correct but I don't think that the e^x function applies to growth of this kind. I know that 'e' is used to calculate interest in bank accounts that is compounded infinitely all of the time. I think it is more correct to look at the growth of a group of people by generation as the previous post by Mathguru.

Shmuel

I have two questions. First, what significance does k have? Second do either of these formulas factor in the death rate?

The explanation I made above does take into account the death rate

Generation People Year
.......1............ 70....... 16
.......2............ 105...... 32

Look at the above example. 70 people makes 35 couples that each had 3 children. Then all of the original 70 died. 35*3 =105. So the numbers above are the number of babies born in that generation. So there were actually 2.6M babies born in year 432.

So the method I employed in my previous post is actually conservative in a way that it assumes that after the babies are born the parents die soon after. You could actually get to 3,000,000 people in 430 years with a lower growth rate than 3 children per couple.

I hope I have clarified.

The "k" is the growth rate. It is rate of increase or rate of decrease in population.

I am not sure if in the formula
N(t) = (No)*e^(rt)
birth rate, death rate, migration, etc., were considered.

I wrote N(t) = (No)*e^(kt).
It should have been N(t) = (No)*e^(rt), where r is growth rate that is constant throughout.

My version B = A*e^(kt) stays.

In reality, r changes year to year. It cannot be possible that every year the rates of birth, death, migration, etc, remain the same.
The popular Population Growth Formula is just a model, so that there can be a formula to use for approximate population growth.

There are many different population growth formulas that are dedicated or based on various conditions.

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If you want to know more on population growth formulas, then search Google, for one, for "population growth formula".