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Calculating the maximum

Ok, a silly meme appeared on a well known social network. It seemed so ridiculous that I chimed in and derided it quickly with a throw-away comment. Then my problem solving brain woke up and thought, "hang on, could that figure be correct?", so I checked my thinking again. I immediately noticed my initial scoffing of the meme was incorrect based on the fact that I had misread it, but then I wondered what the real figure was and not being the best mathematician in the world, I decided to see if I could work it out. Then I got stuck, because other than doing a brute force attempt, I couldn't see a simple way of working it out. So, I thought I'd post here. The distilled meme is along the lines of:

"In six years with 2 litters of 6 - 10 puppies a year, one un-spayed female dog and her un-spayed offspring can theoretically produce 67,000 dogs."

There are some serious assumptions here which are:

She only produces females
All females reproduce at the same rate
All puppies survive

If we assume the maximum litter is 10 each, what is the final figure after a 6 year time span, starting with the first birth of 10 puppies from a 2 year old female? We will assume that the first litter arrives dead on the 2 year age for each dog and they reproduce at the stated rate.

My feeling is that there must be some sort of formula that can be used to work this out. It feels somewhat recursive but I got stuck and it's bugging me.

Anyone? If you get a figure, can you show how you worked it out? I'm stumped.
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Lee Savidge
Asked:
Lee Savidge
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3 Solutions
 
ozoCommented:
let f6 = number of 6month old females
let f12 = number of 12month old females
let f18 = number of 18month old females
let f24 = number of pregnant females
then every 6 months
f6new = f24old*10
f12new = f6old
f18new = f12old
f24new = f24old+f18old

Starting with one pregnant 24 month old at t=0;
That gives a total population of 48631 dogs at t=72 months

But if you start with one un-spayed female dog and her un-spayed offspring, then you can exceed 67,000 in 72 months starting with
one un-spayed pregnanat female dog and her three 6 month old offspring,
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Rgonzo1971Commented:
Hi,

Puppies are first 0 month old

so maybe

f0 = 0
f6 = 0
f12 = 0
f18 = 0
f24 = 1
For Idx = 0 To 6 Step 0.5

f24 = f24 + f18
f18 = f12
f12 = f6
f6 = f0
f0 = f24 * 10

Next

VarSum = f0 + f6 + f12 + f18 + f24

equals 49631

Regards
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ozoCommented:
Rgonzo1971 is correct, the 8 in http:#a40523339 should have been 9
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leonstrykerCommented:
At what age can dogs reproduce? We also need to assume that there is at least one male pup in the first litter and no additional pups will be born until he reaches reproduction age.
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ozoCommented:
The stated assumption was "first birth of 10 puppies from a 2 year old female"
And there had to have been at least one male for that to have happened.
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leonstrykerCommented:
Yes, but is it still around? Based on the question, I would say no, since the rest is based on "one un-spayed female dog and her un-spayed offspring"
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ozoCommented:
The and her un-spayed offspring that we started with could have included a male and an unspecified number of additional females.

In other sources, first litter can occur in less than one year, but breeds with earlier puberty tend to be breeds with smaller litter sizes, so we can get to 67,000 sooner with litters of 6 instead of 10 with earlier litters.
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Lee SavidgeAuthor Commented:
Assume that the females are being impregnated from dogs outside of the problem domain. They give birth at 2 years until the end of the 6th year, so 6 litters each. Also, as I stated, all pups born are females which go through the same cycle.
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ozoCommented:
at 2 years until the end of their 6th year
that's a span of 4 years, which means 9 litters, or if you mean the their 6th year of fertility, that's 13 litters.  But either way, some of the ends of their 6th year would occur after the 6 year span of time from t=0
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Lee SavidgeAuthor Commented:
Apologies, I notived that. They have a 6 year breeding span. I changed it to "the 6th year". So assume that the dogs are 2 when they start and 8 when they finish.
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ozoCommented:
When the first female is 8, her offspring will not yet be 8.
Are we to continue counting until they are all 8, which would then be 14 years from the start?
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Lee SavidgeAuthor Commented:
Just continuing for the breeding life span of the initial female. When her breeding lifespan stops, then the total count needs to be made.
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ozoCommented:
Starting with the first birth of 10 puppies from a 2 year old female, we don't produce 67,000 dogs until 78 months.
If we are allowed to span 6 1/2 years, then we can exceed  67,000 with 8.6 puppies per litter.
If the first litter occurs at 22 months instead of 24 months, then we can get over 67,000 dogs in 72 months, even if we reduce the litter size from 10 to 9.2
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Lee SavidgeAuthor Commented:
Thanks for putting it in a way I could understand.
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