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November 11th, 2014, 06:38 PM   #1
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Post Mathetical Model of Cow herd size after n years

I am trying to figure out a formula (mathematical model) for cow herd size after n years starting from a single cow 4 years old.

Provided:
A cow starts giving child after she becomes 4 years old.
Every years she gives one child.
Take 50% male and 50% female new born cows.
Take lifespan of male as well as female cow as 20 years

Assumptions:
No premature deaths.

I want to construct a mathematical model by which I can know what will be the total number of animals in this herd after 'n' number of years, provided that no animals are killed or sold.

If someone can help it will be nice.

Thankyou,
Damodara Das
damodara.bvks@gmail.com
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November 11th, 2014, 07:45 PM   #2
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Quote:
Originally Posted by Damodara Das View Post
I am trying to figure out a formula (mathematical model) for cow herd size after n years starting from a single cow 4 years old.

Provided:
A cow starts giving child after she becomes 4 years old.
Every years she gives one child.
Take 50% male and 50% female new born cows.
Take lifespan of male as well as female cow as 20 years

Assumptions:
No premature deaths.

I want to construct a mathematical model by which I can know what will be the total number of animals in this herd after 'n' number of years, provided that no animals are killed or sold.

If someone can help it will be nice.

Thankyou,
Damodara Das
damodara.bvks@gmail.com
What follows is untested and may contain errors, omissions and/or over-simplifications etc, but is to give you some idea of how to go about setting up such a model.

Let $C(t)$ be the number of cows in year $t$. Then the births in year $t$ are $b(t)= [C(t)-b(t-1)-b(t-2)-b(t-3)]/2 = C(t-4)/2 $. and the deaths $d(t)=b(t-20)$. So the population in year $t+1$ is:
$$C(t+1)=C(t)+b(t)-d(t)$$
With initial conditions $b(-3)=1$ and zero for all other $t \le 0$ and $b(1)=1$

Notes: more work may be needed on the start up conditions.

CB

Last edited by CaptainBlack; November 11th, 2014 at 08:13 PM.
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November 12th, 2014, 02:45 AM   #3
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Your births equation assumes that no female cows died in the last 4 years.
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November 12th, 2014, 05:30 AM   #4
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Quote:
Originally Posted by v8archie View Post
Your births equation assumes that no female cows died in the last 4 years.
Indeed. See the disclaimer ..

I think the model really needs to model the herd state as [M(t),F(t)] a vector of male and female numbers at each epoc. There is also a problem in that the births are equally likely to be male or female which needs better treatment.

An interesting alternative might be to model the populations with delay differential equations.

On third thoughts we need only model the population of females since the population of males can be reconstructed from it. So now we are (maybe?) interested in the model:

F(t+1)=F(t) + F(t-4)/2 - F(t-24)/2 - F(t-23)/2 - F(t-22)/2 - F(t-21)/2 - F(t-20)

CB

Last edited by CaptainBlack; November 12th, 2014 at 06:28 AM.
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November 12th, 2014, 07:14 AM   #5
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I think the herd size is 1 for 16 years and zero thereafter.

Because if there is only one cow, there is nothing to mate with it.

Unless we aren't counting the bull at all.
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November 12th, 2014, 09:45 AM   #6
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Originally Posted by v8archie View Post
I think the herd size is 1 for 16 years and zero thereafter.

Because if there is only one cow, there is nothing to mate with it.

Unless we aren't counting the bull at all.
I think it was mated before we bought the farm.

CB
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November 12th, 2014, 10:38 AM   #7
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At what age would a bull become able to perform? Because you're still without a bull until one happens to be born and get old enough.
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November 12th, 2014, 07:04 PM   #8
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Originally Posted by v8archie View Post
At what age would a bull become able to perform? Because you're still without a bull until one happens to be born and get old enough.
If we are being serious, even if the first cow gives birth in the first year, there is still a 50% chance that it will be a cow and so there will be no further births. Unless we are assuming there is some means of getting the cows served which does not depend on the bulls of the herd. Or we need to start with a pair rather than a single cow. But in practical terms unless the services of an external bull/s are employed we are running into serious inbreeding problems.

CB
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November 12th, 2014, 07:16 PM   #9
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Indeed.

My (half-serious) point is that the situation needs some clarification at least.
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November 12th, 2014, 08:09 PM   #10
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Originally Posted by v8archie View Post
Indeed.

My (half-serious) point is that the situation needs some clarification at least.
We agree about that. I particularly don't like having to combine a random element (probability for any birth of a male is 0.5, probability of a female is 0.5) with what is essentially a Fibonacci sequence like generation process.

CB

Last edited by CaptainBlack; November 12th, 2014 at 08:11 PM.
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