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November 11th, 2014, 06:38 PM  #1 
Newbie Joined: Nov 2014 From: Baroda Posts: 1 Thanks: 0  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 
November 11th, 2014, 07:45 PM  #2  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
Let $C(t)$ be the number of cows in year $t$. Then the births in year $t$ are $b(t)= [C(t)b(t1)b(t2)b(t3)]/2 = C(t4)/2 $. and the deaths $d(t)=b(t20)$. 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.  
November 12th, 2014, 02:45 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra 
Your births equation assumes that no female cows died in the last 4 years.

November 12th, 2014, 05:30 AM  #4  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
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(t4)/2  F(t24)/2  F(t23)/2  F(t22)/2  F(t21)/2  F(t20) CB Last edited by CaptainBlack; November 12th, 2014 at 06:28 AM.  
November 12th, 2014, 07:14 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra 
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. 
November 12th, 2014, 09:45 AM  #6 
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  
November 12th, 2014, 10:38 AM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra 
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.

November 12th, 2014, 07:04 PM  #8  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
CB  
November 12th, 2014, 07:16 PM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra 
Indeed. My (halfserious) point is that the situation needs some clarification at least. 
November 12th, 2014, 08:09 PM  #10  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
CB Last edited by CaptainBlack; November 12th, 2014 at 08:11 PM.  

Tags 
cow, herd, mathetical, model, size, years 
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