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INTRODUCTION

I desire to model the profitability and growth of capital in dairy farm. Dairy farms generate revenue by two main activities:

  1. Reproduction Of cattle (Population Growth).
  2. Milk Production by the female cows.

Profits generated by both the above activities will be reinvested to enhance the scale of these acts. That is:

reproduction of cattle => population expansion => increased milk => increased induction of new cows => expansion of population expansion........... (& cycle repeats)

CRITERIA

Now we have to model the bovine population's growth. Simple exponential models aren't feasible due to certain assumptions and requirements I have listed below:

  1. All cattle lives from 0 to 7 years of age.
  2. All cattle mature at 3 years of age.
  3. Therefore every cow (Female) gives birth to 1 calf per year for a duration of 4 years i.e from 3 to 7 years of age.
  4. Every calf has equal chances with respect to either gender i.e. half of them will be male and half female.
  5. Additional cows will be purchased & inducted in the farm from dairy milk income (produced by existing cows) every year.
  6. The annually inducted cows will be equal in number to half the cows of preceding year and their age will be 3.
  7. All cows (new or old) have a constant cost c equal to twice the annual net income i (milk earnings) from a single cow. i.e. c=2i. (criterion 7 follows logically from 6.)

Example Of criterion 6 & 7:

If there are 10 cows in starting year i.e. year 0 then 5 additional cows will be inducted in year 1.

ULTIMATE QUESTION

Now starting with x cows at age 3 (in years). How many cows & calves the farm contain after t years?

user579908
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    You also need to specify the ages and genders of the cows in your starting amount $x$ – Marc Jul 27 '18 at 11:29
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    And the ages and genders of newly bought cows – Marc Jul 27 '18 at 11:29
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    @user579 If you're modelling the income from dairy milk, are you modelling the cost of keeping the cows? I think, to begin with, you should ignore the effects of money but then include it in your model later. – Jam Jul 27 '18 at 11:33
  • Consider a spherical cow. B-) – Somos Jul 27 '18 at 12:20
  • @Marc age at of x (starting cows) is 3 and in normal English "cow" refers to feminine domesticated ungulates. The male equivalent is bull. – user579908 Jul 27 '18 at 12:36
  • @user579 I'm NOT modelling the expenses i.e. cost of keeping the cows. The cost is: Labor + food (fodder+grass+hay etc.). Consider the setup free of cost. I'm really modelling the growth of capital invested in the setup & the animals are the vehicles of this capital gain. How? See the above cycle I have elaborated at the question's beginning. – user579908 Jul 27 '18 at 12:42
  • @user579 I have really excluded the effects of money as they depend on local factors. I have myself done those calculations (having obtained the relevant data) and have provided the net income of dairy produce by stating criterion 6. See the annual net income (milk earnings) from a single cow is equal to half the cost of a cow. -Criterion 6 – user579908 Jul 27 '18 at 12:51
  • I didn't study the criteria but, offhand, it looks like a problem that could be solved by setting up simultaneous equations and solving for each variable. Two matrices (nXn and 1Xn) can be solved with Cramer's Rule if the base matrix is not zero. I may be way off base but that was my first impression. Good luck. – poetasis Jul 27 '18 at 14:11

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