Growth Model for Bovines

Question

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)

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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.

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ULTIMATE QUESTION

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