Math Model


Problem: How long will it take to replace an existing herd of Fine Wool sheep with Churro sheep ( and what will be the economic impact of the change ) given the following parameters:

Flock size = 100 head. 90% ewes, 10% rams. All animals will be culled each six years (due to death and replacement). Young lambs will be used as replacement stock, or sold. Each year all stock will be sheared and the fleeces sold.

Breed information:

Fine Wools: lambs sell for $30. Fleeces average 10 lbs. Wool brings $0.70 per lb. Shearing costs $2 per head. Ewes have a 90% chance of a single lamb and a 10% chance of twins. Lambing rate equals 90%.

Churros: lambs sell for $40. Fleeces average 9 lbs and the wool brings $1.00 per lb. Shearing costs $2.50 per head (due to different coats). Ewes have a 50% chance of single lambs, 35% chance of twins, and a 15% chance of triplets. Lambing rate equals 95%.

Basic assumptions:

Both breeds will produce offspring that are 50% male, 50% female. Both breeds can inhabit the same range conditions and have no significant differences in feeding costs or other costs of care. Herd size is limited by available land grazing capacity. Herd breed replacement is based upon the addition of ten churro ewes and one churro ram into the existing herd. Fine Wool mature animals = $125 per head. Churro mature animals = $150 each. Due to the canyon-land terrain of the intended site for the project, it is assumed that the different flocks of sheep can be separated by their breed into isolated areas and thus eliminate all cross breeding.

The model begins with the addition of 10 Churro ewes and 1 Churro ram


{ Comments and explanations in {brackets and italics} }

Flock size = #FW {number Fine Wool} + #C {number Churro} = 100

Initial values: #FW = 89 #C = 11

Number of lambs:

#FWL {number of Fine Wool lambs} = .90 {% FW ewes} (#FW(.9 x 1 {single lambs}) + #FW(.1 x 2 {twins})) x .9 {birth/survival rate}

#CL {number of Churro lambs} = .90 {% C ewes} (#C(.5 x 1 {single lambs}) + #C(.35 x 2 {twins}) + #C(.15 x 3 {triplets} )) x .95 {birth/survival rate}

Wool Income:

Wool Income = (#FW(10 lbs x $.7/lb) - #FW($2)) + (#C(9 x $1) - #C($2.5))



#CR {# Churro replacements} = #C/6 {culls/death}

#FWR {# Fine Wool replacements} = #FW/6 {culls/death }

Churro Growth:

#CEweL {# Churro ewe lambs} = #CL x .5

#CRamL {# Churro ram lambs} = #CL x .5

Churro Increase = #CEweL - #CR

Excess Churro = #CRamL - .1 {% rams in flock} x #CR

#C = #C + Churro Increase

#FWExtras = #FW - Churro Increase {maintains flock size at 100 head}

#FW = #FW - #FWExtras

Limits on Churros:

IF #C > 100 THEN Extra Churro lamb sales = #C - 100

#C = #C - Extra Churro lamb sales

Excess Fine Wools:

Excess FW = #FWL - #FWR


Churro lamb income = Excess Churro x $40

Fine Wool lamb income = Excess FW x $30

Extra Churro lamb income = Extra Churro lamb sales x $40

FWExtras income = #FWExtras x $125

Wool Income = (#FW(10 lbs x $.70 per lb) {fleece wt and price} - #FW($2){shearing cost}) + (#C(9 lbs x $1 per lb) {fleece wt and price} - #C($2.50){shearing cost})

Total income = Wool Income + Churro lamb income + Fine Wool lamb income + Extra Churro lamb income + NFWExtras income

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