Research and Reviews: Dairy

Special Circular 163-99

A Model for Projecting Animal Numbers in a Closed Herd

N. R. St-Pierre 1
The Ohio State University
Department of Animal Sciences

1 For more information, contact at: The Ohio State University, 221A Animal Science Building, 2029 Fyffe Road, Columbus, OH 43210-1095, 614-292-6507; fax: 614-292-1515; e-mail: st-pierre.8@osu.edu

Abstract

A simple model of animal flow from one year to the next was developed. In order to predict the average increase in herd size in a closed herd, the model was built on the principles that (1) all animals in year i must be accounted for in year i + 1, and (2) all animals in year i + 1 were either in the herd in year i, or were born during that year. The model calculates 11 transition coefficients that are factors of culling rate (CR; percent per year), calving interval (CI; months), average age at first calving (AFC; months), dead on arrival (DOA; percent of birth), and death and cull rate of replacement animals (DR; percent of all heifers per year). The model can be cycled n times to forecast animal numbers and average annual herd growth rate (AAGR) over a period of n years. Sensitivity analysis showed that AAGR is most sensitive to changes in CR, followed by CI, AFC, DR, and DOA. The model is easily adapted to forecast animal numbers during periods of expansion in nonclosed herds.

Introduction

The national herd size has increased from an average of 19 cows per herd in 1970 to 74 cows per herd in 1996 (USDA, 1996). Most likely, this trend in increased herd size will continue because of the better efficiency of larger operations and the associated economics of size.

The increase in herd size is achieved in two primary ways. The first way is through a series of size thresholds in which the herd increases its size abruptly by the purchase of animals from outside the herd. This practice is generally most capital efficient because the increased fixed costs (new building) are rapidly spread over many new additional producing units. But it is also associated with a significant health risk (Schaik et al., 1998).

The second option is to expand from within the herd. Although this reduces health risks, it generally implies a poor use of the additional fixed assets while the facilities are being filled with animals. The rate of annual increase in cow numbers is often a significant factor in capital risk during an expansion. Surprisingly, there is no model available to forecast future herd sizes based on herd parameters. This research developed a simple model to project future animal numbers based on parameters readily available from prior production records.

Methods

In Figure 1, a schematic view of the flow of animals from a given year i to the next year i + 1 is presented. Two rather simple principles are involved. First, all the animals in year i must be accounted for in year i + 1. Second, all the animals in year i + 1 were either there in year i or were born during the year. The kj in Figure 1 represents transfer coefficients. They are derived from the processes of aging, culling, drying, and freshening. Solid lines represent the flow of animals from year i to year i + 1. Broken lines represent new animals from freshenings.

Figure 1. Schematic flow of animals from year i to year i = 1, where kj
represents transfer coefficients.

Let CR = culling rate (percent of producing cows per year), then:

     k1 = 1 - (CR/100)
     k2 = CR/100

and k1 + k2 = 1, showing that all producing cows in year i are 
accounted for in year i + 1.



Let CI = calving interval (months) and DOA = dead on arrival, 
i.e., percent of births where the offspring is dead at or near
 birth (percent of freshenings), then:


     k3 = (1 - CR/100) * 0.5 * 12/CI * DOA/100

assuming that 50% of births result in female offsprings. 
The k3 represents the proportion of freshenings from 
producing cows that results in a dead female calf. 
Similarly:

     k4 = (1 - CR/100) * 0.5 * 12/CI * (1 - DOA/100)

and k4 represents the proportion of freshenings from 
producing cows that results  in a live female calf.

Let DR = Death and culled rate of replacement heifers 
(percent of total replacement animals per year), then:

     k5 = DR/100,
     k6 = (1 - DR/100), and
     k7 = DR/100.

Note that in this formulation, the annual death and culling
rate is assumed to be the same for heifers less than 12 
months as for heifers older than 12 months. Accommodating 
a different rate for the two age groups would be 
straightforward. However, most enterprises only know the 
rate for all replacement animals, so the simpler formulation
 was selected.

Let AFC = age at first calving (months), then:

k8 =       12       * (1 - DR/100)
      ____________
       (AFC - 12)

k9 =       12       * (1 - DR/100) * 0.5 * DOA/100
      ____________
       (AFC - 12)

k10 =      12       * (1 - DR/100) * 0.5 * (1 - DOA/100)
      ____________
       (AFC - 12)

k11 =	(1 - (   12      )) * (1 - DR/100)
             ____________
             (AFC - 12).

We can now verify that the two balance principles stated 
earlier are met.

For Principle 1 to apply, the following must be true:

     (1) k1 + k2 = 1,
     (2) k5 + k6 = 1,
     (3) k7 + k8 + k9 = 1.

The first two equalities are easily demonstrated, since:

	(1 - CR/100) + CR/100 = 1, and
	(1 - DR/100) + DR/100 = 1.

The proof of the third equality is not as straightforward.

Let       12       
      _________ = A
       AFC - 12

and (1 - DR/100) = B, then: 

     k11	= (1 - A) * B and
     k8 = A * B.

Therefore: 

k8 + k11 = (1 - A) * B + (A * B)
         = B - AB + AB
         = B

and by back substitution:

     k8 + k11 = (1 - DR/100).

It is now easily demonstrated that:

     k7 + k8 + k11 = DR + (1 - DR/100)
     = 1

and the proof is complete.

The second balance principle implies that the number of 
heifers born must be  equal to one-half of the number 
of freshenings (assuming 50% females at birth). 
This implies that:

     (4) 2 (k9 + k10) = k8 and

     (5) 2 (k3 + k4) = k1 * 12/CI.

Both proofs are relatively easy.

Let      12   + (1 - DR/100) = A,
     _________
      AFC - 12

then: 

     k9 = 0.5 A * DOA/100 and 

     k10 = 0.5 A * (1 - DOA/100).

Then (4) becomes:

     2 (0.5A * DOA/100 + 0.5A (1 - DOA/100))
     = (A * DOA/100 + A (1 - DOA/100)
     = A
     = k8.

A similar proof is used for (5).

We are now in a position to transition all animals from year i to year i + 1. he process is easily repeated to transition animals from year i + 1 to year i + 2, ..., i + n. In doing so, different values can be used for the parameters at different years. That is, the culling rate can be changed from year to year, for example. For planning purposes, this is generally not necessary. Therefore, this implementation assumes constant parameter rates over the years.

Results

Standard herd. As an example, a "standard" herd of 100 producing cows will be used, with 45 heifers less than 12 months of age and 40 heifers more than 12 months of age. The assumed parameters for this standard herd are:

AFC = 24 months, CI = 13 months, CR = 30%, DOA = 5%, and DR = 2.5%

Results for the first 10 years are reported in Table 1 and Figure 2. The average annual growth rate is calculated as follows:

Figure 2. Sample output of herd projections.

Table 1. Dairy Herd Internal Growth Projection Using Input and Output Values for a 100-Cow "Standard" Herd.
	Year											Average Yearly Growth
	0	1	2	3	4	5	6	7	8	9	10	%
Cows	100	109.0	119.1	130.1	142.2	155.4	169.8	185.6	202.8	221.7	242.3	9.25
First Lactations		39.0	42.8	46.8	51.1	55.9	61.1	66.7	72.9	79.7	87.1
% First Lactation		35.8	35.9	36.0	35.9	35.9	35.9	35.9	35.9	35.9	35.9
Heifers, 0-12 Months	45	49.2	53.8	58.8	64.2	70.2	76.7	83.8	91.6	100.1	109.4	9.29
As % of Cows	45.0	45.2	45.2	45.2	45.2	45.2	45.2	45.2	45.2	45.2	45.2
Heifers, 12+ Months	40	43.9	48.0	52.4	57.3	62.6	68.4	74.8	81.7	89.3	97.6	9.33
As % of Cows	40.0	40.3	40.3	40.3	40.3	40.3	40.3	40.3	40.3	40.3	40.3
Culled Cows		30.0	32.7	35.7	39.0	42.7	46.6	51.0	55.7	60.9	66.5
Dead Female Calves		2.6	2.8	3.1	3.4	3.7	4.0	4.4	4.8	5.3	5.8
Culled Heifers, 0-12 M		1.1	1.2	1.3	1.5	1.6	1.8	1.9	2.1	2.3	2.5
Culled Heifers, 12+ M		1.0	1.1	1.2	1.3	1.4	1.6	1.7	1.9	2.0	2.2
Heifers, as % of Cows	85.0	85.4	85.5	85.4	85.5	85.4	85.5	85.4	85.4	85.4	85.4
Inputs Average Culling Rate (%/Year) 30.0 Average Calving Interval (months 13.0 Average Age at First Calving (months) 24.0 Dead on Arrival (% of births) 5.0 Heifer Cull & Death Rate (%/year) 2.5

Let A be the number of producing cows in year zero and B be the number of producing cows in year 10. Then the average annual growth rate (AAGR, in percent per year) is:

AAGR = (1 - exp(log(B) - log(A))) * 100

In this standard herd, AAGR = 9.25% per year.

Sensitivity Analysis

Which parameter(s) is AAGR most sensitive to? This question is easily answered by a sensitivity analysis in which parameters are changed from their standard values one at a time. Table 2 reports the results of a sensitivity analysis when each parameter is progressively changed from +10% to -10% of its standard value. For example, a change of 10% in culling rates means that culling rate increases from 30% (standard) to 33% per year. A 3% unit increase in CR results in a 24% decrease in animal number growth rate (i.e., AAGR drops from 9.25% to 7.03%). These results show that AAGR is most sensitive to CR, followed by CI, AFC, DR, and DOA.

One criticism from the standard sensitivity analysis approach is that the method does not consider the ease or difficulty by which the parameters can be changed in the "real" world. This concern can be addressed by measuring change in AAGR when parameters are moved to their near-minimum or near-maximum biological limits. These values are selected to represent what can reasonably be expected to be minimum and maximum limits on the parameters. For example, it is reasonable to think that culling rates can vary between 20 and 40% per year. Our selection of near-minimum and near-maximum are reported in Table 3. Using these values, the changes in AAGR are reported in Table 4. Once again, these support the drastic effect of CR on AAGR. The DR has a much more significant effect under this approach.

Table 4. Animal Number Growth Rate (Percent Per Year) at Minimum and Maximum Near-Biological Limits of Culling Rate (CR), Calving Interval (CI), Age at First Calving (AFC), Dead on Arrival (DOA), and Death Plus Culling of Replacements (DR).
	CR	CI	AFC	DOA	DR
Maximum1	1.94	7.17	7.48	7.97	4.91
Standard	9.25	9.25	9.25	9.25	9.25
Minimum1	16.84	10.48	10.28	9.76	10.13
1 Refer to Table 2 for the minimum and maximum near-biological limits on all five parameters.

Conclusion

The animal number growth rate of a closed dairy herd can be modeled using five parameters and 11 computed transfer coefficients for each period. Of all herd parameters affecting AAGR, culling rate has the most significant effect. Using standard figures for all parameters, an AAGR of 9.25% per year is estimated, implying that the standard herd can double in size in 7.8 years. The model can easily be modified to accommodate animal purchases.

Addendum

The model is available as a Microsoft Excel® spreadsheet and can be obtained by contacting the author.

References

Van Schaik, G., A. A. Dijkhuizen, G. Benedictus, H. W. Barkema, and J. L. Koole. 1998. Exploratory study on the economic value of a closed farming system on Dutch dairy farms. Vet. Rec. 142:240-242.

USDA. 1996. Milk Production, and Farms and Land in Farms Statistics.