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Modeling cattle production systems: integrating components and their interactions in the development of simulation models C.U. Leon-Velarde1 and R. Quiroz2

Abstract Several simulation models have been built to analyze the complex components of cattle production systems. The modelers usually tend to consider the overall complexity and resources of the systems such as investment, labor, feed, herd structure, and its dynamics. A few models also include physiological status of animals, as well as genetic and health parameters. Based on various models, this paper examines the main components considered to build subroutines to simulate intensive or grazing cattle production systems. The mathematical equations explaining the relationships among and within components can be written in different programming languages (FORTRAN, C++, and spreadsheets, among others). We also describe how models for individual animals or herd can be developed. In the first type of model, each animal within a herd is simulated, considering its productive life cycle and based on nutritional requirements for growth, reproduction, milk production and health. The herd model considers groups of animals instead. Specific management decisions accounted for include age at first service, number of days to mating, number of services, culling based on age and reproduction criteria, and effects on birth rate and mortality. These models can be validated, within a desired precision, using information obtained from commercial farms and/or from production, reproduction, genetics and economic data reported in the literature. The models can be generally used in various management situations as well as in training, extension, and research programs in which various scenarios are considered to help decision making in cattle systems.

INTRODUCTION A cattle production system is a complex system comprising biological, economic and social factors. However, the components and parameters along with their interactions are usually studied separately. Hence, the approach for improving or analyzing cattle systems has been to divide intricate research issues into topics, both simpler and more manageable within each component. The reduction of the system into parts frequently ignores interactions and can result in the loss of vital information for the global 1 2

analysis of the system (Congleton 1984). The use of mathematical modeling methods makes it possible to separate and describe the components, identify and analyze the cattle systems. Therefore, a cattle system is often represented by mathematical models, which are a simplification of reality. We assume that the set of management strategies identified as optimal for the model will similarly prove to be the best when applied to real-world systems (Mayer et al. 1999). Several mathematical models have been published to provide comprehensive descriptions of the biological

Ph.D. Animal breeder; agricultural systems analysis specialist, CIP-ILRI Ph.D. Department of Production Systems and Natural Resource Management; CIP

Proceedings - The Third International Symposium on Systems Approaches for Agricultural Development

characteristics of a herd or other specific components of the systems, such as forage (Rotz et al. 1989), nutrition (Bywater and Dent, 1976), reproduction (Boneschanscher et al. 1982, Oltenacu et al. 1980), health (James, 1977), or genetics (Groen, 1988). Other models emphasize management strategies (Dijkhuizen et. al 1986; Congleton, 1984; Sorensen, 1989) or replacement decisions in relation to production and prices (Van Arendock, 1985; Gartner, 1981; Herrero and Berry, 1982). These aspects play an important role for assessing how biologically sensitive the cattle system is to various production aspects. However, limitations and problems associated with the application of those models, at farm level, occur as they are built to analyze the implications of management changes in average farms or in specific sites and ideal conditions. A cattle system (dairy or beef) model can only be efficient if it accounts for all inputs as a whole. This implies that the biological and productive life cycle of the herd is to be considered, as well as all outputs, and its production level and market return determined (Spedding, 1988). It has been suggested (Blackie and Dent 1974, Leon-Velarde 1991) that structural models should describe the logic within and among components along with the basic parameters of the actual system. This makes it possible to incorporate different management strategies in order to estimate the changes of the system outputs through simulation, allowing the analysis of a particular farm system (Leon-Velarde 1991). This approach can also be used to design optimization methods to analyze cattle systems (Mayer et al., 1996). This paper examines the basic considerations for modeling cattle production system components to simulate a particular dairy or dual purpose systems, suggesting specific mathematical procedures related to cause-effect relationships for individual animal or herd models.

General considerations A system is the integration and interrelation of physical components with a function and an objective. Consequently, modeling a cattle system is only the process of representing an actual system

using components, limits and establishing procedures of cause-effect relationships. The simulation process is built and organized in a dynamic and quantitative way, the knowledge embodied in the model, which includes mathematical programming of the various components, processes and their interactions. In most cases the lack of quantitative knowledge of social aspects, makes it difficult to incorporate them in models and they remain implicit within the farm management decisions and management effects considered in the model. Modeling and simulation are techniques, which enable users to visualize various scenarios of a system with a range of precision as close to the real values as available data permits it. A model can be deterministic or stochastic. In either case, it must allow analysis for decision making on the present and future functioning, based on actual or existing information. The programming can be carried out in spreadsheets or any computer language such as FORTRAN, C++, and Delphi. The definition of the basic and simple parameters of a cattle system are necessary to build a complex model, one must begin with the most basic and elementary in order to define the links between components and their importance in the whole system. This, in turn, determines the equations used in programming and simulation. Programming can be done through a set of integrated equations that can also incorporate stochastic variables, which allow generating probabilistic outcomes (LeonVelarde, 1991). Some authors use deterministic linear programming techniques, Nicholson et al., 1994, Bernet et al., 1999. Figure 1 describes the productive natural life cycle used in the quantification of biological events. It depends on the type of production system. The figure also shows key variables such as intake, weight gain, growth curve, lactation and others that must be described through quantifiable parameters. To initiate a simulation model, one must know the main cause-effect quantitative relationships that impact positively or negatively on production. More often, this knowledge is limited or nonexistent and affects the building of the model. A list of variables necessary to build cattle simulation models (dairy and dual-purpose herds) is included in Table 1.

2

Modeling cattle production systems: integrating components and…

Intake Cost

Calf 0-2 months

Daily weight gain Growth curve Mortality Probability Birth weight

Heifer & Steers 3-12 months

Sex M:F Intake Cost Sale steers

Daily weight gain Growth curve Mortality Probability

Intake Cost Heifer & Steers 13 - First calving

Service age Puberty

Daily weight gain Growht curve

Sale steers

Mortality

Profit

Intake Cows & Sire

Cost

Income

Mastitis (probability) Service/conception; days without pregnancy (use of sire) Milk production (lactation curve) Mortality (probability)

Culling (culling/selection)

Figure 1. Schematic representation of the productive natural life cycle of the animal component

3


Table 1. List of main variables to be considered in modeling cattle production systems (dairy and dual-purpose). Factors, parameters and variables

Unit, expression form

Number of calvings in herd per year Annual average number of suckling calves in herd Number of births per year Number of weaned male calves not needed for replacement per year Number of weaned female calves not needed for replacement per year Age of heifer at service weight Period without pregnancy Age at first calving Length of time before adequate service weight after weaning Calving interval Lactation length Length of non-productive period between calving Interval between calving and pregnancy (conception) Calf mortality (birth - weaning) Mortality between weaning and weight at first mating Mortality between first mating and calving Adult mortality Number of abortions per year (including stillbirths) Female calves weighting less than the group with normal mating Acceptable service weight Calf weaning weight Number of lactation days Birth weight Probability of events of the productive and reproductive cycle Pasture management Farm stocking rate Paddock area Number of paddocks per productive dry cows and heifers Forage growth rate Feed DM available per pasture cycle Dry matter digestibility Dry matter energy (gross, digestible, net) Residual Dry matter (in grazing systems) Supplement energy Supplement protein Mineral compositions (Ca, P, others) Production Milk production per lactation Annual milk production per cow Total annual milk production Average weight of culled cows Growth rate from birth to weaning Growth rate from weaning to pregnancy Average weight of male calves weaned for sale Weight of bulls and heifers for sale Total number of cows culled per year Weight of culled cows

calving/year calves calves/year male calves female calves months months months months months days months days % % % % abortions/year young heifers kg kg days kg

4

LU/ha ha N kg DM/ha/day kg DM/ha % Mcal/kg kg/ha Mcal/kg DM % %; mg/kg kg l/cow/year l/year kg kg/day kg/day kg/calf kg/bull cows/year kg/cow


Integrating components and elements of cattle production systems Forage, herd, and their interactions are components used to model. The cattle production systems are often modeled with four biophysical components. These form the basis for formulating the algorithm equations to be developed in the simulation process. These components explain the functions and objectives of the systems. They are basically: 1. Feed/nutrition is the forage component, which includes pastures, byproducts, crop residues, and supplements; 2. Genetics is the animal component; the physical and bio-economic unit of the system. 3. Health, including the most frequent diseases (e.g. mastitis); and, 4. Reproduction, which is the element of herd dynamics and structure. The economics, which represents cost-benefit relation (Leon-Velarde, et al., 1995a), constitutes an optional component. All are integrated under the general term of animal production system management. In practice, each of these components can take any value within a possible continuum of options, and trying to physically analyze the possible combinations becomes experimentally impossible (Oltenacu et al. 1981). For complex models, a modular approach has proven useful. Modules or subroutines are built, considering input, state, and output variables. The state variables include weight, maintenance requirements, intake, heat regulation, grazing energy expenditure, as well as physiological and production variables. Input variables include forage availability and nutritive value, climate, prices and other inputs. Output variables include milk and meat production and economic parameters of the systems.

The animal component of the system The herd The animal is the main physical component and provides the basis for the dynamics of the production system. It must have all the necessary attributes in relation to inputs including feed, and outputs

including milk, meat and offspring production. Figure 2 describes the categories of a herd considering the animal life cycle and the herd dynamics, using a productive form that describe the relationship within categories and economic aspects. This model allows the simulation of herd productivity with a spreadsheet or programming language considering the specified parameters, which are particular to a system. Four blocks can be defined in the programming, i.e., the productive and non-productive periods of the animal, herd structure, herd productivity, income, cost and profit. The basic attributes of the animal component include weight, age and sex. Other models include lactation stage, the number of days without pregnancy, estrus cycle, duration of gestation and, lactation number (Leon-Velarde, 1991). The breed is generally considered to be implicit because it is included in the equations of the milk production curve and weight gain. It may be difficult to obtain specific equations for each breed or animal type, which explains why the genetic component is usually excluded. Genetics can be modeled separately without taking the feed component into consideration (Meuwissen, 1990). Figure 1, 2, and 3 summarize the steps to be considered in developing the structure of a cattle production system model. Note that there are necessary and specific cause-effect relationships that need to be calculated depending on the model use. For example, the suggestion is to leave some coefficients or parameters as default to allow a wider use of the model, so they can be modified according to the user’s situation. Figure 3. Schematic representation and summary of a dairy production system simulation model (Leon-Velarde, 1991) As far as herd dynamic concerns, parameters related events (Figure 1) such as mortality, gestation, mastitis, etc. must also be taken into consideration using probabilities of occurrence derived from real farm data. For stochastic models, one must determine the nature of the probable distribution density such as binomial and normal distributions, which allows randomization of the simulation process. The common way to simulate a normal distribution is based on:

Value = Average + R 0 ∗σ, where R0 is equal to a random number calculated as R0=2* (RAND –1). However, this equation only considers one standard deviation. To obtain a spectrum of values within the 96.56% of the normal 5


distribution, the following coefficient of 1.82 is useful:

equation

with

the

R = {log10 [(1+ R 0 ) / (1− R 0 )]}/ 1.82 Since the objective of a cattle system is to produce milk and meat, key parameters in the modeling process are the milk production and the animal growth curve. In group models, an appropriate lactation curve is the incomplete gamma curve (Wood, 1967) described as Y = at b e − ct where Y is the milk production on day t, a is associated with peak yield, b represents the I N P U T S

increasing slope, and c the decreasing slope. Other models, McMillan, 1970, Leon-Velarde, et al. 1995b can be used. The compartamental model is described as: Y = me −nt 1− e −p (t − q )

(

where Y is milk yield on day t, m represents a parameter associated with the maximum potential of daily milk production after peak, p represents the slope of the lactation curve during the increasing production, and q is the “lag time” parameter, which may account for lactogenesis prior to calving. These models can be fitted by using non-linear regression techniques (Statistical Analysis System 1994).

Bull calves T4

Male calves T2

P8 PV 3 P6

*

P 2/2

*

C O M P O N E N T S

Culled cows

P5

& F E E D

)

PV 2

P2 Calves P7

* Female calves T1

Dry cows P1

Heifers T3

* +

Total milk

PV 1

P4

P3

* Cost of calves

Total meat

P 2/2

*

Cost of heifers

+

Gross income

Cost of cows

+

Other costs

= Expenses Income

P1 = % cows in production

T1 = Female calves growth rate

P2 = Calving percentage

T2 = Male calves growth rate

P3 =

T3 = Heifers growth rate

Percentage of heifers

P4 = % replacement cows

T4 = Bull calves growth rate

P5 = % culled cows

V

P6 = % heifers for sale

PV1 = Milk price

P7 = % cows for culling

PV2 = Meat price

P8 = Calves for sale

PV3 = Heifers selling price

*

= Milk production

= System’s loss

Figure 2. Diagram of herd dynamics in relation to the economic and production value considering inputs, outputs and coefficients. 6


Start

Livestock parameters: weight, age, physiological stage

Feed parameters: pasture, concentrates, residue, others

Start the cycle, calculation and analysis for each animal or herd group

Calculation: area, milk sale, gross income, profit per kg

Is the animal a a calf?

Calculate growth requirements and nutrient intake

Yes

Print results

Calculate life cycle values for each animal

Has the heifer reached puberty?

Calculate cost of calves and heifers

Yes

No

No

Is the heifer or cow pregnant?

No

Yes

Yes No

Yes

Farm parameters: area, herd, workers, crops, etc.

No

Is the animal a heifer?

Is the cow pregnant?

Collect data for starting life cycle In the system

Yes

No

Is the cow productive?

Economic parameters: labour, feed, fixed costs

Calculate requirements and intake for non lactating empty animals Calculate requirements and intake for non lactating pregnant animals

Calculate cost of cows

Calculate milk production/draw lactation curve

Calculate income from milk production

Calculate requirements and intake for lactating empty animals Calculate requirements and intake for lactating pregnant animals

Figure 3. Schematic representation and summary of a dairy production system simulation model (Leon-Velarde, 1991) In the description of an animal model, each cow must have its own lactation curve, which must be related to the nutrient requirements and physiological stage taking the component feed into account. Since the parameters of the lactation curve are correlated, it is necessary to estimate the parameter of each cow

and then construct the phenotypic variance-covariance matrix of the parameters (V) to obtain a triangular matrix (T) by Cholesky decomposition (Statistical Analysis System, 1994). The matrix V = T’T. Therefore, the simulation for lactation curve of each cow is done by Y = U + T*R, where U is the vector of 7


average parameters, and R is a vector of random normal deviates. There are usually no corrections for lactation when modeling a herd as a whole. In the case of a cow model, the daily milk production is estimated for each cow and corrected by age (Bath, 1985; Gartner, 1981). The following assumptions can be taken into consideration: !

On the basis of 24 months to 72 months of age, milk increases up to 24 percent.

!

From 72 to 120 months (about 5th to 8th lactation) there is no further increase in yield.

!

A decline of 0.5 percent per month occurs on the production after 120 months of age.

!

Between 24 and 72 months of age the correction factor of the milk produced per day can be calculated by (1+ 0.005*(age -24)) where 24 is the base age expressed in months and 0.005 is the coefficient of increment per month.

!

From 120 months of age the value of 24 changes to 144 and 0.005 is negative. Some examples of correction factors for 24, 36, from 72 to 119, and 120 months of age are 1.0, 1.06, 1.24, 1.12, respectively (Leon-Velarde, 1991).

Animal growth Recording animal weight is not a common practice in a cattle system. However, an estimation of weight considering the age of animals can be obtained from secondary information from different farms to construct a specific growth curve. Similarly, to construct the animal growth curve, it is necessary to define the model type. In the case of a cow model, one strategy typically considers the weight from birth to 55 or 60 days calculated by linear regression from farm data. For example, from the Beltsville data (Matthews and Fohrman, 1954), for Holstein cattle, an average of 40 kg at birth and a gain of 0.460 kg.day-1 give an estimation of daily weight gain. Above 55 days, the growth curve needs to be defined with specific data. For the above data, the weight (kg) was calculated as 679.12 (1- e-0.0018 age), (R2 = 0.92) where age is expressed in days, and 0.0018 is the rate of maturing. Similarly the growth of calves and heifers can be based on daily gain in weight. This depends on the composition and quantity of feed intake. It estimates the daily weight gain (G) in

kg.day-1 for calves and heifers until first calving. From data of NRC (1989) the following equation estimates the weight gain (G): G = 0.087 − 0.003W + 0.03E + 1.71P − 0.002E2 + 0.0003WE − 0.006 WP + 0.00004 WEP (R2 = 0.92), where W is weight (kg), and P and E are the total intake of crude protein (kg) and metabolizable energy (Mcal), respectively. The weight change pattern for lactating cows is a function of age, lactation and gestation (Groen, 1988). Since dairy cattle may be growing, lactating, and pregnant at the same time, it is difficult, physiologically, to separate the effects of the weight gain during days of lactation, milk production and intake of nutrients (Leon-Velarde 1991). Consequently, the three effects can be considered separately. The daily gain in weight from calving to mature age can be estimated considering weight difference between two consecutive ages:

(

DGC = 679.12e −0.0018 A 0 1− e −0.0018 (A 1 − A 0 )

)

where DGC is daily weight gain (kg), A0 and A1 are two consecutive ages expressed in days, and 0.0018 is the rate of maturing. The weight change pattern for lactation can be assumed considering a loss of 25 kg of weight during early lactation (75 days), no gain until 150 days and, a gain of weight lost during the remaining days until end of lactation (Groen, 1988). The daily gain in weight for pregnancy (DGP) can be estimated by the equation:

(

)(

DGD = 2453e 0.0116D1 1− e 0.0116 (d0 − d1 ) , R 2 = 0.98

)

This equation was developed from data reported by Swett et al. (1948) considering the difference in foetus weight between two consecutive days of pregnancy where DGP is the daily gain in weight by pregnancy, and d0 and d1 are two consecutive days of pregnancy. At calving, cows lose the weight (kg) that was gained during pregnancy, which includes the weight of calf, fluids and membranes. In this form, the daily weight gain for normal growth, lactation and pregnancy can be combined and accumulated for the growth curve. Based on this, the development of equations for each particular type of animals is necessary. However, the equation can be used with factors developed from farm data.

8


The feed component Forage growth The feed component is given by the forage availability, which is crucial in grazing situations. It depends on water, temperature, and soil nutrients. The model can start from a given forage availability (kg DM ha-1 per cut or per day) in a database on growth rate (kg DM.day-1). Therefore, for each month, forage availability needs to be defined considering the residual forage from previous grazing. A factor of adjustment must be constructed from the database of forage growth, temperature and rainfall. Table 2 shows main adjustment factors for forage availability Barrera and Aguilar (1996), Arce et al. (1994).

Nutrients requirements and feed component The feed component and nutrients required by the animal is basic for any cattle production model. The requirements of the animal and the type of feed, including the various combinations of the forage resources e.g. grazing-forage, crop by-products and residues, as well as supplementation with concentrates and other products determine the main physiological status of the animal within the herd (Figure 4). Forage dry matter intake (DMI) is calculated considering its substitution effect in relation to the total dry matter intake of supplement. The DMI is a function of the animal weight, which varies with breed, type of ration, feed and other variables exogenous to the animal (NRC, 1987). The feed component includes the energy and protein, besides calcium, phosphorus, or other minerals, which are crucial in lactation and reproduction. Since the interactions are difficult to incorporate, the models usually focus on the calorie and protein contents to meet the maintenance, reproduction, production and growth requirements. For modeling intake it is essential to know the type of model, the time interval and the production mode, i.e., intensive (zero grazing) or extensive (pasture) and the type of production (dual or single purpose). DMI intake depends on weight (NRC, 1987, 1989) while consumption depends on the type of feed, climate and breed. It varies from 1.8 to 3.1 kg DM/100 kg of weight, which can also be expressed as a function of metabolic weight (W.75). Potential dry matter intake (PDMI) can be defined as PDMI = 0.03*W, where 0.03 express the percentage of dry matter per each 100 kg of weight (W). This

coefficient varies depending on the forage quality; it usually ranges from 0.018 to 0.031. From the total DMI and its composition, it is possible to separate the energy, protein, calcium, and phosphorus intake. Comparing values with the nutritional requirements and by using the law of the minimum, especially of protein and energy, animals in the model can grow and produce. Table 2 shows equations of nutrient requirements derived from tables of the NRC (LeonVelarde, 1991). Figure 4 describes dry matter intake, showing the interactions between the animal and the pasture. Forage availability and digestibility depend on the period of the year and pasture utilization pattern. Here, grazing pressure plays an important role in the re-growth of paddocks for simulated successive intake periods. In stall feeding, dry matter supply is direct as opposed to the situation in grazing. In both cases, a daily random variation ranging from 3 to 7% can be considered, depending on type of forage, animal and feed period (Cañas et al. 1992). Voluntary DMI in grazing areas is related to forage availability (FA) expressed in kg of dry matter per hectare cut or day, depending of the step basis of the model. This information can be set in a table considering the growth rate of the pasture per month or day depending on data availability. Table 2 shows corresponding equations for forage availability and factors in relation to adjustment by digestibility and selectivity (Arnold, 1970). The selectivity (SF) is a complex factor and difficult to measure. However, it can be estimated from the difference between the digestibility of forage offered and residual. The relation between digestibility of the dry matter consumed and the residual is a function of the forage availability. At higher availability the selectivity factor approaches to 1, and the utilization of the pasture is equal to or greater than 90 %. The relation of forage availability and digestibility to adjust DMI by selectivity measure through digestibility (SFD) was defined by Zoccal (1983): SFD = (FDO 2 - (0.6 * FDO) - 0.16) / - 0.5 where FDO is the percentage of digestibility of the forage offered. The percentage of the pasture utilization (PU) can be calculated as PU= (PDMI* FCDP* current capacity)/ FA. From this equation, some assumptions can be made: if PU < 0.1 the adjustment of selectivity by forage availability (SFA) is equal to one; if PU > 0.5 then SFA equal to zero; and if 0.1240 days

Drying process

no

Figure 5. Diagram of the reproduction process in herd in relation to production and herd increase

13


Daily gestation requirements

Total gestation requirements Foetus composition at calving; NE Mcal/g

Sale

No

Calving Yes

Breeding/ Fattening

Offspring sale

Calf weight Adult size

Age Weight

Herd/Class

Number of calves Milk production

Forage potential intake kg DM/day

Potential intake kg DM/day Milk consumption kg DM/day

Weaning

Digestibility of DM consumed

Forage voluntary intake; kg DM/day ME intake

Availability DM/ha/day

Stocking rate

Maintenance Pasture

Weight change +; –-

Live weight

Figure 6. Relationship between reproduction and herd dynamics based on feed availability and the production and development of growing animals, with regard to breeding, sale or fattering decisions

Economic component

output data from the biophysical components of the model to calculate the economic outputs needed to make decisions.

The farm economy is generally expressed in relation to the total variable costs and the gross income of its products. The aggregate method is used for cow models while the global method is for herd models. Whichever is the case, the difference expresses the gross margin since fixed costs are seldom included. Therefore, modeling the economic component is essential to keep the objective and clearly define the assumptions and restrictions of the model, which must consider that the prices of inputs and outputs could change over the time. Therefore, in models covering long periods of time, the biological product should be the first objective and the economic aspects the second objective with adjustments in terms of net present value. It is worth noting that the economic component uses input and

The common form to express gross margin (GM) is to account for the difference between gross income (GI) and the total annual cost (TAC). The first one includes milk, meat manure and other sold outputs. The latter one includes feed cost, fertilizers, veterinary expenses, labor, reproductive cost, and equipment, among other variable cost. The calculation of net income implies the calculation of fixed cost, which in most of the cases is difficult to address. A simple way to approach it is to include the administration cost (AC), which represents a value that is not accounted for by small enterprises. Therefore, the equation of GI-TAC+AC is closer to net income, which can be obtained from the total cost for each animal added up to determine the income for the herd. Male calves can be sold or kept at birth. 14


Therefore, the total net income for the system can be calculated as the sum of daily income from milk sales minus cost incurred plus income from the sale of male calves, heifers, culled cows and other products. A way to calculate the total cost is the sum of all costs considering the particular programming step basis (daily, monthly). A suggestion is to divide it into three parts. The first part includes feed cost, the second the labor, and the third the other direct cost. The labor cost can be derived based on cattle unit (AU) in man equivalent labor (MEL), which is defined as MEL = 0.432 + 0.024 AU (Leon-Velarde, 1991). For example a small farm of 15-animal unit will require 0.79 MEL (0.432 + 0.024*15), which when multiplied by the labor cost gives the daily labor cost. The direct cost (DC) can be calculated also from cattle units as e.g., DC = 1.72 + 0.018AU 0.00008AU2. Both equations were derived from The Ontario Dairy Accounting Project, 1986. In both cases these equations need to be modified based on specific farm data considering annual increases, inflation rate, prices of milk and meat, and the cost of artificial insemination. Increase in feed and labor cost can be determined a similar way.

Use of models in scenarios analysis Simulation models are built for a specific objective and must be considered as tools. Models can be used in research, to examine possible scenarios for improved resource use. In teaching, they can help in decision making on possible or purely hypothetical situations. Models designed for farm analyses can also be used in extension. These are generally deterministic models, but the program can generate various courses and establish a range of results that can be visualized. One of the possible ways to visualize models used in research, decision-making or teaching is to include or modify two or more conditions of the model in a determined range to create different scenarios (Leon-Velarde y Quiroz 1999). In this case, the model must be able to incorporate a sort of range within the most important factors allowing the analysis of different scenarios. The use of the central composite design allows the construction of response surfaces with two, three or more variables, but only two can be handled at the same time, maintaining the others constant. Thus, a response surface is plotted on the effect of the main factors. The most commonly used second-order response surface

design is the class of rotatable designs. The term rotatable indicates that the variance of the predicted response at some point x is only a function of the distance of the point from the design center, and not a function of the direction. Another property of this design is that the variance of the estimates is unchanged when the design is rotated about the center, hence the name rotatable designs (Montgomery 1984). Among the rotatable designs, the most widely used is the central composite design. This design consists of 2k factorial, augmented by 2k axial points and n0 center points (2k + 2k + n0, where n0 is the number of times the central point is repeated). A central composite design is made rotatable by the choice of a value α. The value of α depends on the number of points in the factorial portion of the design (P = 2k), i.e., α = P1/4 (For instance, if k = 2 then P = 2k = 4 and α = 1.414). The efficiency of the design, estimated as the ratio between the number of parameters to be estimated and the number of treatments, maximizes when k equal 3, i.e., 0.67 (10 parameters and 15 treatments). Table 3 shows, as an example, the combinations of variables considered for seven different scenarios in relation to farm management strategies for cattle studies. The reader can adjust corresponding values or model parameters with his or her own model, and obtain the respective coefficient to construct a response surface (SAS, 1994). The values presented were run with the DAIRYSIM model. Figure 3 describes the flow chart of the model (Leon-Velarde, 1991). The response surfaces in Figure 7 show the effect of the combination of herd size with (a) heifer’s fertility level and weight at first service (370kg) and (b) management effect and a culling rate of 15%.

Concluding remarks Several cattle production system models are available for the reader. The objective was to describe the most relevant equations for modeling a cattle intensive, grazing dairy, or dual-purpose herd system. Therefore, the main cause-effect relationships that exist between the components were taken into consideration so that they can be programmed and simulated. The use of specific parameters will depend on the information availability and on the model objective. Models will be useful insofar as they are accurate in their functional objective as tools for designing scenarios that allow decision making. They can also 15


assist in the identification of knowledge gaps and the development of specific scenarios, research, teaching and extension. One must use models with the under-

standing that they are not an objective in itself, but a way of representing reality.

Table 3. Factors, levels and possible conditions of variation in DAIRYSIM model for simulating various management strategies for stall feeding milk production Various dairy systems1 Code -1 0 1 25 35 45 10 15 20 40 60 80

Variables: management Decisions

-2 -1.68 1.68 2 Herd size (cows)2 (15) 18 52 (55) (5) 6.5 23.5 (25) Culling rate (%)2 Management effect (%) (20) 26.42 93.68 (100) Fertility level (%) Cows (60) 61.6 65 70 75 78.4 (90) Heifers (55) 56.6 60 65 70 73.4 (75) Mastitis (%)3: 1st lac. (10) 9.4 8 6 4 2.6 (2) 2nd (12) 11.4 10 8 6 4.6 (4) 3rd or more (14) 13.4 12 10 8 6.6 (6) (1) 1.6 3 5 7 8.4 (9) Mortality (%)3: cows (2) 2.6 4 6 8 9.4 (10) Heifers 1 year (6) 6.6 8 10 12 13.4 (14) Calves 1 Levels of various dairy systems were codified as -1.682, -1, 0, 1 and 1.682. Figures in brackets are the extremes of the system with code 2. 2 Herd size and culling rates under production are management decisions made directly by the producer. 3 Fertility level is defined as the probability of pregnancy at first service, and declines in following services. Fertility, mastitis (severe cases) and mortality are considered as management effects and are grouped in percentages considering an average of 60% within dairy systems.

35

35 30 25

25

20

10

25

35 Herd size

45

55

%

0 -5 -10 15 [B]

25

35

Herd size

65 45

55

60

80 75 70

Ma na ge me

[A]

15

5

lev el

-

100 80 60 40 20

10

Fe rtil ity

5

15

cts

15

nt eff e

20

Income ($'000/year)

Income $

30

Figure 7. Response surface of profit considering different herd sizes with level of fertility [A] (heier weight of 320 kg to initiate first service) and management effect at 15% culling rate [B] 16


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