 [page 107↓] 
In agricultural production planning, the concepts of technical and allocative efficiency play a key role in guiding decisions relating to resource allocation (Rehman, 1982). Efficient allocation of resources in production requires the definition of an objective, in relation to which different decision alternatives can be evaluated. Technical efficiency relates to the maximum physical output attainable for a given level of production inputs, given the range of alternative technologies available to the producer. Assessing technical efficiency thus can help in choosing the most appropriate technology for production. It should, however, be kept in mind that within the framework of the neoclassical theory of the firm, the achievement of technical efficiency is a necessary, but not sufficient condition to ensure economic efficiency (Ellis, 1993). In livestockrelated research, a confusing multitude of different measures of technical efficiency have been used in the past, some of which have been reviewed by Baptist (1992b) and more recently by Bosman et al. (1997a). The former author identified the failure to distinguish between production level and productivity as defined above as a major deficiency shared by many of the productivity measures commonly applied in animal production science. Typically, to evaluate efficiency at the herd level such indices relate gross output per time unit in terms of monetary value, mass or energy to the number of animals (or their total liveweight, or metabolic liveweight) required to produce it. Examples encompass the socalled ILCA livestock productivity indices (Wilson et al., 1985), which have been widely used for productivity assessments in African livestock production systems. The implicit assumption that is made when using this type of biological efficiency measure is that there exists a constant relationship between the denominator of the index (e.g., the total number or total liveweight of dams that produced the output) and input costs, among which the most important are commonly feed costs (Upton, 1993; Morris et al., 1994; James & Carles, 1996). However, such an assumption is questionable when making comparisons among different species, breeds within species, or between herds of the same breed maintained under different environmental or management conditions. Furthermore, even within a given livestock herd, feed requirements are not constant but vary with age, reproductive status, or life cycle stage. In order to circumvent these problems, Baptist (1992a), Upton (1993), Morris et al. (1994), as well as James and Carles (1996) have proposed the use of feed or feed energy inputs as a scaling factor in calculating livestock productivity measures. When valuing feed inputs in terms of their energy content, the underlying assumption is that, ceteris paribus, once the energy requirements for all physiological functions are met, so are all nonenergy requirements (Konandreas and Anderson, 1982).
However, relating outputs produced to the level of input use is not the only requirement that must be met to obtain reliable estimates of biological efficiency. Two more fundamental aspects which, when ignored in computing efficiency measures, can lead to questionable results in livestock herd productivity assessments, need to be taken into account. These relate to the effects of herd structure and dynamics on the calculation of efficiency measures. Firstly, changes in performance (i.e., survival, reproduction, and yield levels) over successive life cycle stages cause the computation of biological herd productivity measures to depend on herd structure at the time of assessment. Hence, unequal stage abundances among the herds or alternatives which are to be compared will tend to lead to biased assessments. Secondly, allowance has to be made for breeding stock replacements if the herd is to be maintained at its current level of productivity (Upton, 1993). Herd dynamics and structure is determined not only by inherent age or stagespecific fitness traits, but also by the offtake policy, i.e., by the choice that is made between immediate consumption and investment in future herd growth.
Steady state herd growth models can accomodate for the above effects on calculated efficiency measures by considering sustained selfrecruiting herds, in which herd size and structure are maintained in an equilibrium state over time. Previously published approaches to steady state herd productivity assessment are based on an agestructured description of herd dynamics (Put et al., 1987; Baptist, 1992; James & Carles (1996); Upton, 1989, 1993). However, the demography of domestic livestock species may depend on size or developmental stage much more than on age (e.g., parity or size), and often these variables are only weakly correlated with [page 108↓]age. A further limitation of available computer implementations of the concept of steady state productivity assesment in livestock herds (Baptist, 1992a; James & Carles, 1996) is that performance traits cannot be made dependent on the age an animal has reached in its speciesspecific lifecycle. Vital parameters and performance traits of breeding females such as fecundity, litter size, milk yield, liveweight and liveweight gain are assumed to remain constant throughout the productive lifespan. Additionally, these computer programs often totally lack the capability of optimizing the culling policy that leads to a steady state in herd size and structure, e.g. for LPEC (James & Carles, 1996). The program PRY developed by Bapitst (1992a) allows only the optimization of parts of the culling regime (optimal culling age for surplus animals; cullforage threshold for breeding females), so that one has to rely on heuristic rules to optimise other parts (selective culling of breeding females).
In order to refine the concept of steady state herd productivity assessment and improve its flexibility and applicability, the present chapter explores an alternative method of modelling herd dynamics over time and of deriving optimal steady state herd structure and culling policies. The proposed procedure combines results from population dynamics theory (Caswell, 1989), with methods for the factorial assessment of feed energy requirements of livestock, and basic results from optimal harvesting theory of multicohort populations (Williams, 1989). Herd dynamics are described through a deterministic, densityindependent, discrete time model which predicts subsequent stage distributions of a population of breeding females and their male and female offspring for an initial stage/sex distribution and corresponding sets of stage specific survival and fecundity rates. The optimum sex and stagespecific culling policy is derived by translating the population dynamics model into a nonlinear programming problem. The (nonlinear) function to be maximized is the energetic or ecological efficiency (total gross energy produced per unit of metabolizable feed energy input per time unit) of the herding enterprise. Maximization is performed subject to the constraint that total herd size and structure remain constant over time. Although attention is focussed on grazing ruminant livestock, the method can readily be extended to nonruminant livestock species or intensive husbandry systems.
In the following sections, the theoretical background and the procedure used to assess herd productivity is developed. To make the approach transparent, some basic results from population dynamics theory are presented first, beginning with the formulation and parametrisation of an agestructured matrix population model. Stagestructured matrix population models, which form the basis of the present herd growth model, are introduced thereafter, followed by a description of the procedures applied in calculating herd feed energy requirements and energy outputs. Finally, the optimization procedure used to derive optimal culling policies and stage abundances at the steady state of population dynamics are presented, followed by a discussion of the modelling approach. For a detailed description of a practical application of the proposed methodology, reference is made to Chapter 7.
A population can be defined in general terms as a group of individuals coexisting at a given moment. Demographic analysis proceeds by first focusing on the individual (and its demographic attributes) as a natural unit. The fundamental traits considered at the level of the individual are birth, development, maturation, reproduction and death. Developmental processes cause individuals to differ in these traits depending on their age, size or developmental stage. The latter mechanisms can be used to group individuals in a population into several cohorts, which are formally defined as groups experiencing the same event in a particular time period (Pressat (1985), cited in Carey (1992)). The explicit consideration of individual differences by categorizing organisms according to some characteristic leads to the definition and identification of population structure (Caswell, 1996a). The sequence of transitions from one cohort to another and the duration of residence in each cohort define the life cycle of the organism.
The most obvious and widely used classification of individuals in a population is by age cohorts, where exact or chronological ages are grouped into periods or "classes". Classical demographic analysis is based on an age structured approach to population dynamics and uses a system of tabulating agespecific survival and reproduction known as life and fecundity tables (hereafter referred to generically as the life table). In what follows, density dependent effects on vital rates as well as demographic stochasticity, i.e. deviations from model projections due to finite population size will be ignored. The basic entry in the life table is the survivorship function:
A number of other life table statistics, such as survival and mortality rates, the distribution of age at death, life expectancies etc., can be calculated from the survivorship function l(x)(see Caughley (1977) and Carey (1992) for further details). Reproduction is described by the maternity function:
 [page 109↓] 
m(x) is usually expressed in terms of female offspring produced per female aged x time units, but can be expanded to include male offspring in twosex models. For the sake of simplicity, only the female part of the population is considered here. In order to translate the lifetable results into a discrete time matrix population model, the continuous age variable x must be divided into discrete age classes, i=1, 2,..., k of equal duration. By convention, age class i corresponds to ages i—1≤x<i. For a chosen time step, t, the number of individuals in, or abundance of, each age class can then be projected from one time to the next using a set of firstorder difference equations. Let n(t) represent a column vector of the abundances of the k age classes at time t , and denote by f _{i}, i=1, 2,..., k, the number of newborns (or age class 1 individuals) produced at time t+1 per individual present in age class i at time t. The number of individuals in each of the k age classes at time t+1 are
where p _{i} _{—} _{1} is the survival probability of members of age class i—1 over the time interval [t, t+1]. The system of firstorder difference equations specified by equations (3) and (4) can be written in matrix form
or, more compactly
In the context of ageclassified population analysis, the population projection matrix A is often referred to as a Leslie matrix (Leslie, 1945). Iterative postmultiplication of A by the age abundance or population structure vector n(t) projects the future state of the population.
The parameters in the population projection matrix can be derived from life table data. Before proceeding, we make a distinction between birth flow populations, in which births occur continuously over time and birth pulse populations, in which reproduction is concentrated in a short breeding season. These two patterns of reproduction produce very different distributions of individuals within age classes, and lead to different approximations for the survival probabilities (Caswell, 1989). In what follows, births are assumed to occur at one point during the projection time interval so that the population is of the birth pulse type. Note that this assumption does not preclude overlapping of generations, a typical feature of many longlived organisms. Also, it is supposed that within each time interval the population is censused immediately after reproduction. Such a "postbreeding census" implies that all birth events within the interval [t, t+1] occur at t+1, i.e. at the beginning of the next time period. All newborns (age class i=1) are therefore of age x=i—1=0 at census time.
Since the lifetable functions l(x) and m(x) are continuous, the discrete time coefficients in the projection matrix have to be approximated. The derivation of the formulas presented below for estimating survival probability and fertility entries are given in Caswell (1989). When population census is carried out just after reproduction, the survival probabilities p _{i} in matrix A can be estimated by
Note that every individual in age class i is of the same age x=i—1, so that p _{i} represents the probability of survival from age x=i—1 to age x=i. The birth pulse fertilities f _{i} are estimated using the reproductive output produced by an individual of age i upon reaching its ith birthday, and the probability of surviving until reproduction, p _{i}:
 [page 110↓] 
Thus, the fertility coefficients f _{i} are the expected reproductive outputs per individual of age i in the population during each time interval. In livestock herds, fecundity in fertile females can be assumed to be a function of prolificacy, breeding female survival until parturition, and survival of the fetus until birth.
The agestructured population model considered so far implicitly assumes that properties other than age are irrelevant to an individual’s demographic fate. If vital rates (i.e. rates of survival, growth, and reproduction) also depend on factors other than chronological age, these must either be highly correlated with age or the distribution of age cohorts among the relevant categories must be stable. However, the demography of livestock herds, like that of many other organisms, can depend on size or developmental stage much more than on age, and these variables are often only poorly correlated with age (Caswell, 1989; Getz and Haight, 1989). Situations may also occur in which the age of individual animals is difficult to determine accurately, but other characteristics such as body size, reproductive status, or parity number may be more convenient to measure and more pertinent to questions relating to population dynamics (e.g., Wu and Botkin, 1980; Crouse et al., 1987; Escos and Alados, 1994).
A generalisation of the Leslie matrix model, known as the Lefkovitch population projection matrix (Lefkovitch, 1965), allows the categorization of the life cycle of organisms into life stages other than age classes and the projection of future population states. In the Lefkovitch matrix, stage definitions are not required to be related to chronological age of individuals. The method, however, is very flexible since it allows the consideration of individuals classified both by stage and age. This is important in situations where the vital rates change as a function of life cycle stages and of age within stage categories (e.g., Goodman, 1969; Law, 1983). The fundamental assumption of Lefkovitchtype models is that all individuals in a given category are subject to identical mortality, growth, and fecundity schedules (Crouse et al., 1987).
Figure 6.1. Life cycle graph of breeding females of a hypothetical livestock species.  

A graphical description of a hypothetical stagestructured livestock population is given in Figure 6.1. The structure of the socalled life cycle graph (Caswell, 1989) and the resulting transition matrix depend on the projection time interval chosen. Each node n _{i} (i=1, 2, 3, 4, 5, 6) in the graph defines a stage in the life cycle of the population. Let us assume that the life cycle graph describes the dynamics of the female portion of a livestock herd and that breeding females are mated once a year, starting at the end of the juvenile stage numbered 1, and that all individuals are culled (or die) after giving birth a second time upon reaching stage 6 (recall that in a birth pulse population all breeding and birth events within an interval [t, t+1] occur at t+1). Also, the projection time step is assumed to be equal to the one year duration of pregnancy. Then, an arc connecting two nodes n _{i} and n _{j} indicates that an individual in stage i at time t can contribute (by development or reproduction) individuals to stage j at time t+1. Selfloops are of unit time length and indicate that individuals in that stage at time t can contribute individuals to the same stage at time t+1. The coefficients labelling the arcs define the number of individuals to be observed in stage j at time t+1 per individual present in stage i at time t.
Upon reaching reproductive maturity at, for example, 3 years of age, stage 1 individuals may, conditional upon survival, either conceive and grow into pregnancy stage 2, or remain empty and move into stage 3. (Note that all juvenile females are assumed to be used as replacements. How to take into account the decision to rear part of this cohort as surplus animals will be discussed further below). For each stage 2 and 4 individual a number f _{1} and_{} f _{2} of young is expected to be born at the end of the year (note that the survival of [page 111↓]dams enters into the calculation of the f _{i}'s). At the beginning of the following year, immediately before census time, the surviving stage 2 and 3 individuals breed again and, if they conceive, make the transition into stages 4 (for stage 2 individuals) or 2 (for stage 3 individuals). Alternatively, stage 2 and 3 individuals may survive without becoming pregnant and therefore grow into stage 5 or remain in stage 3, respectively. It is assumed here that animals which are susceptible to pregnancy (stages 3 and 5) have to remain at least until the next breeding season in the same stage before they can conceive and move into one of the two pregnancy stages. The average residence in stages 3 and 5 is thus determined by the probability of conception. Individuals in stage 4 survive until the next year with a probability p _{4}, become stage 6 individuals upon giving birth, and are culled (or die) thereafter.
The G _{i} and P _{i} coefficients in Figure 6.1 are transition probabilities defining the probability of surviving and growing into the next stage class, and of surviving and remaining in the same stage, respectively. These depend upon the stage durations, d _{i}, and the stagespecific probabilities p _{i} of surviving from time t to t+1. Except for the probabilities of surviving and remaining in the same stage (P _{i}) and the probability of surviving into stage 6 (p _{4}), all transitions in the graph are conditioned upon conception (β_{i}), so that the final joint transition probabilities are obtained as the product of the marginals, i.e. β_{2} p _{2}; (1—β_{2})p _{2}; β_{i} G _{i}; and (1—β_{i})G _{i} (it is assumed here that the marginal probabilites are independent of each other). Since no selfloop is present at stage 2, in this case transitions are only a function of survival and conception (β_{2} and p _{2}). As before, the f _{i}'s represent reproductive outputs, i.e. the expected number of offspring produced per individual in stage i at time t and observed at censusin stage j at time t+1. The stagestructured transition model depicted in the life cycle graph can be translated directly into the population projection matrix A in equation (5). The projection matrix corresponding to Figure 6.1 is
To summarise, the main differences between age and stagestructured matrix population models are that the stage classes may differ in their duration and that individuals may also remain in the same stage from one time to the next. In order to parameterise the stagebased matrix model, the single time step survival probabilities p _{i} and the fertility coefficients f _{i} can be estimated from longitudinal stagespecific survival and fecundity data. A different approach has to be adopted for estimating the marginal stage probabilities of surviving and growing into the next stage (G _{i}), and of surviving and remaining in the same stage (P _{i}), since they depend on stagespecific survival rates and on the duration of each stage, d _{i}. The stage duration itself can be influenced by physiological or sexual maturation processes. For instance, sexual maturation determines the duration of the juvenile stage (assumed to last 3 years), d _{1 }, in Figure 6.1. For the hypothetical population dealt with here the stage duration is fixed and corresponds to the length of three time steps (3 years) for juveniles, and one time step for all other stages having selfloops. The latter is true because one time step was assumed to be equal to both the duration of pregnancy and the time period between successive breeding seasons. In general, when the stage duration can be assumed to be fixed the transition probabilities P _{i} and G _{i} can be approximated as (Crouse et al., 1987; Caswell, 1989):
Since d _{i}=1 for i=3, 5 in Figure 6.1, the transition probabilities for these stages reduce to
 [page 112↓] 
As with the agestructured population model presented above, numerical projection by repeated matrix multiplication is the simplest form of analysis that can be performed with this type of demographic model. However, projecting a population into the future is of limited value, since the stage abundances at any time will dependent on the initial conditions specified by the stage abundance vector n(t _{0}). More general and widely applicable conclusions can be drawn by applying analytical approaches based on matrix algebra. Eigenanalysis provides expressions for calculating the stable stage distribution, the finite rate of increase of the population, and the reproductive value of an individual in a given stage (see Caswell (1989) and Cochran & Ellner (1992) for further details).
When the projection matrix A is nonnegative, primitive, and irreducible, then it satisfies the necessary conditions for the PerronFrobenius theorem (see Caswell (1989), Chapter 4). According to this theorem, a matrix having these properties has at least one positive latent root. The largest such root, usually denoted by λ, is known as the maximal root or dominant eigenvalue of the projection matrix A. The dominant eigenvalue is a scalar value which expresses the multiplicative effect of the projection matrix A on the stage abundance vector n(t), and thus gives the asymptotic rate of population growth. The right eigenvector w corresponding to λ is defined by
and represents the stable stage distribution to which the population will ultimately converge. Independently of the initial stage structure the population abundance vector n(t) will, after a sufficient number of projection time steps, approach the stable stage distribution w, where each stage class increases in size by a factor equal to λ each time period. The reproductive values of the stage classes are given by the left eigenvector v corresponding to λ. The left eigenvector satisfies the equation
The reproductive value of a stage class is a measure of the potential contribution of an individual in that stage to future population growth, and is a function of the amount of future reproduction, the probability of surviving to realize it, and the time required for the offspring to be produced (Caswell, 1989). For comparative purposes the elements of the right eigenvector w are rescaled so that they sum to 1 and express the proportional abundance of individuals in each stage class at equilibrium. Likewise, the reproductive values in v are usually expressed in relation to the newborn stage whose reproductive value is set to 1.
An important part of the analysis of the projection matrix is to investigate how the finite rate of increase λ would be affected by changes in the vital rates or, alternatively, how important each stage class is in determining population growth rate. These information are valuable for assessing the impacts upon λ of errors in estimation, alternative management strategies, and environmental perturbations (Crouse et al., 1987; Caswell, 1989, 1996b). The sensitivity of λ to a change in the a _{ij}th element of A when all other elements are held constant is
where v _{i} and w _{i} are the ith element of the reproductive value vector and jth element of the stable stage distribution vector, respectively, and <w,v> is the scalar product of the two vectors. The sensitivity of λ gives the effect of a small additive change in one of the vital rates. The effect of a small proportional change in a vital rate can be assessed through the elasticity of λ:
The e _{ij} sum to 1 and also measure the proportional contribution of each of the matrix coefficients to overall population growth rate (Caswell, 1996b).
For analytical purposes, a number of useful ageclassified statistics can also be derived for stagestructured matrix population models (see Cochran and Ellner, 1992). These comprise agebased life history traits such as the mean age at first reproduction, the probability of surviving until time at first reproduction, and the net reproductive rate, as well as agebased measures of population dynamics such as the generation time. The computational formulas for these parameters are given by Cochran and Ellner (1992). The mean age at first reproduction is the average age at which a newborn individual enters a stage with positive fecundity. It can be conceived as an average over a cohort of newborns which do not all necessarily follow the shortest possible path to a stage with positive fecundity (e.g., individuals may stay for several time steps in stage 3 in [page 113↓]Figure 6.1). The net reproductive rate, usually denoted by R_{0}, is the expected number of offspring produced by an (female) individual over its lifespan. And the generation time can be defined as the mean age of the parents of offspring produced in the current time period, once the population has reached stable stage distribution.
So far, the exposition has concentrated on describing the approach used to model the dynamics of the breeding female part of a livestock herd. Principally, incorporating stages for female and male surplus animals into the lifecycle graph in Figure 6.1 is straightforward. Since surplus animals have by definition zero fertility they form sequences of stages with only oneway communication, i.e. there is no pathway back to the part of the life cycle that does reproduce. Therefore, it is relatively easy to describe the dynamic behaviour of the surplus part of a herd, and both stage and/or age classified matrix methods for population analysis can be used to model the dynamics of surplus cohorts over time, as demonstrated
Figure 6.2. Modified life cycle graph of a hypothetical livestock species taking into account male juveniles (stage 1m) and female and male surplus stages (2f and 2m). The parameter σ denotes the fixed proportion of female immatures that are to be reared as breeding females. A sex ratio of unity was assumed at birth.  

in Figure 6.2. In order to be able to incorporate a female surplus cohort into a matrix population model, one has to introduce an additional parameter, say σ, which represents the fixed proportion of females surviving the juvenile stage which are to be reared as breeding females. This introduces an element of decisionmaking which usually is absent in conventional applications of matrix population models. Further below, the decision to rear female youngstock as breeding females or surplus animals will explicitly be taken into account in the derivation of steady state optimal culling policy and herd structure. Note, however, that the analytical approaches based on Eigenanalysis described above cannot be applied to such a transition matrix, since it necessarily contains stages that make no contribution to some other stages, and hence is reducible.
For the purpose of illustrating the calculation of total feed energy required at the herd level per projection time unit, only the body functions maintenance, growth, lactation, and pregnancy are considered here. However, the principles behind the formulas presented can easily be extended to other functions such as draught power or wool and hair production. Ideally, total forage consumed by the herd per time unit, as well as quality of forage on offer should be measured in the field as a basis for calculating the input in terms of [page 114↓]metabolizable energy (ME) utilized for achieving an observed level of performance. This, however, seems to be impracticable and thus an alternative approach has to be devised. James and Carles (1992) proposed to estimate total feed intake in terms of metabolizable energy indirectly, by using standard ration formulae to calculate the quantity of metabolizable energy animals must have obtained in order to achieve the observed level of performance. This procedure can be adopted in the present context to estimate feed energy requirements per time unit of livestock herds.
In assessing feed energy requirements, an account has to be taken of the fact that not all individuals present at the beginning of a time interval survive until its end. ME requirements per animal over a single time interval have therefore to be corrected for mortality. This requires the knowledge of survival curves for each life history stage. Then, an appropriate assessment of stagespecific ME requirements for each body function over the interval [t, t+1] consists in summing together the daily requirements for each body function for all days a within [t, t+1], weighted by the probability of surviving until day a, t≤a≤t+1. However, if necessary the projection time interval can also be devided into intervals of length l, covering several days or weeks. The expected ME requirement for maintenance, growth, lactation, and pregnancy R _{ij}, of an animal in stage i at time t over the projection time interval [t, t+1], of length L days, can be calculated as
where a = is the number of the time interval of length l (in days) since the start of interval [t, t+1], with 1≤α≤L/l. r _{ij}(a) = is the mean ME requirement of an animal in stage i over period a for the jth body function. p _{i}(a) = is the conditional probability that an individual in stage i alive at the start of time period a within [t, t+1] survives until the end of a. j = maintenance, growth, lactation, or pregnancy. 
The r _{ij}(a)'s have to be computed from stagespecific growth and lactation curves. The time axis of each curve first must be discretized into n consecutive time periods. Stagespecific average daily values for body weights, body weight gains, and lactation yields (Y _{ij}(a)) within each period a can for example, be obtained by linearly interpolating estimates at the start and end of the period:
Note that interpolation becomes necessary only when l>1day. Finally, the mean ME requirement of an animal in stage i over period a for the jth body function can be calculated with the following formula:
where ME _{j} = is the daily ME requirement per unit of body function j. 
A slight modification of equation (15) is necessary for stages with selfloops, since here, stage durations exceed the length of one projection time step. The effect is that stage abundances at the start of each time interval consist of a mixture of individuals recruited from other stages and of individuals that survived the previous time step and remained in the present stage. However, the expected ME requirements over[t , t+1] for an individual present at time t can be approximated from total ME requirements for the entire stage duration and the probability of surviving and remaining in the same stage, P _{i}:
where a' = is the number of the time interval of length l (in days) since the start of stage i,with associated stageduration d _{i}, and 1≤ a‘≤ d _{i}/l. r _{ij}(a') = is the mean ME requirement of an animal in stage i over the a'th period for the jth body function. p _{i}(a') = is the conditional probability that an individual in stage i alive at the start of time period a' within d _{i }survives until the end of a'. j = maintenance, growth, lactation, or pregnancy. 
 [page 115↓] 
As before, equations (16) and (17) can be used to compute the r _{ij}(a)'s. Thus, on average, a fraction of (1P _{i}) of the expected ME requirements per stagemust be available, over a single projection time interval, to animals in stages with positive probability of remaining in the same stage in subsequent time periods.
Energy released from mobilization of body reserves can be taken into account when calculating total ME requirements for each life history stage per time unit. In grazing livestock, substantial weight losses usually occur soon after parturition, but are also common in situations of low or declining quantity and quality of forage on offer. Similarly, the expected amount of energy available to youngstock from milk consumed until weaning could also be accounted for in calculating total metabolizable feed energy requirements per progeny present at the beginning of the juvenile stage. Both of these quantities would effectively reduce the calculated total amount of ME from feed resources that must have been available to cover observed maintenance, pregnancy, and/or lactation requirements.
To simplify the presentation, herd output is solely defined in terms of total meat and milk offtakes for human consumption per projection time interval, valued at their gross energy contents. Details of the method for deriving optimal culling policies are given in the next section below. For the purpose of describing the method of estimating energy contents in outputs, suffice it to say here that live animal offtakes by stage class, u _{i}(t), are assumed to take place at the beginning of each time interval. For each life cycle stage, body weight estimates for animals present at the beginning of each time step are needed.
Using gross energetic values for protein (GE _{P}) and fat (GE _{F}), the following formula is a simple method for estimating gross energy yield per live animal offtake in life cycle stage i
where EBWT _{i} = is the empty body weight (kg) of an animal at the beginning of stage i. GE _{EBWT},_{i} = is the gross energy yield in empty body per live animal offtake in stage i. PPFAT _{i} = is the proportional fat content per kg EBWT _{i}. PPROTEIN _{i} = is the proportional protein content per kg EBWT _{i}. 
Similarly to the assessment of energy requirements, milk yield and milk offtake per animal present at the start of a time interval [t, t+1] in lactation stage i has to be adjusted for mortality. Upon making appropriate assumptions with respect to the dry matter content and energetic value per unit weight of fresh milk, energy yield from milk offtake per animal present at the start of the time interval [t, t+1], in lactation stage i,can be assessed as
where a' = is the number of the time interval of length l (in days) since the start of lactation stage i,with associated stageduration d _{i}, and 1≤a‘≤d _{i}/l. y _{i}(a') = is the total milk yield per animal in stage i in period a'. p _{i}(a') = is the conditional probability that an individual in stage i alive at the start of time period a' within d _{i }survives until the end of a'. O _{m} = is the milk offtake rate per projection time period. P _{i} = is the probability of surviving and remaining in the same stage i at the beginning of the next projection time period. 
 [page 116↓] 
The mathematical programming model for evaluating steady state culling and replacement policies, as well as overall productivity in livestock herds, is based on the matrix population model described above. The problem of finding the best combination of offtake decisions for each of the stages in the life cyclethat maximizes a given index of performance, and satisfies a specific set of constraints, falls into the realm of population harvesting theory (see Getz and Haight, 1989, for a comprehensive exposition of this subject). Both age and stagestructured matrix population models have been widely used to address questions relating to optimal harvesting of populations (e.g., Usher, 1966; Beddington and Taylor, 1973; Doubleday, 1975; Rorres, 1976, 1978; Magshoodi and Grist, 1995; Forsberg, 1996; Jensen, 1996). Doubleday (1975) was the first to show that, under certain conditions, harvesting problems in matrix population models are essentially linear, and can therefore be formulated as standard linear programming problems.
Within the framework of steady state herd growth modelling dealt with here, the management control (i.e., removing or culling individuals from the herd) imposed on herd dynamics must be such that herd size and structure remain unchanged, or in a steady state, over time. Subject to this constraint, a stagestructure and culling policy may be chosen to maximize a specific index of performance. Since the matrix population model describing herd dynamics is discrete in time, it is necessary first to specify exactly when the removal of individuals occurs with respect to the process of reproduction and stage transition. Two culling schedules are available; the first is to remove animals before reproduction, and the second to cull animals immediately after they have reproduced. With respect to the discrete time scale, prereproductive culling occurs immediately after the start of a new time interval at time t, whereas postreproductive culling occurs at the end of a time interval, just prior to time t+1.
As has been shown by Doubleday (1975), postreproductive culling is generally more efficient than prereproductive culling, since in the former case individuals are allowed to reproduce just before they are removed from the herd. However, for the specific life cycle of grazing animals considered here, a postreproductive schedule would lead to unreasonable management policies. For instance, it would be possible to remove breeding females at the end of a pregnancy stage, just after parturition. Therefore, in what follows a postreproductive culling regime is assumed to be imposed.
Let us first suppose that we are concerned with the question of choosing a set of offtake strategies to maximize a given objective functional over some time period [0, T]. Then, a culling policy consists of a sequence of offtake decisions u(t) made in each time period [t, t+1], so that the dynamics of the herd can be described by
Where n(t) and u(t) are respectively, a kdimensional column vector of stage abundances, and a kdimensional column vector specifying the number of individuals removed from each life cycle stage i (i=1, 2, ...k) at time t. A denotes the stage transition matrix as defined, for example, by (8). For consistency, it is required that the number of animals removed from each stage be less than or equal to the stage abundance:
In (22), offtakes are constrained to be nonnegative, thus implying that we are considering a closed herd without, for instance, the possibility of introducing animals from external sources into the herd. Further below, this condition will be relaxed. The next step is to formulate an appropriate objective function relating to the management of a herd modelled by equation (21). Generally, the performance index, I, in each time period will be a function of both stage abundance and offtake at time t:
Since the problem is a dynamic one, the objective functional will then consist of maximizing the sum of individual rewards I(n(t), u(t), t) obtained in each time period over the planning horizon T:
In summary, the optimization problem may then be written as
 [page 117↓] 
The value of J in this problem depends on the length of the planning horizon T, and on the initial stage distribution vector of the herd, n _{0 }(Mendelssohn, 1976; Getz and Haight, 1989). The dependencies on T and n _{0} may be removed upon imposing further constraints on the stage abundance vector n(t). For example, an additional management objective might be to keep total herd size and structure at the same level for all t∈ [0, T], which is also the condition required to model steady state herd productivity as defined previously, so that
In fact, upon introducing this steady state condition, the dynamic optimization problem (25) reduces to a static one, in which the single period optimality index I(n(t), u(t), t) is to be maximized (Williams, 1989). Furthermore, the dependence upon the initial stage distribution can be eliminated by requiring that the herd be kept at a stationary population level c:
The optimization problem becomes
Note that this problem can only be solved for u(t) if A has a dominant eigenvalue that satisfies λ≥1. The solution, or optimal stationary culling policy, u* to the optimization problem is such that the herd is kept in stationary state with respect to herd structure and size (Doubleday, 1975; Williams, 1989). In addition, the problem can be solved by standard linear programming methods, as long as the objective functional I[n(t), u(t), t] is linear. The formulation in (28) is thus a straightforward way of determining simultaneously the optimal culling policy that maximizes a given performance index and the associated stable stage distribution of the herd that will be achieved at a specific level of total herd size.
The more general type of constraints as in (27) allows the taking into consideration of such things as input resource constraints, production costs, stages from which no animals are to be removed, minimum offtake levels from specific stages, and so forth. Rorres (1976) has shown that these constraints are crucial in determining the structure of the optimal offtake vector u*. Particularly, if a number q, q=1, 2,...., m, 0≤m≤k, of constraints of the type
(where the d _{i} and e are a given set of k coefficients and a given scalar value, respectively) are imposed, at most q of the k stages will be partially culled. For all other stages, either all individuals present will be removed or none at all.
 [page 118↓] 
The general problem formulation in (28) can be extended to allow for controlling whether an immature female is to be reared as surplus or as breeding female replacement. For an immature female, let us suppose that this decision is made when it has survived to the end of the immature stage, denoted by stage subscript i=1 f in Figure 6.2. Let s denote the number of immature females surviving to this time point which are to be reared as surplus animals, and consider the following (abridged) transition matrix describing the dynamics of the female part of the herd only:
The second stage (second row and column in A) represents the first surplus animal cohort. In contrast to the lifecycle graph in which the parameter σ causes a fixed proportion of stage 1 f animals to move into stage 2 f, the structure of transition matrix A above is such that there is no pathway into the surplus stage. Let us assume that the transition probability from the juvenile into the first surplus animal stage is the same as that from the juvenile into the first pregnant or barren stage. The matrix element at the intersection of the second row and the first column of A in (30), which is equal to zero, would then be replaced by G _{1f}. Then, the following modification of the system dynamics equation (21) introduces the decision to rear a number s(t) of immature females as surplus at time t:
where in the present example the diagonal matrix D has the form
and the vector s(t) has components
The structure of matrix D and vector s(t) depend on the lifecycle graph and the time point at which the [page 116↓]decision to rear immatures as surplus stock is made, and do not need to take on the specific form given here.
The objective function used to evaluate herd productivity is an efficiency ratio, defined as the total gross energy output of the herd devided by its total metabolizable feed energy requirements at the stationary state. Based on equations (15), (18), (19), and (20), the objective function is given by
where index values i=1,2,...,h1 refer to stages other than female surplus stages, i=h indexes the first, and all other values i>h the subsequent female surplus stages. Note that when the optimal solution is such that s(t) is set to zero, all remaining female surplus stages must also have zero abundance in order to satisfy the herd dynamics (31) and steady state (26) equations. To summarize, the herd productivity assessment model can be stated as
for a production system in which the decision to rear immature females as surplus stock is made at the end of stage i=i*. Due to the nonlinearity of the objective function, this formulation is a nonlinear programming problem, which can readily be solved using any standard nonlinear programming software. However, problem setup and editing is greatly facilitated when using a spreadsheet software, in combination with a nonlinear programming addin package such as the Premium Solver (Version 2.0, Frontline Systems, Inc., 1996).
The nonnegativity constraint on offtake numbers can be relaxed in cases where the dominant eigenvalue of the transition matrix is λ<1, i.e. when the herd is not sustainable due to low survival rates in youngstock or breeding females, and/or insufficient reproductive performance of breeding females. As mentioned before, in these cases the problem (35) cannot be solved. In order to maintain herd size, animals have to be imported, either through purchases or borrowings. In general, herd decline is linked to insufficient availability of breeding female replacements to maintain the size of the mature breeding stock. The nonnegativity constraint on offtake numbers can be relaxed to allow for an inflow of female replacements into the herd. In order to account for the capital costs involved, each female replacement animal added to the herd could be valued at the total metabolizable feed energy input required to raise one surviving female replacement, based on the following formula:
where subscript s denotes the stage at the end of which immature females are reared as replacements, and other parameters are defined as in (15).
 [page 120↓] 
Herd productivity assessments and energetic efficiency
This work has favoured the use of energetic efficiency as a choice criterion because it provides a common basis on which biological productivity can be assessed and compared among different treatments, management strategies, or among husbandry systems operating in different ecosystems. Clearly, productivity can be measured in many different ways, depending on the type and quantification of input and output flows considered. For purely descriptive purposes, the attractiveness of energetic efficiency stems from its close relationship to the notion of feed conversion efficiency as used in livestock production contexts, as well as from its possible interpretation as a measure of foodchain efficiency (i.e., energy consumption by trophic level n, divided by energy production by trophic level n—1), which may be used to identify the physical or biological barriers to higher productivities (Duckham, 1971; Western, 1982). Another advantage is that different livestock products such as meat, milk, and draught power can readily be aggregated in energetic terms, while monetary evaluation of the same products may prove to be difficult when dealing with subsistence producers, or when assigning market prices to livestock products is rendered difficult by the presence of noncompetitive markets. The same is true with respect to important production inputs.
However, an important point to remember is that the use of technical efficiency measures is founded on the notion of certainty and is most often employed in a profit maximizing sense (Anderson et al., 1977). James and Carles (1996) for example explicitly utilize this frame of reference for livestock productivity assessments in arguing that “the most general objective of grazing livestock production, at both farm and national levels, is to obtain maximum economic margin from the available forage resources” (p. 273). Serious doubts may be raised about the general validity of this assumption. Firstly, a characteristic common to all agricultural and livestock production systems, especially when they operate under dryland conditions, is the pervasive influence of stochastic factors on the future state of the system, and thus on outcomes of decisions. In the presence of such uncertainty, available empirical evidence suggests that farmers and livestock producers will tend to be risk averse in making decisions (Anderson and Dillon, 1992). Secondly, in pursuing livestock keeping activities producers may derive other benefits in addition to the production of goods such as meat, milk, draught power and live animals for sale or home consumption. Livestock may also play an important role as a source of financing and insurance, particularly in situations where financial markets are imperfect and opportunities for risk management through formal insurance are not available (Bosman et al., 1997a). Hence, when there is considerable uncertainty about the consequences of an action and, at the same time, producers pursue multiple, possibly conflicting goals, it appears to make little sense to assume an unqualified goal of expected profit maximisation (Anderson et al., 1977). Certainly, these considerations will apply to many traditional livestock production systems in developing countries.
In essence, relative to the task of productivity indexing of livestock production systems the important point to be made here is that recommended improvements to existing management practices that aim at increasing livestock productivity must be consistent with the decisionmaker’s beliefs and preferences. In other words, the most energetically efficient management alternative is not necessarily the one that has also to be preferred by livestock producers. Maximisation of energetic efficiency can help identify the management alternative with the highest biological potential, but this alternative will not necessarily correspond to the choice which maximizes the decisionmakers’s subjective expected utility. Livestock producers may continually have to trade off the various benefits derived from their herds when making management decisions. For instance, as pointed out by Bosman et al. (1997a), insurance and financing goals may induce producers to keep ‘unproductive’ animals in the herd, thereby reducing biological productivity.
Representation of herd dynamics and derivation of optimal culling regime
The general methodology for assessing biological productivity at the herd level developed in the present work is basically an extension of that proposed by Baptist (1992b). While this author assumes that the same inherent survival, fecundity, development rates, as well as yield levels are applied to all breeding and (except, of course, for fecundity rates and milk yields) surplus females alike, the stagestructured population dynamics model employed here is much more flexible since it permits the modelling of life histories in which vital rates and production traits vary with age, stage, or a combination of both factors. Generally, the appropriate definition of relevant lifecycle stages is entirely dependent on the species and on the production context considered. Perhaps most importantly, it has been shown that assessing biological herd productivity can be formalised in a nonlinear programming model which determines the optimal stagespecific culling policy that maximizes the chosen biological productivity criterion, subject to the constraints that herd size and structure remain constant. Of course, the choice of the objective criterion to be maximized is not limited to the ecological efficiency index employed here, but could as well be an economic one. For instance, if all [page 121↓]outputs considered were valued in monetary terms, the productivity criterion would represent the economic value of production obtained per unit feed energy input and time (James & Carles, 1996). A major distinction of the proposed methodology compared to other available procedures for the assessment of herd productivity at the steady state of population dynamics resides in the fact that it can readily be used for carrying out a full economic activity analysis of livestock operations. While the computer programs LPEC and PRY mentioned above are fairly flexible with respect to the definition of the productivity criterion, they cannot explicitly take into account the objectives, decision alternatives, technical and resource constraints, as well as managerial limitations that typify the herding enterprise considered.
For comparative studies, the importance of adopting an optimality approach in order to obtain a common basis on which alternatives can be compared with respect to their performance in terms of the chosen productivity criterion must be emphasized. Other authors, such as James and Carles (1996) and Upton (1989, 1993), suggested to use observed or estimated offtake rates for a given management alternative or production system and they did not attempt to simultaneously optimise herd structure and offtake rates with respect to a specific performance criterion. Clearly, a distinction has to be drawn between evaluating potential biological (or economic) productivity, and the level of productivity that can be achieved under the prevailing management practices observed in a specific production system, particularly with respect to offtake rates. Herd productivity is greatly affected by the choice of culling policy, not only directly through the removal of individuals, but also indirectly through its impact on herd structure. Of course, herd management affects herd productivity not only through offtake decisions but, if potential herd productivity is to be assessed, at least this major source of variation should be controlled for in comparative assessments. Productivity assessments may be biased downwards if they are based on observed or estimated, suboptimal offtake decisions. An additional motivation to use an optimality approach to determine optimal culling policy and herd structure stems from the fact that accurate data on offtake rates by animal category and time unit are generally difficult to obtain in production system studies.
Although the relevance of selective culling rates for the determination of productivity levels in livestock herds has clearly been recognised by Baptist (1992b: 265), his simulation approach based on life tables brakes down with respect to the identification of optimal culling policies for breeding females, because “too many combinations of different ages are possible”. Indeed, the different ways in which herd composition can be changed is virtually infinite and therefore all possible strategies cannot be compared using simulation runs. Whenever a key measure of economic or biological performance can be nominated, applying an optimisation algorithm is a far more powerful approach that enables the optimal policy to be precisely calculated. Note that the general type of linear control system formulation for multicohort herds adopted in this work has previously also been proposed by Upton (1989). Surprisingly, however, he did not envisage to apply an optimisation algorithm for the simultaneous derivation of the optimal steady state culling policy and herd structure. Similarly to the above cited statement made by Baptist, he argued that the principle difficulty resides in the fact that there are too many different possible combinations of herd structures and offtake rates that need to be compared, and concluded that "selection of a particular solution is a question of judgement" (Upton, 1989:161). The present work has shown that more rigorous methods are available to circumvent this problem.
Alternative model parameterisations
In the example used to illustrate the developed herd productivity assessment procedure, it was assumed that the hypothetical livestock herd is of the birth pulse type, i.e. that reproduction is concentrated in a short breeding season. This requires specific approximations for model parameters which may not hold for continuously reproducing livestock species or aseasonally managed livestock herds. Caswell (1989) gives approximations for birthflow survival probabilities and fertilities. Alternatively, continuous reproduction in domestic livestock herds can be accomodated for by making appropriate changes to the corrresponding lifecycle graph, based on the following reasoning. The major difference between an aseasonal and a seasonal breeding regime resides in the length of the time interval between successive mating events. Whereas in its simplest from controlled breeding logically implies a one year breeding interval and thus a fixed time period between successive mating events, this parameter may vary considerably under unrestricted breeding. It follows that in the aseasonal case birth intervals are a function of both the probability of conception and the timespan between consecutive mating events. For fertile breeding females the latter parameter determines when, on average, the first postpartum service occurs. It also determines the average waiting time until next service for breeding females that failed to conceive.
In general, the shortest period from birth to conception corresponds to the time until occurrence of the first oestrus postpartum. Expected birth intervals for various combinations of values for species specific conception rate and period between successive matings can be calculated in order to identify a reasonable value for the latter parameter. The following formula can be used to estimate average birth intervals (T _{b}) [page 122↓]based on conception rate (β), gestation period (T _{g}), and period between successive mating events (T _{m}):
where n denotes the projection time step. It is assumed here that all parameters are population averages and remain constant over time. Depending on the determined value of T_{b}, a projection time step of less than the length of the gestation period may be required to model the dynamics of the livestock species of interest. While the general methodology does not impose any restrictions in this respect, such a choice may cause a significant increase in model complexity in terms of the number of different stages and associated model parameters that need to be estimated.
The second important assumption that was made in this work concerns the duration of individual stages in the life cycle, which affects the way stage transition probabilities are approximated. Stage duration was assumed to be fixed, and situations may arise where this assumption is considered inappropriate for an accurate representation of population dynamics. Several alternative approximations for stage transition probabilities, which depend on the withinstage age distribution and the distribution of stage durations among individuals, were presented by Caswell (1989).
Sensitivity analysis
A valuable feature of the procedure for productivity indexing employed in this work is the possibility of conducting sensitivity analyses. Sensitivity analysis serves two main purposes. Firstly, as tools for model assessment sensitivity calculations may help in identifying critical components of a model and can be used for model validation. Generally, particular emphasis should be given to the appropriate specification of critical parts of the model structure and to accurately estimating parameters to which model results are very sensitive. Secondly, sensitivity analyses are important management tools which can help to indicate the type of controls that should be applied in order to most efficiently bring about desired changes in system response.
In the present setting sensitivity analyses can be carried out in two different ways. One possibility is to conduct a sensitivity analysis within the nonlinear programming framework. Here, the most basic form of sensitivity analysis information is provided by the dual values or shadow prices of binding constraints, but these are of little interest in situations where only steady state constraints with respect to herd size and structure are included in the optimization models studied. However, sensitivity of model results to individual parameters can be checked by rerunning the original model consecutively after gradual changes in a single parameter have been made over some reasonable range. Simultaneous changes in several exogenous model parameters such as mortality, fecundity, and developmental rates can also be investigated through parametric programming (Hillier and Lieberman, 1986).
Alternatively, one can conduct asymtpotic analyses on the transition matrix for the female part of a livestock herd and investigate its longterm dynamics. Because the dominant eigenvalue, λ, and the rigth and left eigenvectors, w and v, are properties of the matrix entries (i.e., reproductive, survival, and developmental rates) rather than initial conditions, they can be used as demographic statistics. The computation of eigenvalue sensitivities and elasticities pertains to perturbation analysis, a special form of sensitivity analysis that examines the impact of changes in vital rates on asymptotic population growth. In livestock herds, the asymptotic growth rate determines potential offtake rates and therefore is an important variable affecting overall biological herd productivity. An important advantage of elasticities of λ with respect to individual matrix entries is that they sum to one and thus can help to identify those vital rates to which λ is most sensitive. For instance, this is a property that is not shared by sensitivity analyses carried out through parametric programming. Elasticities can also be calculated with respect to parameters other than matrix entries themselves, but in this case they also do not sum to one and cannot be interpreted as contributions to asymptotic population growth rate. Note that the interpretation and utilization of eigenvalue elasticities in matrix population models is analogous to the concept of elasticity as used in microeconomic theory. Using matrix algebra, eigenvalue elasticities are easy to calculate once the eigenvectors of the projection matrix are known (Caswell, 1996b). Cochran and Ellner (1992) have made available a FORTRAN 77 program which calculates a number of measures for the analysis of population projection matrices, including the complete eigenvalue spectrum as well as other, agebased life history parameters. The same authors also suggested the use of bootstrapping methods to make rigourous statistical comparisons of these parameters.
To summarize, the information that can be derived from a stagebased demographic description using matrix [page 120↓]population models appears to be much richer than that usually obtained from conventional sensitivity analyses such as those proposed by Peacock (1987) or Upton (1985, 1989). The latter authors proposed to separately vary (vital) parameters by one standard error and to investigate the effect of such a change on outcomes of interest. This, however, is a rather large change in parameter values. In order to be valid the entire approach must be based on the unrealistic assumption that all model parameters are independent from each other. In contrast, the concept of eigenvalue sensitivity is based on infinitessimal parameter alterations, and under these conditions the independence assumption is much less likely to be violated.
The approach to steady state herd productivity assessment proposed in this work is very general and can easily be applied to a wide range of domestic livestock species and production systems. Stagestructured matrix population models can be generalized to complex life cycles in which individual animals can be classified by several factors other than age. The proposed method represents a readily accessible tool for constructing and analysing herd dynamics models that reflect important biological factors determining survival and reproductive rates in domestic livestock species. In many situations, these vital rates depend on size, developmental, or reproductive stage much more than on age. The use of stagestructured matrix population models in conjunction with nonlinear programming for the derivation of the optimal steady state herd structure and culling policy removes most limitations of previously published procedures for steady state herd productivity assessment. This includes the possibility of taking into account stagedependent vital rates and production performances, and the determination of optimal yield and stagespecific selective culling rates. Compared to the work of James and Carles (1996) and that of Baptist (1992a), an attractive feature of the proposed optimality approach to steady state herd productivity assessment is that it reduces the influence of subjective judgements on model results, an element which certainly is involved when using heuristic culling rules, or observed or estimated offtake rates. Furthermore, the entire procedure is easily implemented in any standard commercially available spreadsheet program. The only specialized software required is a nonlinear programming addin package for solving the optimisation model.
© Die inhaltliche Zusammenstellung und Aufmachung dieser Publikation sowie die elektronische Verarbeitung sind urheberrechtlich geschützt. Jede Verwertung, die nicht ausdrücklich vom Urheberrechtsgesetz zugelassen ist, bedarf der vorherigen Zustimmung. Das gilt insbesondere für die Vervielfältigung, die Bearbeitung und Einspeicherung und Verarbeitung in elektronische Systeme.  
DiML DTD Version 3.0  Zertifizierter Dokumentenserver der HumboldtUniversität zu Berlin  HTML generated: 08.10.2004 