 [page 125↓] 
A major constraint on biological productivity of grazing livestock under semiarid conditions in northern Kenya is the marked seasonal fluctuations in the availability and quality of rangeland resources (Schwartz & Schwartz, 1985). With the increasing trend towards sedentarization of pastoral herds and households, the search for an alternative to herd mobility as the most effective adaptive management strategy to attenuate the effects of seasonality in production resources is an issue of growing concern (Roth, 1995). Continuous breeding throughout the year, which is made possible due to the nonseasonal reproductive behaviour of local goat breeds, is typical for pastoral goat flocks in northern Kenya. However, this strategy tends to produce poor survival as well as reproductive and productive performances whenever late pregnancy and birth fall into periods with suboptimal forage availability (Wilson et al., 1985; Delgadillo & Malpaux, 1996; Mellado, et al., 1996; WalkdenBrown & Restall, 1996). In general, an aseasonal breeding regime implies that nutrient requirements of the flock remain relatively constant over the year. With the transition of pastoral households and their herds to sedentism, a significant increase in the likelihood of the occurrence of serious imbalances between peak nutritional requirements of livestock herds and seasonal feed supplies can be expected.
The results of an experimental study conducted in northern Kenya on the effects of controlled seasonal breeding on biological performance traits of pastoral goat herds has revealed that such a management practice could effectively reduce the impact of fluctuations in nutrient supply on youngstock survival, mainly through concomitant improvements in birth weights and milk production of dams (Chapters 3 and 5). In contrast, the effects on reproductive and growth performance were less clearcut (Chapters 2 and 4). As a synthesis of the previous results, the present chapter is concerned with the assessment and comparison of overall biological herd productivity achieved in goat herds subjected to controlled breeding at different time points in a year. The first hypothesis for testing is that there is an optimal period in a year to which breeding can be restricted to improve overall biological productivity of goat flocks. Secondly, a simulated, aseasonally reproducing herd is used as a reference in testing whether controlled breeding in goat herds is superior to uncontrolled breeding under semiarid rangeland conditions in northern Kenya. Thirdly, the question of whether a restricted, onceayear breeding regime entails a significant increase in production risk is addressed. In addition, hypothesis testing is carried out in relation to two different levels of milk extraction for human consumption. It is important to take such variations in production goals into account because, traditionally, goats play a crucial role in buffering insecurities and imbalances in food supply to pastoral households and are reared for dualpurpose meat and milk production, while the transition towards a more sedentary lifestyle increases the pressure on pastoral producers to specialise and commercialise their mode of production.
Productivity assessments are made using a newly developed approach to steady state herd productivity assessment (Chapter, 6). The procedure is based on a stagebased description of population dynamics and uses nonlinear programming to derive the steady state herd structure and culling policy that maximizes overall energetic efficiency of pastoral goatkeeping. The efficiency of energy transformation (in percent), which has also been called ecological efficiency by Coughenour et al. (1985), from metabolizable feed energy into gross energy available for human consumption is a relevant performance index under the assumption that feed is the main limiting resource for grazing livestock production (James and Carles, 1992). Also, using gross dietary yield as an aggregate measure of output, instead of production or economic margin measured in monetary terms, is probably more appropriate in situations where profit maximisation cannot be assumed to be the primary objective of the decision maker. This is often the case in traditional, subsistence oriented pastoral production systems (Behnke, 1985; Upton, 1989).
 [page 126↓] 
Experimental data
Data for the parameterisation of the herd productivity assessment models studied in this work were obtained from the results of an experiment conducted between January 1984 and January 1988 at the Ngare Ndare Research Station of the University of Nairobi in Isiolo District, northern Kenya. The experimental design and herd management were described in detail in Chapter 1. The experiment consisted of 145 does of the SEA type, which were maintained under simulated pastoral management conditions and used for a total of 381 exposures. These were distributed among 18 consecutive breeding groups consisting of approximately 18 does each. Yearround mating, kidding, and weaning was achieved by consecutively introducing a buck into each of the 18 breeding groups for a period of two months duration. 381 kids were born during the experiment, and weaning occurred at 16 weeks of age. Three complete production cycles, ranging from mating until the time at which youngstock had reached an age of two years were obtained for five of the six consecutive twomonth breeding periods per year generated by the experiment. Due to incomplete records for the third repetition of the sixth mating period, which was initiated in December 1986, only two complete cycles were available for this period. The experimental treatment thus consisted of six different mating periods or seasons, the first taking place from February to March (labelled as mating season 1) and the sixth taking place from December to January (mating season 6). Mating seasons 4 and 5 (August to October, and October to December) had to be assumed to have taken place over a period of three months due to a delay of one month which occurred in setting up the first group of mating season 4 in 1984.
Statistical analyses were carried out on all traits required to parameterise the herd productivity assessment models for each of the six mating seasons (see section model parameterisation and description of scenarios performed). For details on the statistical modelling procedures employed, reference is made to Chapters 2 through 5. However, an important point to note here is that the adoption of longitudinal data analysis methods is crucial in order to obtain reliable estimates of survival rates, of liveweight growth and development over time, as well as of lactational milk yields.
A matrix population model for pastoral goat herds
The principles outlined in Chapter 6 were utilized to formulate a population dynamics model for pastoral goat herds. The model describes the dynamics of a herd of breeding females and their female and male offspring. Male breedinganimals are not considered explicitly, and it is assumed that their abundance at any time is sufficiently high so as not to limit reproduction. Thus, all male offspring produced are expected to be reared as surplus animals. The model also allows for rearing immature females as surplus animals which are culled at an appropriate age, instead of using them as breeding female replacements. The decision as to whether an immature female is to be reared as female replacement or surplus stock was supposed to take place at the age of 10 months. Births are assumed to occur at one point during the projection time interval so that the herd is of the birth pulse type. Finally, it is supposed that within each time interval the population is censused immediately after reproduction.
The model was intended to be amenable to both the analytical methods for matrix population models, and to assessments of herd productivity using mathematical programming techniques (Chapter 6). The former requires the transition matrix to be irreducible, whereas the latter does not. As discussed in Caswell (1989, 1996b), a reducible transition matrix contains at least one stage that cannot contribute, by any development path, to some other stages. For instance, this occurs with the nonreproducing or surplus male and female parts of the population which 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, the surplus stages of both sexes were omitted from the productivity assessment model for carrying out matrix analyses. This, however, does not affect or invalidate the analytical results obtained, since, in the present context, the dynamics of the breeding herd are independent of those of surplus animals.
The life cycle graph describing the dynamics of the breeding female portion of the herd is shown in Figure 7.1. The projection interval from [t, t+1] is five months and approximates the gestation period for goats reported in the literature (e.g., Devendra and Burns, 1983). The population is divided into two immature stages (birth to 5, and 5 to 10 months of age), one replacement stage, and a total of 38 breeding female stages. The latter are structured according to the reproductive status (pregnant, nonpregnant, lactating) and the parity number of breeding does. It takes one year to complete a full reproductive cycle, which is split up into pregnancy and lactation stages for fertile does, and two consecutive nonpregnant stages for temporarily infertile animals. It is assumed that mating occurs once a year and is concentrated in a short breeding season. The selfloops attached to the lactation and some of the nonpregnant stages indicate that the duration of these stages exceeds the 5 months projection interval. Stage durations are fixed for all stages. Coefficients [page 127↓] labelling arcs in Figure 7.1 specify the recruitment rates into each of the i stages. The β_{i}, P _{i}, and G _{i }parameters denote, respectively, stagespecific marginal probabilities of conception, surviving and remaining in the same stage, or of surviving and growing into one of the next stages. The parameters p _{i} are stagespecific probabilities of survival over one time interval; the f _{i}'s represent stagespecific fecundity rates. The subscripts np(k, l) for nonpregnant stages identify the kth nonpregnant stage for an animal of parity l.
Figure 7.1. Life cycle graph for breeding does. The reproductive cycles for does in parities 2, 3 and 4 are identical to the one depicted for primiparous does. Surviving breeding does are culled (or die) after completing the fifth lactation.  

The life history of a breeding female begins with its birth into the juvenile stage (0 to 5 months of age), then it either dies or becomes a weaned female young five months later. The decision to rear an animal as a breeding female replacement or as surplus is made at ten months of age. First mating occurs at the age of fifteen months or approximately 65 weeks. This is in agreement with the average age at first breeding of 67.8 weeks (±12.6) observed during the experiment.
The parity number of does is increased by one whenever a pregnant doe survives until parturition. A fertile doe completes a full reproductive cycle at the end of the seventh lactation month, i.e. at the beginning of the next breeding season, which is also the onset of a new reproductive cycle. Breeding does which were mated for the first time in their parity class and failed to conceive move through two nonpregnant stages (0 to 5 months and 5 to 12 months) and are rebred the following year. If they again fail to conceive, the process is [page 128↓]repeated for an additional year (12 to 17 and 17 to 24 months nonpregnant). Nulliparous females are allowed to remain reproductively inactive for a maximum of 36 months, and are culled (or die) if no pregnancy is detected at the end of the third consecutive infertile cycle. The infertile state for breeding females of parity ≥ 1 is limited to two consecutive cycles, or a maximum period of two years. Principally, it is straightforward to extend the time period potentially spent by an animal as reproductively inactive by simply adding further nonpregnant stages to the life cycle graph. However, the associated increases in model size and data requirements are hardly justified if, as it is the case in pastoral goat herds, the rate of nonconceivers is of the order of 515%, such that very few animals remain infertile after three unsuccessful consecutive matings (e.g., 0.34% if 15% of all animals present at mating fail to conceive).
The cull for age threshold was set to the period after completion of the fifth lactation, both for computational and data analysis reasons. The decision to ignore breeding female stage classes beyond the fifth parity was supported by the fact that only 5 out of the 287 fertilisations in the experiment occurred the fifth or higher parity stage. Similar reproductive lifespans of SEA goats have been reported by Peacock (1984) and Wilson (1992).
The life cycle graph in Figure 7.1 was used to model each of the six different mating season groups of the experiment. The experimental design did not include an "aseasonal" control group to which the restricted breeding regimes could have been compared. However, the matrix population modelling technique offered the opportunity to at least approximate such a breeding strategy by suitably modifying the life cycle graph in Figure 7.1. The construction of the lifecycle graph of an aseasonally managed pastoral goat herd was based on the reasoning discussed in Chapter 6. There, it was argued that the major difference between aseasonal and seasonal breeding regimes resides in the length of the time interval between successive mating events, and that in the aseasonal case kidding intervals are a function of both the probability of conception and the timespan between consecutive mating events. In goats, the shortest period from kidding to conception corresponds to the time until occurrence of the first oestrus postpartum, which implies a minimum kidding interval of about 175 days (Devendra and Burns, 1983). Wilson (1992) reported mean kidding intervals for goats of the SEA type of 306 days (Maasai), 297 days (Mubende), 233 days (Mashona) and 14 months (Boran).
Using equation (37) of Chapter 6, expected kidding intervals for various combinations of values for conception rate and period between successive matings were calculated in order to identify a reasonable value for the average period between successive mating events. The figures in Table 7.1 suggest that an average period between successive mating events of about 122 to 152 days returns estimates of kidding intervals which compare fairly well with the range reported by Wilson (1992). A period of 152 days or roughly five months was finally chosen for modelling the aseasonal breeding regime. The main reason for this was that selecting a shorter time period would have resulted in a projection time step less than five months, and thus in a more complicated model structure of the model due to the corresponding increase in the number of stages. An average period of five months between successive matings leads to the life cycle graph shown in Figure 7.2. Note that all selfloops vanish, since the period between successive mating events is equal to the projection time step. Conditioned upon survival, nonpregnant does breed on average every fifth month, and the first postpartum service for fertile does occurs after the fifth month of lactation. The number of nonpregnant stages by parity class is the same as for the seasonal breeding life cycle, although the maximum duration of reproductive inactivity is reduced to 30 (parity 0) and 20 months (parity ≥ 1). The population dynamics of the surplus female and male portions of a herd were modelled according to a simple agestructured life cycle (Figure 7.3). The life history is identical for both breeding regimes. Surplus animals are kept in the herd until they reach a maximum age of 35 months, after which they are culled. The p _{i} coefficients labelling arcs denote age/sexspecific survival probabilites over a five month time interval.
 [page 129↓] 
Figure 7.2. Life cycle graph for breeding does, aseasonal breeding regime.  

 [page 130↓] 
Figure 7.3. Life cycle graphs for surplus animals.  

Estimation of feed energy inputs
Metabolizable energy (ME) requirements were assessed according to the factorial method, in which requirements for maintenance, growth, pregnancy, and lactation are estimated separately, and then summed to give the total net requirement (ARC, 1980). Values of daily ME requirements for maintenance, growth, pregnancy, and lactation were based on those recommended for goats by NRC (1981). A 50 percent increment was applied to the basic maintenance requirements of animals older than 16 weeks of age, as suggested by NRC for medium activity levels under grazing conditions on semiarid, slightly hilly rangeland pastures. Information on weight development, weight gain, and lactation yield in all life cycle stages were obtained from growth and lactation curves that were fitted to experimental data (Chapters 4 and 5).
Recall that in modelling herd growth, a postbreeding census was assumed, and therefore projected stage abundances refer to those observed at time t in each time interval [t, t+1]. Consequently, an account is taken of the fact that individuals present at the beginning of a time interval survive until its end. ME requirements per animal over a single time interval were therefore corrected for mortality. Given that longitudinal stagespecific survival data were available, it was possible to estimate 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] would have consisted 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 the end of day a. However, to reduce the amount of calculation involved, it was decided to divide the five months (or roughly 152 days) projection time interval into intervals of length l=14 days (breeding female stages) and l=56 days (youngstock, replacement, and surplus animal stages).
With repect to energy requirements for pregnancy, the approach suggested by NRC (1981) was adopted, which is to increase daily ME requirements per animal by a fixed value of 5.94 MJ during the last two months of pregnancy. Energy released from mobilization of body reserves was also taken into account in calculating total ME requirements for each life history stage. 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. For time periods during which animals were losing weight, the energy released from body reserves and used to meet maintenance, pregnancy, and/or lactation requirements was estimated as follows.
Body tissue was assumed to have an energy content of 20 MJ/kg, with a coefficient of conversion to net energy (NE) of 0.82 (Konandreas and Anderson, 1982). Based on these assumptions, net energy available per kg body tissue mobilized was approximately 16.4 MJ. Upon making the additional assumption that the average efficiency of conversion of feedME into NE is 0.56, the amount of feedME replaced per kg body tissue mobilized was equal to 29.3 MJ ME. Whenever animals were losing weight, this value was used to replace ME requirements for growth. Negative growth rates thus effectively reduced the total amount of ME [page 131↓]from feed resources that must have been available to cover observed maintenance, pregnancy, and/or lactation requirements. Based on a similar reasoning, the expected amount of energy available to kids from milk consumed until weaning was accounted for in calculating total metabolizable feed energy requirements per kid present at the beginning of the youngstock stage (0 to 5 months). The expected milk produced by lactating does until weaning was estimated from stagespecific lactation curves and survival rates of does until weaning. From these estimates, the amount of milk extracted for milk yield measurements during the experiment had to be substracted. Since milk yield measurements were made once every 14 days and implied complete removal of a day's yield, the average experimental milk offtake was about 1/14th of total milk produced until weaning. The procedure employed to calculate expected gross energy yields from milk produced is described in the next section. Conversion of gross energy content into ME content per kg milk was performed using a digestibility coefficient of 0.93.
Estimation of herd outputs
Herd output was defined in terms of total meat and milk offtakes for human consumption per projection time interval, valued at their gross energy contents. The quantity of milk extracted for human consumption over one projection time step was assumed to correspond to a fixed proportion of expected total milk produced by animals in lactation stages (see the section on model parameterisation and description of scenarios performed further below).
For each life cycle stage, body weight estimates for animals present at the beginning of each time step were available from corresponding stagespecific growth curves (Chapter 4). The main problem resided in estimating the gross energy content per kg of empty body and carcass weight in different life cycle stages. Information on this subject is very scarce for tropical goat breeds in general, and are virtually nonexistent for goats of the SEA type. Most of the published experiments relate to temperate breeds (e.g., Panaretto, 1963; Jagush et al., 1983; Brown and Taylor, 1985; Sanz Sampelayo et al., 1990; Benjamin et al., 1993). The few examples on tropical breeds that could be found were those of Gosh and Moitra (1992) on Black Bengal goats, Aganda et al. (1989) on Sahel and Maradi goats, and Viljoen et al. (1988) on Boer goats. The last study analysed body composition of does over a relatively wide range of ages and live masses for does, and the results reported therein are used here as a basis to estimate gross energy contents per kg of empty body and carcass weight in different life cycle stages.
Figure 7.4. Relationships between degree of maturity and protein and fat contents in empty bodies of goats (data from Viljoen et al., 1988).  

In their paper, Viljoen et al. provided data on age, live masses, carcass analysis, and gut contents of 12 Boer goat does. Live masses ranged between 20 and 60 kg, and ages between 7 and 44 months. The amounts of body protein and fat measured on the 12 experimental animals given in this paper were reexpressed as a [page 132↓]function of the degree of maturity, assuming that the mature live weight in the population of Boer goat does from which the experimental animals were sampled was approximately 60 kg. This corresponds to the lower limit of the range of mature live weights (60 to 75 kg) reported by Devendra and Burns (1983) for female Boer goats. Average empty body weight in the experiment published by Viljoen et al. was 84% of live mass, and thus an empty body weight at maturity of 50 kg was assumed. After expressing all measured empty body weights relative to the empty weight of mature animals, the following regressions were fitted to data on proportions of protein and fat contained in empty live masses:
Figure 7.4 gives a plot of the relationships between degree of maturity and proportions of protein and fat per unit empty body weight. For instance, protein and fat contents in empty body at maturity were estimated as 15.3 and 30.4 percent, respectively. In the absence of any further information, the above relationships were assumed to hold true for both female and male animals.
Data on empty body and carcass weights for SEA goats were available from an experiment conducted simultaneously to the present study at the Ngare Ndare research station (Hofman an Schwartz, 1987). From these data, average empty body weight was estimated as 82.0 percent (SD 4.35, N=48 goats) of live mass, and dressed carcass weight as 59.6 percent (SD 5.15, N=48 goats) of empty body weight. No significant differences were found between female and male animals. The live weight of mature SEA goats was assumed to be 38 and 48 kg (31.1 and 39.3 kg empty body weight) for females and males, respectively. Then, based on a gross energetic value for protein (GE _{P}) and fat (GE _{F}) of, respectively, 38.12 MJ/kg and 24.52 MJ/kg (Zygoyiannis and Katsaounis, 1986), gross energy yield per live animal offtake in stage i was calculated as
where EBWT _{i} = is the empty body weight 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. GE _{carcass,i} = is the gross energy yield in dressed carcass per live animal offtake in stage i. PPFAT _{i} = is the proportional fat content per kg EBWT _{i},_{}obtained from equation (44) upon expressing EBWT _{i }in terms of degree of maturity. PPROTEIN _{i} = is the proportional protein content per kg EBWT _{i},_{}obtained from equation (45) upon expressing EBWT _{i }in terms of degree of maturity. 
Whenever average empty body weight at entry into a stage class exceeded the assumed weight at maturity, degree of maturity was set to 1 in calculating gross energy yields.
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 was adjusted for mortality. Goat milk was assumed to contain 4.48 g/kg fat, 3.65 g/kg protein, and 5.02 g/kg lactose, giving a total dry matter content of 13.15 g/kg milk (Jagush et al., 1983). Using the energetic values for protein and fat given above, and an energetic value of lactose of 16.54 MJ/kg, results in a gross energy content of milk (GE _{m}) of 3.43 MJ/kg.
Model parameterisation and description of scenarios performed
As mentioned above, the parameterisation of the individual herd productivity assessment models was based on results of statistical analyses of experimental data obtained on stagespecific conception, fecundity and survival rates, as well as on liveweight development and milk yields in each of the six different mating season groups. Since the ultimate objective of the productivity assessment exercise was to evaluate differences in energetic efficiency achieved under each of these treatments, some form of statistical [page 134↓]significance test was required. This, of course, presupposed that more than one observation in each treatment group was available. Fortunately, as indicated above all mating season groups were observed over three consecutive production cycles, except for mating season six, which had only two replications. The procedure then involved obtaining estimates by mating season group and production cycle for each parameter. Several of the statistical models used were of the mixed model type, with production cycle being treated as a random effect, so that these estimates implied a narrow inference space. However, for all parameters, mean values by mating season group, but averaged over production cycles, were also estimated. These were used to construct an "average" model for each mating season group, on which some of the matrix analyses were performed. In sum, a total of 23 different herd productivity models were parameterised in this way.
Parameterisation of the aseasonal reference model was accomplished by fitting the same type of models as above, but excluding the mating season treatment effect. Parameter values for the aseasonal model thus correspond to expectations obtained by treating the experimental data set as if no restriction on the timing of breeding had been implemented throughout the year. Four different aseasonal herd productivity models were constructed; one model for each production cycle, and, as for the mating season groups, an additional "average" model that combines the information from all three production cycles.
Parityspecific estimates of survival rates, body weights, and milk yields of does were only available up to parity class four. Because very few animals lactated more than four times during the experiment, observations of such animals were pooled with those of parity stage four animals. Consequently, parameter values for parity class five in the life cycle graphs (Figures 7.2 and 7.3) were identical to those for parity class four.
Two different types of scenarios were investigated with respect to milk offtake rates. In baseline model runs, milk offtake corresponded to the amount extracted for milk yield measurements during the experiment, or roughly 1/14th of total milk produced in each lactation stage. Pastoral goat herds are generally reared as dual purpose meat and milk herds, and milk offtake rates for human consumption affect both survival and growth rates of kids.
Based on the assumption that the dual purpose mode of production would affect the relative ranking of mating seasons with respect to productive efficiency, a scenario with increased milk offtake for human consumption was designed. No information on milk offtake rates per lactation in pastoral goat herds in Kenya was available, and use was made of an expert opinion. H.J. Schwartz (personal communication) suggested that roughly one third of total lactational milk yield could be considered as an upper limit for milk offtake rates in pastoral goat herds. Thus, in the increased milk offtake scenario an offtake rate for human consumption of 33 percent (including the experimental offtake) of all milk produced by does in the different parity stage classes was assumed.
All parameter values in herd productivity models for the increased milk offtake scenario were the same as those for the baseline scenario, except for those relating to kid survival and body weight development. Milk yield of does until weaning, categorized into five classes (10 kg apart), was one of the predictor variables used in the statistical analysis for estimating these two traits. Therefore, using appropriate estimable functions of the fixed effects (i.e., linear combinations of the model parameters), it was possible to estimate kid survival probabilities and body weights for reduced milk availability until weaning. To accomplish this, estimated mean milk yields in each of the mating season groups and the aseasonal control were used as a reference from which the 67 percent yield levels were computed. Next, interpolation weights between lower and upper class midpoints of the reduced yield classes were calculated; these were then used as coefficients for the levels of the milk yield fixed effect in estimable functions. A total of 27 additional (including the aseasonal control group) runs of the herd productivity model were required to obtain the results of the increased milk offtake scenario.
In order to avoid problems relating to demographic stochasticity, all model runs were performed using a steady state herd size of c=10.000 animals in equation (27) of Chapter 6. For both scenarios, energetic efficiency as defined in Chapter 6 was used as the objective criterion to be maximized. Two additional productivity indices were computed from the solution of the nonlinear programming model. These were a reproductive performance (RPI) and flock productivity (FPI) index, similar to those reported by Bosman et al. (1997):
 [page 134↓] 
where RPI _{} = is the reproductive performance in terms of weaned empty body weight (kg) per doe and five months time unit at the stationary state. n _{young [5;10)} = is the number of animals present in the yougstock stage (5 to 10 months) at the start of each time period. EBWT _{young [5;10)} = is the empty body weight of youngstock aged five months. n _{i}(t)_{} = is the number of individuals present in breeding female stage i at the start of each time period. 
where FPI _{} = is the flock productivity in terms of net empty body weight (kg) offtake per kg herd metabolic empty body weight and five months time unit at the stationary state. u _{i}(t)_{} = is the number of stage i animals removed from the herd at the start of each time period. EBWT _{i} = is the empty body weight of an animal in stage i at the start of each time period. n _{i}(t)_{} = is the number of individuals present instage i at the start of each time period. 
Tests of significance of differences between energetic efficiency, finite rate of increase (λ) of the herd, reproductive efficiency (RPI) and flock productivity (FPI) achieved in each of the treatment groups were based on a mixed model ANOVA, with mating season as a fixed and production cycle as a random effect.
Eigenvalue elasticity was used to investigate the proportional contributions of the five different types of matrix entries of the female herd projection matrix (fecundities; survival rates; probabilities of surviving and growing into pregnant stages; probabilities of surviving and growing into nonpregnant stages; and probabilities of remaining in the same stage) to asymptotic herd growth, and thus energetic efficiency once an optimal culling regime has been established. Sensitivity analysis of the nonlinear optimization models was used to study the overall impact of changes in conception rate and in age at first breeding on herd productivity. This was motivated by the fact that the latter factors have previously been argued to be major determinants of goat herd productivity under semiarid conditions in Africa (Peacock, 1983; Wilson, 1989). Both types of sensitivity analyses were performed for the baseline scenario only. They were not carried out for the increased milk offtake scenario because none of the parameters of the corresponding models were derived from field data, but rather were obtained from statistical models which, strictly speaking, can only be considered to provide reliable parameter estimates for the baseline scenario.
Baseline scenario
Relevant performance measures summarizing results of the herd productivity assessment procedure are presented in Table 7.2. Note that in all model runs the criterion to be maximised was the overall energetic efficiency. Values for the indices of reproductive performance (RPI) and flock productivity (FPI) were calculated from the solutions to the optimization problems. For the sake of clarity, the solutions themselves, i.e. optimal stage abundances and offtake policies at the steadystate, are not presented for each production cyclexmating season group combination, but only for "average" models which are based on mean parameter estimates by mating season group, pooled over production cycles (Table 7.3). Values for the potential population growth rate (λ), generation time (A), net reproductive rate (R _{0}), as well as the probability of surviving until the age at first reproduction (Pmat) in Table 7.2 were obtained from matrix analyses.
Figure 7.5 illustrates the variability in selected performance measures across production cycles. Parameter estimates obtained for mating season group 1 and 6 in the first and third production cycle, respectively, were such that sustainability at the flock level could not be achieved (λ<1), and inflows of replacement females of about 17 and 2 percent of their equilibrium stage abundances were necessary to meet the steadystate conditions
 [page 135↓] 
 [page 136↓] 
Figure 7.5. Values of selected production efficiency measures by mating season group for the baseline scenario. (EBWT = Empty body weight).  

with respect to herd structure and size. Overall, the latter two groups displayed the poorest productivity. The main reason was the small number of female offsprings by which a newborn female individual was expected to be replaced by the end of its life (mean R _{0}=1.46 and 1.26 for groups 1 and 6). This variable is strongly affected by the probability of survival until maturity (mean Pmat <0.5 in both cases), fecundity and breeding female survival rates.
The entries in Table 7.2 clearly show that mating season group 4 achieved the best performance with respect to all measures, except Pmat, which was highest for group 3. At 2.63 percent, energetic efficiency in group 3 was only slightly lower than that for group 4 (2.78 percent), followed in rank order by group 5 (2.44 percent) and the aseasonal reference herd (2.40 percent). Expressed in relative terms, the efficiency of energy conversion in the latter was about 87 percent of that attained in group 4, but this difference was not statistically significant. Markedly lower energetic efficiencies of less than even 2 percent were determined for groups 1 and 2, however the poorest rate of energy conversion of 1.47 percent was achieved in mating season group 6. Pairwise comparisons revealed that energetic efficiency figures for mating season groups 1 and 6 were significantly lower (p <0.05) than those for groups 3 to 5 and the aseasonal reference herd, whereas for group 2, statistical significance was reached only in comparison to group 4.
Figure 7.5 illustrates the variability in selected performance measures across production cycles. Parameter estimates obtained for mating season group 1 and 6 in the first and third production cycle, respectively, were such that sustainability at the flock level could not be achieved (λ<1), and inflows of replacement females of about 17 and 2 percent of their equilibrium stage abundances were necessary to meet the steadystate conditions with respect to herd structure and size. Overall, the latter two groups displayed the poorest productivity. The main reason was the small number of female offsprings by which a newborn female individual was expected to be replaced by the end of its life (mean R _{0}=1.46 and 1.26 for groups 1 and 6). This variable is strongly affected by the probability of survival until maturity (mean Pmat <0.5 in both cases), fecundity and breeding female survival rates. The entries in Table 7.2 clearly show that mating season group 4 achieved the best performance with respect to all measures, except Pmat, which was highest for group 3. At[page 137↓] 2.63 percent, energetic efficiency in group 3 was only slightly lower than that for group 4 (2.78 percent), followed in rank order by group 5 (2.44 percent) and the aseasonal reference herd (2.40 percent). Expressed in relative terms, the efficiency of energy conversion in the latter was about 87 percent of that attained in group 4, but this difference was not statistically significant. Markedly lower energetic efficiencies of less than even 2 percent were determined for groups 1 and 2, however the poorest rate of energy conversion of 1.47 percent was achieved in mating season group 6. Pairwise comparisons revealed that energetic efficiency figures for mating season groups 1 and 6 were significantly lower (p <0.05) than those for groups 3 to 5 and the aseasonal reference herd, whereas for group 2, statistical significance was reached only in comparison to group 4.
Marked differences among mating seasons were also apparent with respect to the variability in energetic efficiency. Coefficients of variation of more than 38 percent were calculated for herds in which mating occurred between February and May (groups 1 and 2), these ranged between only 9 and 17 percent for does joined between June and November (groups 3 to 5). Fairly high levels of variability in herd performance were also observed for group 6 and the aseasonal reference herd (about 29 and 23 percent, respectively).
Reproductive performance(RPI) ranged between 2.2 (group 1) and 4.2 kg (group 4) of weaned empty body weight per doe over a five months period. Likewise to energetic efficiencies, the highest reproductive performances were achieved in mating seasons 3 to 5 and in the aseasonal reference herd. The lowest values were again calculated for groups 1, 2, and 6. A similar distribution pattern was characteristic of the flock productivity index (FPI), that ranged between 0.23 (group 6) and 0.4 kg (group 4) of empty body weight offtake per kg herd metabolic empty body weight.
Besides yield levels in terms of milk production and body weight development, the most important factor that determined energetic efficiency at the herd level was the potential or asymptotic rate of population growth, λ. This parameter subsumes the joint effects of vital rates, i.e. reproduction, survival, and development, on population dynamics, and therefore offtake rates. As may be seen from Table 7.2, mean energetic efficiencies and asymptotic growth rates followed almost the same patterns, and indeed correlation analysis revealed a strong linear relationship between both entities (r=0.98, p <0.001). A logical consequence of this is that energetic efficiency must also be closely related to the net reproductive rate (r=0.96, p <0.001), since the latter is, by definition, one of the two factors determining λ (ln λ=ln R _{0 }/ T , T=time required for a population at stable stage distribution to grow by the net reproductive rate). When converted to annual percentage growth rates (100x[λ^{2.4}—1]), joining goats in season 4 would result in an annual increase of goat flocks of almost 25 percent, compared to only about 5 percent in season 6 (p <0.005). These, of course, are theoretical values which would only be achieved if vital rates remained constant over time, but they nevertheless indicate that the choice of the breeding season may have a large impact on herd growth and therefore offtake rates and production efficiency. The annualised potential rate of increase for the aseasonal reference flock (approximately 18 percent) was about 73 percent of that achieved in mating season group 4, but this difference was not statistically significant.
Consistent with expectation, generation time was markedly shorter in the aseasonal reference flock (38.5 months) than in seasonal breeding groups except for group 1 (p <0.05), due to the shorter period between successive mating events than that implied by a seasonal breeding regime. The mean generation time of 39.5 months for group 1, however, also emphasizes the effect of conception rates, which was the highest in this group, on A. This is best seen when comparing mean generation time in the latter group with that of mating season 2 (43.1 months), in which the average probability of conception was particularly low (82.7 percent). Given that the difference in doe survival over lactation and pregnancy stages between both groups was small, leading to similar stable stage abundances in breeding female stages, the lower probability of conception implied that, in mating season 2, a larger proportion of replacement and breeding females would be expected to spend one or more production cycles in an unreproductive stage, thus increasing the average age at all kiddings. With a mean generation length of 48.1 months, this effect was even more pronounced in group 6. Estimated probability of conception was also low in this group (86.1 percent), and its impact on generation time was compounded by the fact that higher stable stage abundances of older does were predicted than in the first two mating seasons, because of lower mortality rates over lactation and pregnancy stages for higher parity does.
 [page 138↓] 
 [page 140↓] 
Average offtake rates per 5 months time unit in relation to total herd size at the stationary state ranged between 8.7 percent in mating season 1 and 17.2 percent in mating season 4. Offtake rates in groups 3 to 5 were found to be significantly higher than in groups 1, 2, and 6. Variability in offtake rates was again markedly higher in the latter groups and in the aseasonal reference herd. In order to accommodate variations in the size of the male herd, in Table 7.2 offtake rates are also given in relation to the steadystate abundance of breeding females. For instance, because of the larger proportion of males maintained in the first two mating seasons (Table 7.3), their performance relative to the other mating season groups was slightly improved when compared on the basis of offtake per breeding female in the herd. Nevertheless, ranking of mating season groups with respect to both offtake rate measures was almost identical to that described previously for herd performances in terms of energetic efficiency.
An important result emerging from the herd productivity assessment procedure was that in all mating seasons, optimal offtake per animal in the herd per time unit exceeded the respective asymptotic population growth rate (λ), as derived through matrix analyses. To clarify this issue, values of λ (expressed in percent increase per 5 months period), offtake rate by mating season group, and as well as of the maximum proportional offtake rates that would keep the herd level constant from one period to the next are plotted in Figure 7.6. (In case of a herd subjected to a prereproductive harvesting schedule, maximum proportional offtake rate can be calculated as [100×(λ—1)]/λ, if population growth rate is greater than or equal to 1). Potential population growth rates were between 38 (mating season 3) to as much as 84 percent (mating season 6) lower than optimal offtake rates determined by the herd productivity assessment procedure. Similarly, an offtake policy which harvests an equal proportion of individuals from each stage class would result in offtake rate levels of only 16 (mating season 6) to 56 percent (mating season 3) of those determined by nonlinear programming. Hence, by comparison with the situation without active management controls (as described by average R _{0} and λ values obtained through matrix analyses), optimising herd structure and culling policy with respect to energetic efficiency resulted in stationary state stage abundances which maximized net reproductive rates, and therefore offtake per time unit.
Figure 7.6. Values of λ (in percent increase per 5 months time unit), optimum offtake rate, and maximum proportional offtake rate according to mating season group for the baseline scenario ("average" models).  

In general, keeping breeding females in the herd beyond the third parturition was found to be adequate only in situations where asymptotic population growth rate was lower than 5 percent per five months period (groups 1, 2 and 6). In all other cases, breeding females were culled after completing their third lactation, i.e. upon entering parity 3 pregnancy or open stages (Table 7.3). Also, nonpregnant breeding females were only kept in herds with insufficient growth potential (groups 1, 2 and 6), but this was limited to does with at most one prior lactation. Culling policies for nonpregnant does in the aseasonal herd differed from this general pattern because of the shorter time period between successive mating events. Here, the optimal culling strategy consisted of keeping reproductively inactive breeding females at least for another 10 months or two matings [page 141↓](nulliparous does) or for at most 5 months or one mating event (primiparous does) in the herd, while in higher parity stages does were immediately culled when they failed to conceive.
An additional result to note is that in none of the herds modelled female youngstock were reared as surplus females (Table 7.3). In all cases considered it was more efficient to rear them as replacements and to exploit their potential to contribute to future herd growth and to the production of male surplus animals. The optimal culling age for the latter ranged between 5 and 25 months, and there was a tendency for an increase in optimal culling age with decreasing potential population growth rate and, hence, energetic efficiency (groups 1, 2, and 3). Optimal decisions rules for mating season 6 differed from this trend due to the excessive mortality in youngstock after weaning. However, it should be emphasized that decisions relating to the management of surplus male stages are not independent of the performance, in terms of energetic efficiency, achieved by the female part of a herd. This leads to very complex decision rules which are difficult to predict beforehand.
Based on "average" herd growth models for mating seasons 1 and 4, Figure 7.7 illustrates how optimal culling ages for male surplus animals may differ depending on the level of energetic efficiency achieved in the remaining herd. The plot on the left shows that considering gross energy outputs produced and total metabolizable energy consumed in the surplus males herd only, energetic efficiency was a strictly declining function of culling age. This is to be expected, since cumulative metabolizable energy requirements grow at a much faster rate than gross energy accumulation in the body with increasing age. From this perspective, fattening of male animals beyond five months of age is certainly not optimal when the conversion efficiency of metabolizable feed energy to gross energy content in edible products is to be maximized. However, when metabolizable energy costs required and gross energy outputs produced by the female part of the herd, which is supplying male young for fattening, were taken into account, the picture changed dramatically.
Figure 7.7. Effects of culling age of male surplus animals on energetic efficiency determined at the male herd and total herd level for mating season groups 1 and 4 ("average" models). Note that energetic efficiencies for the newborn stage could not be calculated by considering the male herd only since, at birth, cumulative metabolizable energy requirements were assumed to be equal to zero.  

The plot on the right in Figure 7.7 shows that for a herd in which the process of producing male animals for fattening is rather inefficient, such as in mating season 1, the optimal culling age should be extended well beyond 15 months of age. In contrast, the optimal culling time point is reached much earlier at around 5 months of age in mating season group 4, which is the most efficient one with respect to reproductive performance.
[page 142↓]Sensitivity analysis of the results of the baseline scenario
Elasticities of λ with respect to changes in fecundities, survival rates, and probabilities of surviving and growing into pregnant stages for “average“ projection matrix pertaining to the six mating season groups and the aseasonal reference herd are shown in Tables 7.4 and 7.5. Note that because elasticities sum to one, for each mating season treatment the relative contribution of the matrix elements to λ can be compared. The three groups of parameters shown in the above figures together account for 86 (mating season 6) to 93 percent (aseasonal herd) of the contributions of all matrix entries to population growth. The other two groups, i.e., probabilities of surviving and remaining in the same stage and probabilities of surviving and moving into nonpregnant stages (elasticities of λ to these entries are not shown graphically), are thus relatively unimportant parameters in terms of their impact on λ. The largest changes in asymptotic herd growth can be expected to occur when survival rates of female youngstock and fertile female replacements are perturbed (Tables 7.4 and 7.5), either due to environmental factors, or as a consequence of management interventions. About one third of all contributions to l come from these three matrix coefficients. An increase in fecundity rates, in contrast, can be seen to have a rather limited impact on herd growth and productivity (Table 7.4).
In order to enhance goat herd productivity in the current production system, management interventions should therefore not primarily be directed towards increasing reproductive output of breeding females. Another consequence of the low elasticities of λ to perturbations in fecundity rates is that model predictions generated in this study can tolerate somewhat larger uncertainties in these parameters than in juvenile and subadult survival. These findings are contrary to the results of a sensitivity analysis reported by Peacock (1984), which was aimed at identifying major factors affecting levels of biological productivity in Maasai goat herds. According to this study, the number of kids born per year was a major determinant of biological productivity. The impact of changes in the probabilities of surviving and moving into pregnancy stages are somewhat more difficult to interpret, since these entries depend on both the probability of conception and that of surviving over the corresponding stage. However, it can be seen from Table 7.5 that the probability with which female replacements survive and conceive plays an important role in determining overall herd growth rate. The impact of this type of matrix entry declines over successive parturitions due to the reduction in residual reproductive value with advancing parity stage.
 [page 143↓] 
Given that probabilities of surviving and moving into pregnant stages are derived parameters, it is worthwhile investigating the impact of the underlying vital rates on population growth. In the present context, it was of particular interest to elucidate the potential effect of perturbations in conception rates on potential herd growth and productive efficiency. This is because conception rates can be difficult to assess in the field, and it is therefore important to obtain some information on how errors in the parameter estimation may affect model predictions. Results of the sensitivity analysis of conception rates are presented here for the baseline scenarios of mating season groups 1 and 4 only, being representative of goat herds with low and high potential growth rates and productive efficiencies. The corresponding nonlinear optimization models were rerun with the original conception rates altered, respectively, by –30 to +3.5 percent, and by –30 to +12.5 percent (Figure 7.8). In absolute terms, this amounted to testing the impact of a change in conception rates over the range from 67 to 100 percent for mating season group 1, and from 62 to 100 percent for mating season group 4. Note that, in contrast to eigenvalueelasticities, the soobtained proportional changes in population growth and productive efficiency plotted in Figure 7.8 do not sum to 1 over all underlying variables, nor do they represent the contributions of the individual variables to λ in the sense used before (see Caswell (1989:136) for further explanations).
Overall, responses in terms of productivity tend to be lessthanproportionate to the perturbations in conception rates, while population growth rates appears to react more than proportionately to variations in this variable. The response curves in Figure 7.8 also reveal that the effects of changes in conception rates on herd dynamics and productivity depend on the initial herd growth rate. Response are more pronounced in initially slowly growing herds than in fast growing herds. Thus, management interventions that rely upon manipulating conception rates in order to improve herd productivity will be less effective at high than at low levels of herd growth. For the presently chosen lifecycle structure of goat herds, estimation accuracy of conception rates appear to be more important in determining asymptotic herd growth than productive efficiency. As indicated by sensitivity analysis, predicted productivity levels will be affected by less than ±10 percent as long as errors in estimation of conception rates range between ±15 to ±20 percent.
 [page 144↓] 
Figure 7.8. Effect on asymptotic herd growth rate of proportional changes in conception rate by +3.5 to –30 percent (group 1, corresponding to conception rates of 100 to 67 percent) and +12.5 to –30 percent (group 4, corresponding to conception rates of 100 to 62 percent). Calculations for mating season groups 1 and 4, based on “average” projection matrices (baseline scenario).  

Sensitivity analysis with respect to age at first breeding was based on the same approach as that used for conception rates, and involved changing the original parameter values by –33.3 to +33.3 percent or, in absolute terms, from 10 to 20 months at one monthly steps. Results of this part of the analysis are again presented graphically for mating season groups 1 and 4 in Figure 7.9. Generally, changes in age at first breeding tend to have a larger impact upon herd growth rate than upon herd productivity. Reducing age at first breeding from 15 to 10 month (a 33.3 percent decrease) increased energetic efficiencies by only 12.1 (mating season 1) and 17.8 percent (mating season 4), while the same parameter changes had an almost proportionate impact upon herd growth. Similarly to conception rates, the effect of a given proportionate change in age at first breeding on herd productivity is clearly dependent on the initial magnitude of herd growth rate. If λ is large (mating season 4), decreasing developmental time will have a larger effect on overall productivity than at low levels of λ (mating season 1).
 [page 145↓] 
Figure 7.9. Effect on asymptotic herd growth rate of proportional changes in age at first breeding over the range from 10 (a decrease of 33.3 percent from an initial value 15 months) to 20 months (an increase of 33.3 percent from an initial value 15 months). Calculations for mating season groups 1 and 4, based on “average” projection matrices (baseline scenario).  

It should be stressed that the above results were obtained by making the assumption that changes in age at first breeding neither affect vital rates, especially conception, fecundity, and survival of breeding females, during the first, nor over subsequent reproductive cycles. If, however, strong interactions among life history traits exist, even large changes in age at first breeding might fail to produce the responses in asymptotic growth and herd productivity as depicted in Figure 7.9. A reduction in age at first breeding, in particular, is likely to embody a tradeoff between current reproduction and future survival and /or fecundity patterns, a fact which has not been taken into account in conducting the sensitivity analysis. Data from the current experiment indicated that poor body weight development of does may be detrimental to their survival and productive performance, the most important of which is milk production. Insufficient milk availability until weaning, in turn, has been shown to expose kids to a very large risk of death. As youngstock survival is of paramount importance to herd dynamics and productivity, reducing the age at first breeding may prove to be a far less attractive management strategy than has previously been suggested, for instance, by Wilson (1989). More specifically, changes in developmental time will tend to have the smallest effect on herd growth rate and productivity in slowlygrowing or stationary herds in which juvenile mortality is already high.
Increased milk offtake scenario
Means values of performance measures by mating season group for the increased milk offtake scenario are given in Table 7.6. The increased milk offtake for human consumption implied an increase in juvenile mortality, as reflected by the uniform decrease in probabilities of surviving until maturity compared to those reported for the baseline scenario. Consequently, potential population growth rates and net reproductive rates were also lower than in the baseline scenario.
 [page 146↓] 
 [page 147↓] 
Mating season groups 1, 2 and 6 were most affected by the simulated change to a dual meat and milk production system, with mean values of λ being reduced by 65, 40, and as much as by 251 percent, respectively. The average decrease in λ was lowest in group 3 with only 6 percent, followed in rank order by group 4 (10 percent), 5 (16 percent), and the aseasonal reference flock (18 percent). Significant differences in λ were only detected between the latter groups and mating season 6 (p <0.05) which, on average, was not sustainable (λ=0.966) and an inflow of replacement females of about 13 percent of their equilibrium stage abundances was required to achieve constancy of herd size and structure. Upon shifting stable stage abundances towards higher parity does, the increase in juvenile mortality also led to a general increase in generation time, which, again, was most pronounced in group 6 (48.1 vs. 53.5 months).
By comparison with the baseline scenario, average herd productivity in terms of energetic efficiency improved between 26 percent in group 1 and 36 percent in group 6 (Figure 7.10), while the relative ranking of mating season groups with respect to this criterion remained unaffected by the simulated change in production strategy. Significant differences in energetic efficiency occurred only between mating season group 6 and groups 3, 4, 5 and the aseasonal reference flock (p <0.05).
Figure 7.10. Changes in energetic efficiency, RPI, and FPI in the increased milk offtake scenario relative to the baseline scenario.  

As one might expect, reproductive performance (RPI) and flock productivity (FPI) were negatively affected by higher juvenile mortality rates and reduced growth rates in kids. RPI and FPI values were 10 (group 5) to 42 percent (group 6) and 6 (group 4) to 25 percent (group 6) lower than in the baseline scenario, respectively (Figure 7.10). However, for both indices the resultant ranking of mating season groups was not affected, except that the lowest RPI value was observed in group 6 instead of group 1, as in the baseline scenario.
There was no clear cut pattern of change in offtake levels in relation to total herd size, and the differences between both scenarios were generally small. In contrast, offtake levels in relation to the breeding female herd size were consistently lower in the increased milk offtake scenario. On the one hand, this was due to the fact that fewer male animals survived to culling age. On the other hand, higher juvenile mortality rates also reduced the number of female youngstock available as replacements and, concomitantly, led to increases in breeding female herd sizes. Generally, there was a tendency to replace breeding females, at least partly, in later parity stages than in the baseline scenario. This was mainly due to the necessity to compensate for the reduced potential population growth rates.
 [page 148↓] 
Comparison of estimated levels of energetic efficiency
This section discusses the range of energetic efficiencies obtained in this work to those published in the literature for other grazing systems. Analyses of energy flows and efficiencies of energy transformation from one trophic level to another were first introduced by ecologists to describe ecosystem organization and functioning (Odum, 1969), but more recently have also been used to investigate energetic pathways and conversion efficiencies in and among pastoral grazing ecosystems (Coughenour et al., 1985; Ellis et al., 1979; Western, 1982), and as a means for comparing levels of biological efficiency achieved in different livestock production systems (de Ridder and Wagenaar, 1986). Some of the energetic efficiency figures obtained in the latter studies can be used as benchmarks against which the results of this work can be compared (Table 7.7). All figures were originally given in terms of gross energy output for human consumption in relation to gross energy intake of livestock. Since in the present study forage input was defined in terms of metabolizable energy intake, the published ratios were converted to this scale by dividing them by a conversion factor of 0.405, obtained upon assuming an average digestibility of 0.5 and a ratio of metabolizable energy to digested energy of 0.81 (Konandreas and Anderson, 1982).
From Table 7.7, it can be seen that the energetic efficiencies computed for both the baseline and increased milk offtake scenarios are within the range of values reported for other pastoral production systems. The dual purpose meat and milk production scenario uniformly increased efficiencies by 26 to 36 percent compared to the baseline scenario. Nevertheless, the herd of SEA goats studied in this experiment seemed to demonstrate relatively low forage energy conversion efficiencies when compared to commercial cattle ranching systems. Production efficiency is probably constrained by environmental factors such as poor forage quality, water and heat constraints on intake, and sparse and unequal distribution of forage and water resources which cause high energy investments in selfmaintenance (Coughenour et al., 1985; Western, 1982). The solutions to the herd productivity assessment models considered in the present work revealed that milk composed between 2 and 4.5 percent of total gross energy output for human consumption in the baseline scenario, and between 9.9 and 15.3 percent in the increased milk offtake scenario. By contrast, for the Turkana pastoral production system Coughenour et al. (1985) estimated that 70.1 percent of pastoral food energy extracted from sheep and goat herds were obtained in form of milk. On the one hand, the large discrepancy between these estimates may be due to the fact that Turkana pastoralists achieve much higher milk offtake rates from their sheep and goat herds than the one third of total lactational milk yield assumed in the increased milk offtake scenario. On the other hand, when considering the comparatively low rate of ecological efficiency achieved in Turkana smallstock herds (Table 7.7), it may also indicate that the production objective of Turkana herders differs from that assumed in assessing herd productivity in this work. Maximizing energetic efficiency subject to keeping total herd size constant invariably implies that animals are disposed of at the optimum biological moment. By contrast, from the perspective of pastoral producers a substantial part of the benefits of livestock keeping may be derived from the maintenance and accumulation of large animal biomasses. Ultimately, one of the main incentives for pursuing these goals is likely to be the need to reduce the risk of losing the entire herd. This aspect of pastoral herd management obviously has not been accommodated for in the herd productivity assessment procedure.
[page 149↓]Optimum mating season
The results of the herd productivity assessment lead to the conclusion that under the prevailing environmental conditions, confining breeding in pastoral goat herds to the period from June to July (mating season 4) confers a distinctive advantage in terms of all relevant biological efficiency parameters considered. From a practical standpoint, however, there is probably little basis on which to differentiate between mating seasons 3 and 4, the latter performing only slightly better than the former in terms of all performance criteria calculated. The present results, therefore, concur with Wilson (1984) and Wilson et al. (1985) who found that dams which gave birth in the hot dry season in monomodal rainfall regimes in Mali and Sudan and in the analogous short dry season in southern Kenya showed the best performance as measured by a productivity index defined as average total litter liveweight at 150 days produced per doe and year (the socalled ILCA Index I, named after the former International Livestock Center for Africa). Thus, there appears to be some justification for the management practice of Maasai pastoralists in southern Kenya to try to limit breeding activity in their goat flocks to the beginning of the dry season in June, so that births occur during the short dry season (de Leeuw et al., 1991).
Contrary to expectations, the relative ranking of mating season groups was not affected by the simulated increase in milk offtake for human consumption. Such a change in management strategy reduces reproductive performance (as measured, for instance, by the RPI criterion) through a negative impact on kid survival, which increases at an accelerating rate with declining milk availability for suckling kids. On the other hand, these losses tend to be offset by the rise in energy output from the system due to the higher milk offtake rates for human consumption. Based on these considerations, a change in relative ranking of mating season groups with respect to overall energetic efficiency could have been expected to arise among group 4 on the one hand, and groups 3 and 5 on the other, given that mean total lactational milk yields in the latter two groups exceeded that in group 4. However, this difference in milk yields, as well as the relatively larger increase in kid mortality in mating season 4 associated with the reduced availability of milk for youngstock were not sufficient to compensate for the initial gap in productive efficiency between groups 3 and 5 on the one hand, and group 4 on the other.
The results of the increased milk offtake scenario clearly show that productivity measures such as RPI and FPI cannot be used to assess technical efficiency of livestock production systems when a significant proportion of total benefits are derived in terms of products other than live animals for sale, slaughter, or herd accumulation. This was illustrated by the effect of a reduction of milk availability for suckling kids, which, through higher mortality and slower growth, lead to a uniform decrease in both the RPI and FPI indices, while, by contrast, the simultaneous rise in energetic efficiency adequately reflected the positive effect of such a shift to a dualpurpose production system on biological herd productivity. By the same token, this study confirms the contention frequently advanced by ecologists that, while milk consumption diverts energy from youngstock and reduces their survival and growth rates, energy transfer to humans through milk, or through a combination of milk and meat is still (energetically) more efficient than transfer through meat alone (Coughenour et al., 1985; Western, 1982; Western and Finch, 1986).
If maximum energetic efficiency is to be achieved, breeding of goats in the period from December to May should be avoided, irrespective of the amount of milk extracted for human consumption. The poor performance of mating season groups 1, 2 and 6 was mainly due to elevated kid and doe mortality rates. In all three groups, significant parts of the lactation stage coincided with the long dry season, during which feed availability was insufficient to cover the nutrient requirements of dams and their progeny. This effect was particularly pronounced in mating season group 1, in which kids were born at the onset of the long dry season, resulting in very low milk yields of their dams. The latter example also illustrates that there is little point in trying to enhance reproductive performance in terms of, for instance, the number of kids born per doe exposed, as long as a reasonably high rate of survival of newborns beyond the juvenile stage cannot be guaranteed.In a similar vein, the sensitivity analysis to be presented below shows that kid survival is the single most important vital parameter determining the productivity of goat herds under the prevailing environmental conditions.
Compared to the six seasonal breeding groups, the simulated aseasonal breeding management system performed remarkably well. In terms of energetic efficiency, it ranked as the fourth best management alternative closely behind mating season group 5, and even outperformed the latter with respect to the reproductive and flock performance indices. These results were obtained by assuming that all vital and production parameters corresponded to the mean taken over all mating season groups, and that the average kidding interval in an aseasonally managed goat herd is approximately equal to 10 months. Hence, it was implicitly assumed that expected values of all vital and production parameters would remain unaffected by the increased frequency of parturitions. This type of interaction is likely to occur under semiarid production conditions, but does not appear to have been documented in the literature. Similarly, although an average kidding interval of 10 months seems to be reasonable in view of previously reported figures (Wilson, 1992;[page 150↓] Wilson and Light, 1986), it is nevertheless clear that an aseasonal breeding programme is unlikely to result in a uniform distribution of conceptions throughout the year. Model parameters may therefore be misrepresented by setting them equal to expected values, averaged over all mating season groups. This is because such an approach is valid only under the assumption that the same number of does are present at the beginning of each of the six different mating periods in a year.
Principally, information on the distribution of animal abundances among the six consecutive mating periods that would be achieved if the mating season specific vital parameters were to remain constant over time can be obtained by simulating the transfer of pregnant and barren females from one mating period to another. Using data on average conception rates and kidding intervals in each mating season group, this can be done with a simple compartment model.
In order to further elucidate the effect of aseasonal breeding, such a simulation was performed by assuming that, in each mating period, does which did not conceive would be rebred in the next period two months later, while pregnant does would be bred again after 10 months, at the end of the current reproductive cycle (i.e., a constant kidding interval of 10 months was assumed, irrespective of breeding time point). Note that the effect of doe survival on the distribution was not taken into account. The structure of the model is described in Appendix 1. Based on the set of average conception rates estimated for the various mating season groups, the following relative distribution of pregnancies among the six mating periods was obtained: period 1, 17.1 percent; period 2, 15.2 percent; period 3, 17.7 percent; period 4, 17.5 percent; period 5, 16.7 percent; and period 6, 15.8 percent. Taking doe mortality into account would substantially alter this distribution, but the numerical example suffices to show that it will most likely not be uniform. Hence, the model results obtained by assuming an equal distribution of breeding females can only give a rough approximation to the level of productivity that would be achieved under aseasonal breeding management.
Although results from the aseasonal herd model have to be interpreted cautiously, there appears to be slight evidence in support of the conclusion that the increase in biological productivity associated with a shift from continuous to controlled breeding probably is much smaller than has previously been suggested. Two considerations lead to this conclusion. Firstly, seasonal breeding logically implies onceayear reproduction, and the advantages in terms of increased reproductive and productive performance, as well as higher survival rates of kids and does, may well be offset by the higher frequency of parturitions achieved under uncontrolled breeding. Secondly, aseasonal breeding generates an unequal distribution in the number of does available for breeding throughout the year. The number of does mated in periods that lead to low biological productivity may be less than proportionate, as it was the case in the simulated distribution presented above. Hence, this effect would also tend to reduce losses incurred due to production cycles which take place under suboptimal production conditions. In this regard, generalizations are difficult to make because the distribution of breeding females at consecutive points in time within a year entirely depend on the expected value of vital parameters. Clearly, more research is needed in order to obtain data from which the biological productivity in aseasonally managed goat herds under the prevailing environmental conditions can be assessed. The final point to be made is that, as noted before, from the pastoral producer’s perspective an aseasonal breeding management may generate benefits which have not been captured by the energetic efficiency criterion used to make comparisons among management alternatives. For instance, an aseasonal breeding regime may be valued for its ability to produce a constant flow of goods throughout the year, in contrast to the pulses of output occurring under controlled reproduction.
Comparison to previous productivity assessments in African goat herds
Comparison of the present productivity estimates to the results of previous productivity studies in tropical goat herds is difficult because of the lack of standardization in defining and assessing biological productivity in the literature (see Baptist, 1992b, for an account of the diversity of approaches that have been employed). The most widely used measures for rating the biological performance of livestock herds in the tropics still seem to be the socalled ILCA indices as defined in Wilson et al. (1985), and the modifications thereof proposed by Peacock (1987) and, more recently, by Bosman et al. (1997a). An attempt at comparing the performance of the goat herds studied in this work to the productivity reported for some other goat production systems in Africa is made in Table 7.8. The comparison is based on the three indices proposed by Wilson (1985), as well as on the reproductive (RPI) and flock performance indices (FPI) as defined by Bosman et al. (1997a).
 [page 151↓] 
For the present experiment, ILCA indices were computed from raw, unadjusted data and are only given for the baseline scenario, since parameters for the increased milk offtake scenario were derived from statistical models fitted to the same data. ILCA productivity indices were calculated on the basis of individual observations per female breeding animal. Does whose litters died before weaning (or before reaching the timeendpoint at which weights used in calculating in the index are measured) are usually taken into account by setting their productivity indices to zero (Wilson et al., 1985), and this was also the procedure adopted here. Values for the reproductive performance (RPI) and flock productivity indices (FPI) by mating season were transformed to annual rates and calculated in terms of live weights instead of empty body weights so as [page 152↓] to allow comparisons to be made to the results reported by Bosman et al. (1997a). (Note that corrections for the time required to produce the first litter and for net inventory changes as made by these authors were not necessary in the present study, because these adjustments were implicitly made by imposing steadystate herd size and structure).
Table 7.8 shows that in terms of reproductive performance (RPI) the results for the baseline scenario are comparable to the values reported by Bosman et al. (1997a) for West African Dwarf goats. However, the figures for the best performing mating season groups (3 to 5, and aseasonal) tended to exceed those observed in the latter study. The picture is reversed when comparisons are made on the basis of the flock productivity index (FPI), probably due to the lower mature weight of the West African Dwarf goats. In this case, does of the West African Dwarf breed that were kept onstation outperformed the mating season 4 flocks by about 14 to 18 percent. The figures for the milk offtake scenario clearly indicate that, as noted before, the use of RPI and FPI should be limited to productivity ratings in singleproduct, meat production systems, since they cannot account for production benefits other than meat or live animals. Although not computed for the increased milk offtake scenario, a similar negative bias in the performance of dualpurpose herds would also occur if comparisons based on the three ILCA indices were to be made.
With respect to the ILCA index I, the figures calculated for mating season groups 3, 4, 5, and the simulated aseasonal reference herd exceeded that recorded for SEA goats in southern Kenya (Wilson, 1984; Wilson et al., 1985) and in Zimbabwe (Ndlovu et al., 1996). As before, the relationship was reversed when doe liveweight was used as the denominator (indices II and III). With respect to index I, the Sudanese Desert goats studied by Wilson (1984) appear to be far more productive than all other production system and breed combinations included in Table 7.8. However, it should be noted that in addition to the failure to account for products other than meat and live animals, the indices employed in Table 7.8 do not adequately correct for feed availability and/or differences in mature size. In this respect, Ogink (1993:6) points out that the practice of using the breeding female’s liveweight as the denominator in productivity indices tends to overvalue small individuals because their production level per unit weight is higher as a result of their smaller size. Hence, with regard to their use for comparative purposes, these indices cannot be expected to provide reliable estimates of the biological production potential of different breeds maintained under the same environmental and managemental conditions, or of the same breed maintained under either different environmental conditions, different management systems, or combinations of both factors. Based on these grounds, the statement made by Wilson (1984: 251) that the Maasai herders of Kenya “show results greatly inferior” to the husbandry systems in Sudan and Mali appears to be unsubstantiated, since the ILCA index I utilized in this report corrects neither for differences in mature size between breeds nor for differences in available feed resources between the three husbandry systems. As pointed out by James and Carles (1996), comparisons of productivity among different production systems are likely to be very misleading if they do not take into account differences in feed inputs.
Optimum decision rules
A close relationship has been found between flock performance in terms of energetic efficiency on the one hand, and asymptotic growth rate and net reproductive rate of the flock on the other. Similarly, the calculated measure of flock performance (FPI) and of reproductive performance (RPI) appeared to be an increasing function of asymptotic population growth (Fig. 7.11). Thus, for a given range of alternative management strategies, each described by a specified herd projection matrix, it would seem that the strategy corresponding to the projection matrix with the largest dominant eigenvalue (λ) will also tend to be the one for which biological efficiency of the herd is highest under optimal culling. Interestingly, qualitatively similar relationships are often supposed to hold true with regard to the selection for optimal life histories in natural populations. For example, a fundamental assumption of research conducted in the field of lifehistory theory is that each lifehistory strategy can be characterized by its population growth rate (λ), and that natural selection usually leads to a strategy which maximizes this growth rate, or average fitness of a population (Caswell, 1980, 1982; de Kroon et al., 1986; Houston and McNamara, 1992; McNamara, 1993).
Figure 7.11 also shows that the apparent equivalence between the management strategy (i.e., the optimum choice of mating season in the present context) which maximizes energetic efficiency and that which yields the largest rate of herd growth per time unit is not affected by an increase in milk output for human consumption. This suggest that, at least for this type of goat production system, the major factors determining biological productivity pertain to vital rates (i.e., fecundities, survival, and developmental rates), and not yield levels (i.e., growth and lactation performance). Hence, a tentative conclusion would be that a simplified herd productivity assessment procedure could consist of ranking alternative management strategies or production systems according to their associated asymptotic population growth rate. The information required to carry out such an assessment reduces to the sets of survival, developmental, and fecundity patterns generated by the different alternatives. It should be emphasized, however, that the equivalence [page 153↓]between the most biologically efficient alternative and that which produces the largest longterm population growth rate is conditional upon implementing a stagespecific optimal culling policy which maximizes energetic efficiency, subject to keeping herd size and structure constant over time. The possibility of increasing yields through stageselective culling has been demonstrated by comparing model results to a proportional culling regime which removes equal fractions from each stage group. Essentially, this finding may be regarded a standard result from optimal harvesting theory in age and stagestructured populations. Mathematical proofs of the superiority of age or stageselective harvesting policies over proportional ones are provided by, among others, Beddington and Taylor (1973), Doubleday (1975), Reed (1980), and Rorres (1976). Beddington and Taylor (1973) and Doubleday (1975) also point out that this superiority is preserved irrespective of whether the objective functional is expressed in terms of number of animals or some other yield, for example body weight, biomass, economic value, or any linear combination of the proportions harvested from the separate groups.
Figure 7.11. Relationship between asymptotic population growth rate (λ) and energetic efficiency (EE), reproductive performance (RPI), and flock performance (FPI). Secondorder polynomials in λ were used to fit trendlines. The plots are based on the results from the productivity assessments for all mating seasonxproduction cycle combinations, including results for the aseasonal reference group as well as for the seven “average” model runs.  

At first sight, offtake rates exceeding potential population growth per time unit, as presented in this study, may seem to be unrealistic. However, this is only true when an equal proportion of animals is to be removed from all stages, thus leaving overall herd structure unchanged from one time period to the next. In contrast, nonproportional culling policies such as those shown in Table 7.3 do affect herd structure and, by shifting relative abundances in favour of the most productive stages, herd growth. The effect of stageselective culling upon population growth is perhaps best illustrated by looking at the relative increase in herd size over the first projection time period when a herd previously driven to equilibrium size and structure through optimal culling is suddenly allowed to grow unrestrictedly. These figures can be calculated by postmultiplying the transition matrix A by the stable stage distribution vector n* from the herd productivity assessment procedure (equation (25)), so as to obtain a new stage distribution vector n´. n* and n´ can then be used to calculate the relative increase in herd size after removing management controls. For example, the firststep increases in herd size calculated for the “average” models of mating season groups 1 and 4 and theaseasonal reference flock were, respectively, 14.1, 23.1, and 19.6 percent, while the corresponding [page 154↓]asymptotic rates (λ), to which the rate of herd increase would have decayed subsequently, were much lower at 4.6, 8.3, and 7.6 percent. This suggests that maximisation of herd energetic efficiency through optimal culling implicitly maximizes the rate of selfrenewal of the herd. Indeed, in studying harvesting patterns that maximize the yield per recruit in agestructured populations, Gray and Law (1987) were able to show that, usually, yield is maximal when the selfrenewal of the culled population is at its greatest.
In the present setting, this meant that all available female youngstock were used as replacements. In terms of energetic efficiency, it was never profitable to keep female youngstock as surplus for fattening. Instead, model predictions indicated that it should always be more advantageous to exploit their reproductive potential and, hence, their capacity to contribute to future herd growth and to the production of surplus male animals, before using them as a source of energy output from the system. Of course, optimal decision rules may be radically different when productivity assessments are based on criteria other than energetic efficiency, and when prevailing market demand and prices for livestock products, as well as resource constraints within the herding enterprise are taken into account. This does, however, neither invalidate the results, nor the general approach to livestock productivity assessment presented in this work. To the contrary, the procedure presented is perfectly amenable to incorporating such extensions and to carrying out a full, economic activity analysis of livestock operations.
It should be emphasized that other available procedures to steadystate productivity assessments in livestock herds such as those proposed by James and Carles (1996) and Upton (1993), would not have been able to arrive at decision rules similar to those presented above, since these procedures determine offtake rates, and thus optimal herd structures, as residuals after all replacements necessary to maintain herd size constant over time have been made. This approach to steadystate modelling invariably generates decision rules which (in order to maintain herd size constant) use only part of the newborn females for breeding, and which do not fully exploit the potential for increasing herd productivity through actively manipulating herd structure. Very different predictions with regard to the biological efficiency of livestock herds may thus result when using the procedures proposed by the previously cited authors.
With respect to optimal culling stage for fertile breeding females, different rules emerged depending on potential herd growth rate. In herds with potential rates of increase per year in excess of 18 percent (mating seasons 3, 4, 5, and the aseasonal reference flock), maximum energetic efficiency was achieved when breeding females were culled after their third lactation. This may seem somewhat surprising, because third parity does were predicted to have the highest fecundity rate, irrespective of mating season. However, there is a tradeoff between reproduction and survival of does over consecutive production cycles, such that a higher efficiency is attained by culling does at an earlier stage, even though there is some loss of reproduction. In contrast, for herds with [page 155↓]insufficient growth potential due to high kid and/or doe mortality (mating seasons 1, 2 and 6) culling policies had to be adjusted so as to meet the steadystate constraints, and does tended to be kept in the breeding herd at least until they had completed their fourth lactation. Optimal culling policies for nonconception showed similar differences between mating season groups as those described for fertile breeding females. In herds with adequate growth potential and seasonal mating, does were culled immediately after failure to conceive, irrespective of parity stage (mating seasons 3, 4, and 5). In contrast, with decreasing availability of female replacements, temporarily infertile firstbreeders were at least kept until the next mating season one year later (mating seasons 1, 2 and 6). Finally, when the frequency of breeding was increased (aseasonal herd), the expected energy yields generated by a successfully established pregnancy tended to outweigh the energetic costs associated with maintaining a nonpregnant breeding female until the next breeding event, such that reproductively inactive does with at most one prior kidding were given the chance to breed at least for one more time. Similarly to the pattern described for mating season groups 3, 4, and 5, all nonconceiving higher parity does were immediately culled in the aseasonal reference herd.
An interesting pattern emerged with respect to optimal culling ages for surplus male animals, in that these appeared to depend upon the level of energetic efficiency achieved in the breeding herd. Generally, optimal culling ages increased with decreasing productivity of the breeding herd, which itself was determined by fecundity and survival parameters. Note that this was possible due to the fact that the optimization model employed did not constrain the available quantity of forage, but only the total size of the herd. Although this assumption seems to be unrealistic, it has nevertheless been stated previously that standing biomass on semiarid rangelands in Kenya often is not the primary limiting factor for livestock production, but rather the low levels and digestibility of energy and crude protein (Pratt and Gwynne, 1977; Western and Finch, 1986). If the simplifying assumptions underlying the present productivity assessment procedure are accepted, the decision rules it generates seem to contradict the common belief that, in order to assess overall herd productivity, surplus animals (males and females), and breeding females and their replacements can conveniently be considered as two separate herds (Matthewman and Perry, 1985). This is mostly based on the argument that gains and losses from surplus animals do not affect the productivity of the breeding female herd, and that inefficiencies in the breeding female herd have an effect on the surplus herd only through the number of youngstock available for fattening, and possibly through less than optimum growth rate due to insufficient milk production. It is further assumed that maximum biological productivity can be achieved upon optimizing these individual components separately. However, the results of the present study clearly showed that the assumed separability between individual herd components may not hold, and that the optimal culling age for surplus animals may depend upon the level of energetic efficiency achieved in the breeding herd. The main reason for this is that indices of biological productivity are intrinsically nonlinear (and thus nonadditive) in nature, since they represent ratios of outputs to inputs. Hence, overall (optimum) herd productivity must be determined by simultaneously choosing appropriate culling ages/stages for both surplus and breeding animals, and by summing all outputs produced and inputs required in the production process prior to calculating productivity measures.
The consequences of ignoring these relationships may be far reaching, as illustrated by an interesting example from animal breeding reported by Ogink (1993: 4). In broiler production, animals have for long been selected for high growth rate, high feed efficiency and high lean proportion at a fixed slaughter weight. This has favoured the selection of large sized animals, since, at equal weight, larger sized animals are physiologically less mature and thus tend to display a faster growth rate, higher feed efficiency, and higher lean proportion than genetically smaller sized (and thus more mature) animals. However, this breeding strategy has also led to very large sized parental stock which is expensive to maintain in terms of feed energy consumption and thus has a depressing effect upon overall energetic efficiency. If the breeding strategy would have been based on an evaluation of overall energetic efficiency of the production process, instead of exclusively focusing on the surplus animal part of it, breeders probably could have avoided these negative effects by defining sizeindependent selection traits for improving feed efficiency and lean proportion.
With regard to the comparison of optimal culling ages of surplus male animals between mating seasons 1 and 4, it is worth noting that the much higher optimal culling age in mating season 1 is entirely due to the suboptimal performance of the breeding female herd, and not to low productivity of the surplus male herd, as indicated by the plot on the left of Figure 7.7. Energetic efficiency of the breeding female part of the herd is impaired by excessive kid and doe mortality rates, which lead to a shortage of female replacements. Consequently, there is also little scope for improving herd structure, and, by this means, herd reproductive rate through stageselective culling. In conclusion, situations such as those characterized by herds pertaining to mating season 1 may provide a rational explanation for the often observed phenomenon that ‘unproductive’, i.e., older surplus animals, are kept in pastoral flocks. This may be a complementary motive to that of insurance or financing (Bosman et al., 1997), though it would require the assumption that, principally, pastoralist producers aim at maximizing energetic efficiency of their goat herds.
Production risk
The major risk faced by pastoral producers in semiarid areas is the considerable variation in yields obtained from livestock. This variation is primarily of climatic origin, which not only causes large seasonal fluctuations in the availability of production resources, but also leads to large differences in seasonal resource availability between individual years. The unpredictability in yields faced by pastoral producers will necessarily affect their decisionmaking process and, hence, their herd management strategies. As has been stated previously by Mace and Houston (1989), the most rational option for pastoral households operating in unpredictable environments appears to be the maximisation of their longterm viability. Clearly, then, evaluating improvements to existing management practices must take into account both the mean level of output, as well as the variability in output generated by individual management alternatives. Additionally, in agricultural production, risk (or variation) is now often identified as the most important factor in the adoption process of new technologies and management practices (Anderson et al., 1977; Anderson and Dillon, 1992), such that this aspect deserves special attention.
In the present work, production risk was described in terms of coefficients of variation pertaining to individual herd productivity measures (Tables 7.2 and 7.6). In light of the importance of the subject, this is a rather cursory treatment, and a more elaborate approach building upon the proposed procedure for herd productivity assessments will be discussed further below. Nevertheless, results of the baseline scenario indicated that the best performing seasonal mating groups also displayed the lowest variability in productivity as measured by energetic efficiency or FPI (groups 3, 4 and 5). With respect to RPI, rankings were somewhat different, because of the strong influence of early kid growth and survival on this index. On the basis of the present results, the hypothesis that restricted breeding entails an increased variability in pastoral goat herd productivity has to be rejected. The aseasonal reference herd showed a fairly large variability in energetic efficiency of 23.1 percent, compared to only 12.8 percent for the best seasonal breeding group 4. Since the aseasonal model was constructed from mean parameter values taken over all mating season groups, the large variability in performance observed in some of the mating season groups was in part carried over to the aseasonal reference herd. Clearly, joining does in the period from December to April has substantial negative impacts on both the predicted level of, and the variability in, productive efficiency. The main reason for the [page 156↓]large variation in energetic efficiency of mating season groups 1, 2, and 6 is that their productive performance was highly dependent upon good forage availability towards the end of the long rainy season, a period during which moisture supply and thus pasture forage production is particularly unreliable.
The change in the production system as simulated by the increased milk offtake scenario did not affect the rank orders of the various management alternatives with respect to variation in performance indices. There was some evidence that the shift to a dualpurpose production system might reduce variability in goat herd performance at high levels of energetic efficiency (mating season groups 3 and 4), despite the negative impact of an increased milk offtake for human consumption on kid growth and survival. Similarly, although a relatively large inflation of kid mortality occurred in some of the mating season groups (groups 1, 2, and 6) due to the reduced milk availability until weaning, increases in the coefficients of variation of energetic efficiency were relatively small when compared to the results of the baseline scenario. This indicates that the more efficient conversion of feed energy to milk as opposed to liveweight may have a buffering effect on overall herd productivity.
As indicate above, describing variation in terms of coefficients of variation may be considered an inadequate assessment of the inherent variability attached to individual management alternatives. Like sensitivity analysis, coefficients of variation which were computed from a rather small number of observations (two for mating season group 6, and three for all other groups) present a very limited view of the wide range of possible outcomes for each management alternative. More formal methods of risk analysis, such as stochastic dominance analysis, are available which provide more insight into the ‘riskiness’ of management alternatives. Generally, formal risk analysis methods involve the specification of probability distributions of uncertain or stochastic variables that are deemed to have a major effect on outcomes of interest, (i.e. herd productivity in the present context) in order to generate (cumulative) probability distributions of outcomes for each of the considered decision alternatives. This is usually done by simulating or calculating (for each alternative) outcomes of the performance measure for a large number of possible combinations of values for the stochastic variables. The outcome distribution itself already constitutes valuable information which can be used to compare decision alternatives based on criteria such as: mean, variance and range of the performance measure; the probability that the performance measure takes on specific values, or that it either falls below or exceeds certain critical thresholds. Alternatively, risky decision alternatives can also be compared in terms of full distributions of outcomes, not just in terms of moments, using stochastic dominance analysis (Hardaker et al., 1997). Stochastic dominance analysis is intended to be used for ranking decision alternatives in situations where the preferences of the decision maker cannot be elicited and described. Based on increasingly stronger assumptions concerning the decisionmakers preferences (i.e., the decisionmaker has positive marginal utility for the performance measure and is risk averse) stochastic dominance criteria of varying degree can help in identifying a risk efficient set of decision alternatives. A full treatment of stochastic dominance analysis is given in Hardaker et al. (1997). The essential prerequisite for the mentioned methods of risk analysis is to generate a probability distribution of the performance measure under each management alternative. In what follows, a short description is given on how such a distribution could be generated within the framework of the productivity assessment procedure proposed in this work.
Principally, one can apply resampling techniques such as bootstrapping to construct distributions of the vital rates (conception, fecundity, and survival rates) and yield levels (milk and growth performance traits) for each of the mating season treatments. Two different ways of obtaining bootstrap distributions for these parameters are available (Neter et al., 1996). The first procedure obtains bootstrap replicates of predicted values of all required vital rates and production parameters directly from the estimated models by adding randomly sampled residuals to the predicted values. The other is to randomly select individuals for a number of m samples, and to fit the same statistical models m times using these samples to the individual traits, such as was done in this study for obtaining confidence intervals of survival curves. The latter is the only practical way, since we need to obtain estimates for all traits from a common sample. In the first place, then, this approach requires sampling a number m of new bootstrap data sets from the original experimental data for each of the 17 breeding groups. Note that the bootstrapped data sets must be drawn from the original data set according to the original sampling design (McPeek and Kalisz, 1993). The m different estimates obtained for each trait and mating season could then be utilized to parameterise m different herd productivity assessment models for each treatment group. Although the general procedure for generating the bootstrap distribution of performance measures is relatively straightforward, the computational burden involved to obtain a large number (m≈100) of bootstrap estimates for each model parameter and solving the same number of nonlinear optimization models can be quite formidable. In the present context, it would be impossible to carry out the individual steps manually, and the entire procedure would need to be automatised by programming it using, for example, the SAS programming language. An advantage of using the SASSystem is that both the required statistical as well as the nonlinear optimization routines are provided as builtin procedures.
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