a simulation model of long-term climate, livestock and vegetation interactions on communal...

Ecological Modelling 183 (2005) 211–230

A simulation model of long-term climate, livestock and vegetationinteractions on communal rangelands in the semi-arid Succulent

Karoo, Namaqualand, South Africa

B.D. Hahna,∗, F.D. Richardsona, M.T. Hoffmanb, R. Robertsc, S.W. Toddc, P.J. Carrickb

a Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch 7701, South Africab Department of Botany, Institute for Plant Conservation, University of Cape Town, Private Bag, Rondebosch 7701, South Africa

c Kirstenbosch Research Centre, National Botanical Institute, Private Bag X7, Claremont 7535, South Africa

Received 11 July 2003; received in revised form 21 June 2004; accepted 12 July 2004

Abstract

The communal areas of the semi-arid (200 mm) winter-rainfall Namaqualand region of South Africa are heavily utilised bya large number of subsistence farmers who keep sheep and goats. For several decades, farmers have maintained, on average,more than twice the number of animals on the rangeland than the number recommended by the Department of Agriculture forneighbouring commercial enterprises. As a result of the continuous high stocking densities, significant changes in the vegetationhave taken place. Farmers are now heavily reliant on an annual flush of vegetation following winter rains to keep their livestocka long-termc nagements stainabilityc nsistentw iews ofra rocesses.©

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live. Using a combination of empirical data and consensus agreement, a computer model is developed to describe thelimate, livestock and vegetation interactions on the communal rangelands and to investigate the impacts of a range of matrategies. The model suggests that the system (including livestock) is sustainable although not stable, and that its suould be due to climatic variability. The model supports the view that when livestock numbers vary in a manner coith recorded observations herbivory has little long-term impact on productivity of the system. This supports recent vngelands in semi-arid and arid environments where non-equilibrium conditions are thought to dominate ecosystem p2004 Elsevier B.V. All rights reserved.

eywords:Semi-arid rangeland; Vegetation change; Goat production; Desertification

∗ Corresponding author. Tel.: +27 21 650 3191;x: +27 21 650 2334.E-mail address:[email protected] (B.D. Hahn).

1. Introduction

Land degradation is the reduction or loss of biolocal or economic productivity arising from inapproprialand use practices (Anonymous, 1995). Overgrazingby domestic livestock is one such inappropriate lause practice and has been widely cited as an impo

304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2004.07.028

212 B.D. Hahn et al. / Ecological Modelling 183 (2005) 211–230

cause of land degradation in Africa (Tainton, 1999).For Namaqualand, a region covering 50,000 km2 inthe semi-arid western part of southern Africa, heavy,continuous grazing by domestic livestock, especiallyin association with communal land tenure, is perceivedas the most important threat to the region’s biodiversity(Cowling and Pierce, 1999; Hoffman and Ashwell,2001). Several studies have recently documented thisimpact, which is alarming, since Namaqualand is con-sidered the richest desert in the world and forms oneof 25 internationally recognised biodiversity hotspots(Myers et al., 2000). For example,Todd and Hoffman(1999) have shown at one site in Namaqualand thatheavy, continuous grazing leads to the replacement ofperennial succulent and non-succulent palatable shrubsand grasses by annuals, geophytes and unpalatableperennial shrubs. This, they maintain, has importantimplications for livestock production since few herdsare able to sustain themselves during the hot, dry sum-mer months characteristic of the region and conditionsbecome exacerbated during drought periods. Associ-ated changes in the composition of insect assemblages(Seymour and Dean, 1999), birds and small mammals(Joubert and Ryan, 1999) and a reduction in soilproductivity as a result of heavy, continuous grazing(Allsopp, 1999) have also been described for theregion.

In addition to these empirical studies, severalmodels have been developed which simulate thedynamics of the vegetation of these semi-arid areas.T stateasal angea eenr on.S batesi lears odsa thea sus-t1 oft se ofp anyc ingr eigh-

bouring farms held under private tenure are consideredexamples of where this has already occurred (Hoffmanand Ashwell, 2001). This view, however, has been chal-lenged in recent times by another which suggests thatin semi-arid and arid areas livestock and vegetation aregenerally not at equilibrium because of highly variableclimatic conditions. Consequently, livestock impactsare negligible (Scoones, 1995) even in communally-managed areas (Shackleton, 1993). Hoffman et al.(1999)present data that suggest an element of sustain-ability in the Namaqualand system. Resolution of thisdebate is important as very different policy implicationsarise depending on one’s position along this continuum.

Sullivan and Rohde (2002)suggest that scientistsshould refine and expand knowledge of specificrangeland production systems to improve ecosystemsustainability and the welfare of pastoralists. Oneway to do this is by modelling the response of thevegetation in a region to both climatic and livestockinfluences over time. A central question in this debateis the influence of climate and livestock and theinteractions between them on vegetation compositionand production. The purpose of this paper is to describeone such model (essentially a long-term ecosystemmodel using output from a short-term productionsystem model), which may be used to investigate theinteractions between rangeland dynamics, rangelandstate and livestock production for a semi-arid regionof Namaqualand. We demonstrate the validity of themodel in tracking recorded data, and show how it mayb tions,s ion.T dingl thati .

2

einc apeP rox-i a.s.l.w re-d m-b rm encedi al-

he models range from conceptual (generalised)nd transition models (Milton and Hoffman, 1994) topatially-explicit, mechanistic models (e.g.Wiegandnd Milton, 1996; Wiegand et al., 1995). They focus

argely on processes associated with vegetation chnd do not focus explicitly on the interactions betwangeland condition, rainfall and livestock productiuch an emphasis, however, is central to recent de

n the range science literature which incorporate ctatements about rangeland condition, livelihond sustainability. At one end of a continuum isrgument that heavy grazing is detrimental to the

ainability of livestock production systems (Tainton,999) and will result in the permanent degradation

he natural environment and an associated collapeople’s livelihoods. Grazing systems evident in mommunal lands of southern Africa, where stockates are generally 1.85 times greater than on n

e used to examine the effects of management opuch as stock reduction, on livestock producthe model also suggests that the system (inclu

ivestock) is sustainable although not stable, andts sustainability could be due to climatic variability

. Study area

Paulshoek is one of nine villages in the Leliefontommunal area and is located in the Northern Crovince of South Africa. Paulshoek occupies app

mately 20,000 ha and is situated at about 1000 mith a mean annual rainfall of 200 mm which falls pominantly in the winter months from May to Septeer. Temperatures rarely exceed 37◦C in the summeonths and sub-zero temperatures can be experi

n winter. The terrain is mountainous and rocky,

B.D. Hahn et al. / Ecological Modelling 183 (2005) 211–230 213

though well-drained deep soils derived from gneissboulders and bed rock material occupy the river valleys.In the lower-lying areas of Paulshoek the vegetation isdominated by a mixture of perennial, low-growing (lessthan 1 m) leaf succulent shrubs and deciduous and ev-ergreen shrubs. When heavily utilised, perennial plantsare replaced by annuals, geophytes and the perennialtoxic shrub,Galenia africana. Higher-lying areas aredominated byElytropappus rhinocerotis(renosterbos),a 1.5 m unpalatable shrub.

The commons is managed under a communal landtenure system wherein any bona fide inhabitant ofthe village is entitled to use the natural resources ofthe commons. The area is used primarily for raisinglivestock although crop production and the use ofmedicinal plants and wood for energy also occur. Thevillage of Paulshoek contains about 1000 people with28 stock posts scattered around the commons. The areasupports more than twice the number of livestock asrecommended by the National Department of Agricul-ture as the sustainable carrying capacity for the region.

3. Model description

The model has three basic interacting components:rainfall, vegetation and livestock.

3.1. Rainfall

w lo-c l forS al-i h ap talf itha otalf a-b ngf e ofa thep

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om-i ner-

Table 1Rainfall data from Springbok

Rainfall (mm/year) Probability (%)

70 1.7790 3.54

110 6.19130 8.85150 6.20170 9.73190 8.85210 13.28230 12.39250 7.08270 7.08290 2.65310 3.54330 2.66350 0.88370 1.77410 0.89450 0.88470 0.89490 0.88

ally described (Todd and Hoffman, 1999) as woodyperennial shrubs palatable to livestock (WP), leaf suc-culent shrubs, primarily in the family Mesembryan-themaceae (ME), an unpalatable shrub guild which isdominated byG. africana(kraalbos) (GA) and annu-als and geophytes (AG). Trees and grasses are rela-tively rare in Namaqualand and are excluded from thisstudy.

Units used are percentage cover, i.e. the percentageof the total area covered by a particular guild. Since thequantities eaten by livestock are measured in units ofmass and not two-dimensional cover, units of percent-age cover are converted into (edible) biomass, and viceversa.Table 2gives the standing biomass of each guild,as well as its edible fraction. So, for example, if the WPcover is 20%, this converts to an edible biomass, WPe,

Table 2Coefficients for conversion of percentage cover of each guild tobiomass and edible forage

WP ME GA AG

Potential standing biomass(kg/ha) (biomass whencover = 100%)

11368 6580 9000 500

Proportion of edibleforage (%)

15 10 2 75

Annual rainfall is generated randomly fromTable 1,hich is based on data from Springbok, a centrallyated town in Namaqualand. Mean annual rainfalpringbok is 212 mm (n= 122 years). The season

ty of the rain is also random. It can be early, witrobability of 20% (more than 40% of the annual to

alling from February to May) evenly distributed, wprobability of 60% (more than 40% of the annual t

alling from April to September) or late, with a probility of 20% (more than 40% of the annual total falli

rom August to November). There is a 16% chanc“summer bonus” (more than 35 mm of rain for

eriod December to February).

.2. Vegetation

The model assumes four plant guilds which are dnant in Namaqualand and which have been ge


Table 3Annual guild response to rainfall

Category WP ME GA AG Category ceiling (mm/year)

Early rain 1 −2 −3 −4 0 752 −1 −1.5 −2 3.75 1503 0.5 0.5 1 7.5 2504 1 1 2.5 13.75 3255 2 1.5 4 20

Even rain 1 −2 −2.5 −3 02 −0.5 −1 −1 6.253 0 0.25 0.5 12.54 1.5 2 2.5 18.755 2.5 3 3 25

Late rain 1 −1 −2 −3 1.252 0 −0.5 −0.5 103 1.5 1.5 2 17.54 2 2 3 22.55 3 2.5 4 25

Summer bonus 0.5 0 0 5

Values reflect changes in percentage cover.

for WP of

WPe = 0.2 × 0.15× 11, 368× 20, 000 kg.

Values inTables 2–5were derived in part fromCarrick(2001) and from unpublished data collected in thefield. Where data were not available ecologists activelyworking in the area were asked to estimate values anda consensus “expert” value was derived. Each guild’sprimary response to rainfall, in terms of an increase ordecrease in percentage cover, is shown inTable 3. Cate-gory 1–5 inTable 3is selected according to whether theannual rainfall generated is less than 75 mm, between75 and 150 mm, between 150 and 250 mm, between250 and 325 mm, or greater than 325 mm. The growthresponse generated in this way is interpolated linearlywithin categories and added to each guild’s percentagecover.

The growth response generated fromTable 3is qual-ified by a competition factor arising from competitionboth within and between guilds. In order to find thecompetition factor we first calculate the “cover seen”by a guild, i.e. a guild’s perception of its own coverand that of its competitors. The entries inTable 4givethe contribution of the row guild to the column guild’sestimate of its own cover in the range 0–1. Larger val-ues indicate similar niches; smaller values indicate less

Table 4Inter-guild competition: contribution of the row guild to the columnguild’s estimate of its own cover in the range 0–1

WP ME GA AG

WP contribution to cover 1 0.3 1 1ME contribution to cover 0.3 1 0.3 1GA contribution to cover 0.5 0.7 1 1AG contribution to cover 0 0 0 0

Larger values indicate similar niches; smaller values indicate lesscompetitive effect on the column guild.

competitive effect on the column guild. The entries inTable 5(which is diagonally symmetrical by definition)give the percentage overlap in cover values.

The guild “cover seen” by WP (gWP) and ME (gME),for example, is calculated as

gWP = 1 × WP(1− 0.008)+ 0.3 × ME(1 − 0.17)

+ 0.5 × GA(1 − 0.043)+ 0 × AG(1 − 0),

Table 5Cover overlap (%) used in calculating guild cover seen by each guildfor estimation of competition, as described in the text

WP ME GA AG

WP 0.8 17.0 4.3 0.0ME 17.0 3.9 10.0 0.0GA 4.3 10.0 21.1 0.0AG 0.0 0.0 0.0 0.0


Fig. 1. Calculation of competition factorc from guild cover seeng.

gME = 0.3 × WP(1− 0.17)+ 1 × ME(1 − 0.039)

+ 0.7 × GA(1 − 0.1) + 0 × AG(1 − 0).

For each guild we then calculate the competition factorc from the guild cover seeng as follows (Fig. 1):

g < a : c = 1;

a < g < b : c = g

a − b+ b

b − a;

g > b : c = 0,

wherea andb are the values of the guild cover seenwhere competition is initiated and is at its maximumrespectively (seeTable 6). The growth response gener-ated fromTable 3is then multiplied by the competitionfactorc if the growth response is positive, or dividedby (1 +c) if it is negative.

Finally (except in the case of annuals), if the adjustedgrowth response is positive, it is further multiplied bya regrowth factorr, to account for the effect of defo-liation due to severe grazing in the previous year (seeAppendix A). The cover of the three perennial guilds isthen incremented by the final growth response, whereasin the case of annuals, their cover is set to the finalgrowth response.

Table 6Interguild competition: constants used in calculating the competitionf

a 0b 5

If there are livestock present, the cover of each guildis reduced by the amount eaten by livestock (see nextsection).

3.3. Livestock

Livestock (goats) are divided into three age classes:kids, yearlings and two-year-olds/adults (two-year-olds become adults at some point in their secondyear—see next section). Half the kids are males; thesemales are removed as a matter of course when theybecome yearlings.

Livestock numbers are updated annually using con-ception rates, birth rates and survival rates calculatedfrom livestock weights, which are in turn calculatedfrom available plant material. It is more accurate toestimate live weights in relation to availability of edi-ble material rather than a two-stage approach (i.e. liveweights from plant material eaten, plant material eatenfrom available edible material) which may propagateerrors (Lynam et al., 2002). Milk yields (with and with-out hand-milking) are also calculated. The methodol-ogy employed is described in detail byHahn et al.(1999) and Richardson et al. (2000). The short-termproduction system model of individual plants and ani-mals referred to by these authors is run repeatedly withdifferent scenarios (rainfall, initial conditions of guildsand livestock weights, stocking rate) in order to gen-erate relatively simple regression equations for thesequantities (seeAppendix A).

aley allyr andm em usedt s onp

3

n-c thef sione iveni

• ing

actorc shown inFig. 1

WP ME GA AG

(competition initiated at) 20 10 15 4(maximum competition at) 40 40 40 8

Rangeland production is measured in terms of mearlings removed, old does sold, animals optionemoved in excess of a maximum stocking rate,ilk yield (optional hand-milking). Bulk runs of thodel over hundred-year periods may then be

o determine the effects of these optional controlroduction.

.4. Sequence of events in a model year

During a production cycle lasting one (noalendar, breeding) year the model works throughollowing sequence of events (details of regresquations for birth and survival rates, etc. are g

n Appendix A).

Annual rainfall is generated randomly, includseasonality and possible summer bonus.


• Surviving mature does from previous year are parti-tioned into those which had 0, 1 or 2 kids, accordingto survival rates for these categories.

• Surviving yearlings are graduated into two-year-olds, and partitioned similarly to mature does (two-year-olds and mature does are collectively referredto as goats).

• Number of surviving kids from last year’s maturedoes and yearlings is calculated (assuming that a kidcannot survive if its mother does not). This assump-tion is based on output from the production systemmodel (Richardson et al., 2000) that predicts thatthe majority of kid deaths occur during the first fourweeks of life.

• If the number of goats is low (e.g. less than 200)only 5% are sold; otherwise a proportion, 1/s, aresold wheres is the “sell-by date” in years. All salesare added to the offtake (production) total.

• Half the surviving kids (females) become yearlings;the remainder (males) is added to offtake.

• If there has been an extinction (i.e. number of goatsplus yearlings less than 1) the range is restockedwith, say, 100 (barren, because easiest to implement)mature does.

• If the number of goats plus yearlings exceeds the (op-tional) maximum stocking rate allowed, the excess(the same proportion of each category—referred toas “capping”) is removed and added to offtake. Cap-ping does not mean the adult population size is fixed;rather it may vary but never exceed the maximum cap

• srely.de-y

• nse

• heail-

ern

and Tothill, 1993, suggest that dry season losses bydecomposition are 0.04 per month, i.e. about 0.22over six months—this givespM = 0.78) and WPe isthe edible biomass of WP.

• MEu, GAu and AGu are calculated similarly (exceptthat the factorEr is omitted in the calculation ofAGu, because annuals grow from seed every yearand the fraction of edible material remains constantfrom year to year).

• Conception rates (Cy—conception rates are success-ful conceptions that lead to the birth of live offspring;conceptions that lead to abortion or absorption of thefoetus are excluded) and birth rates (By) for year-lings are calculated from regression equations withthe provisos thatBy ≥Cy (births may not be less thanconceptions) andBy ≤ 2Cy (triplets or more are notallowed). Yearlings are partitioned into three groups,according to how many kids they have:y0 (barren),y1 (single kid) andy2 (twins). The number of year-lings with twins,y2 is calculated as

y2 = (By − Cy)y

(wherey is the number of yearlings). The number ofyearlings with single kids,y1 is calculated as

y1 = (Cy − (By − Cy))y.

The number of barren yearlings is theny0 =y− y1 − y2.Conception rates and birth rates for two-year-olds

e re-ionedups

• age-s areear-

• htsurees-

• ture.

• eadle in

fol-WP

value.The number of goat unitsug is defined aug =g+ t + 0.8y, whereg, t and y are the numbeof goats, two-year-olds and yearlings, respectivThe stocking rate, goats per hectare, is thenfined asgh =ug/A, whereA is the area under stud(20,000 ha).Guild cover is updated according to growth respoto rainfall, competition and regrowth factor.Guild cover is converted to edible biomass. Tamount of edible biomass (of WP for example) avable to a goat unit is calculated as

WPu = ErpMWPe

gh,

whereEr is the effect of rainfall on edible matt(seeAppendix A), pM is the maximum proportioof the edible biomass that can be grazed (De Leeuw

and mature does are calculated from separatgression equations, and the age-classes partitaccordingly. Number of kids born to all these grois totalled.Two-year-olds are absorbed into the mature doeclass. Mature doe (barren and breeding) weightrecalculated as weighted averages over two-yolds and mature does.Milk consumption, survival rates and weigare calculated for kids (of yearlings and matdoes—including two-year-olds now) from regrsion equations.Amounts of each guild eaten (biomass) by madoes (and their kids) and yearlings is calculatedMost plants in the WP guild are deciduous. The dleaves are edible and are assumed to play a rosustaining livestock during the driest part of thelowing year. The amount of carried-over dead


Table 7Initial perennial guild covers (%) for three initial vegetation conditionscenarios

WP ME GA

Pristine 27.6 12.0 1.2Standard 14.2 6.9 12.1Degraded 1.2 1.5 28.7

available next year to a goat unit is calculated as

WPde = 0.25(WPe − WPT)

gh,

where WPT is the total amount of WP eaten this year(assuming that 75% of this material is lost by tram-pling, insects, wind and other decay—this argumentdoes not apply to the other guilds).

• The regrowth factorr (seeAppendix A) is calculatedfor use in next year’s guild growth.

• Guild biomass is updated according to the amounteaten and converted back to percentage cover.

• Milk yields (for human use) are optionally calculatedfrom regression formulae.

• New weights for yearlings and does are calculated.• Survival rates for yearlings and does are calculated.• Doe and yearling weights are updated, i.e.wd(k) are

replaced byw′d(k), etc.

• Kid weight is updated as weighted average over kidsof mature does and kids of yearlings.

3.5. Initial conditions

A number of variables in the model must be initial-ized before it can be run. For simplicity, it starts with nobarren yearlings or mature does. A given initial numberof “non-kids” is distributed as follows: does with onekid (40%), does with twins (20%), yearlings with onek

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ce-nT ow.

4. Results and interpretation

The model is implemented as a Visual Basic pro-gram (available from BDH) linked to Excel for easyhandling of input and output.

4.1. Single replicates

Before attempting to compare the output of singlemodel runs (replicates) with what little data is availablefor this system, it is helpful to run the model undersome sets of extreme conditions to see if it behavesin a realistic manner and in accordance with real data.Output from two such runs are shown inFigs. 2 and 3. Inboth cases, the vegetation starts in the standard scenario(Table 7), and the same random sequence of rainfall,seasonality and summer bonus is used. InFig. 2, wherethere are no animals on the range, WP and ME increasewhile GA decreases; the range ends in a pristine state(Table 7). In Fig. 3, the range is subjected to heavyovergrazing. GA increases while WP and ME die off,leaving the range in a severely degraded state after 100years (Table 7). These scenarios are consistent withdata collected at a heavily grazed site and at a lightlygrazed or protected site in Namaqualand (Todd andHoffman, 1999).

Table 8 shows recorded rainfall, seasonality andsummer bonus data from Springbok for the 30-yearperiod 1971–2000. The model may be run over thisp tputf val-u ely,i ckd kms a re-m thei icha f thisn the3l ther hea d inF m isa s inr d arep

id (30%) and yearlings with twins (10%).Does all have a weight of 46 kg initially; yearlin

nitial weight is 60% of this; kid initial weight is 16 kghese reflect realistic animal weights measured inegion.

Initial survival rates are 0.952 for does and 0.or all other age classes.

For the perennial vegetation guilds, three initial sarios are generally considered, as shown inTable 7.he derivation of these scenarios is described bel

eriod of time using the recorded rainfall data. Ourom such a run, starting with 1600 non-kids andes of 20, 12 and 10 for WP, ME and GA, respectiv

s shown inFig. 4(a) compared with recorded stoata from Paulshoek, a communal village about 90outh of Springbok. The model tracks the stock datarkably well when one considers that, apart from

nitial vegetation conditions referred to above (whre only estimates, there being no recorded data oature), the only input from the real world during0-year period is the rainfall data inTable 8. In fact, the

arge variations in model output caused by forcingainfall to bealwayseither early, even or late (while tmount of rain is still randomized), as demonstrateig. 4(b), seem to suggest that this rangeland systelmost entirely climate driven. Dramatic increaseecorded stock, as in year 12, cannot be natural anrobably due to buying in of stock.


Fig. 2. Model output—vegetation guilds with no animals (standard initial scenario).

Fig. 3. Model output—vegetation guilds with 5000 animals (fixed) (standard initial scenario).


Fig. 4. Model output (a) against recorded stock and rainfall, (b) for different rainfall scenarios.


Table 8Recorded rainfall data from Springbok: 1971 is year 1

Year Rainfall(mm)

Season: 1, early;2, even; 3, late

Summer bonus:1, yes; 2, no

1 200.0 2 02 165.0 2 03 134.0 2 04 218.0 3 15 192.0 1 06 335.0 2 17 240.0 2 18 106.0 3 19 116.0 3 0

10 190.0 2 011 178.0 3 012 200.0 2 013 276.0 1 014 109.0 1 015 199.0 2 116 209.0 2 117 172.0 2 018 151.0 2 019 175.0 1 020 160.0 2 021 228.9 2 022 189.0 2 023 291.0 3 024 212.3 2 125 263.0 3 026 375.5 3 027 161.0 1 028 61.5 2 029 166.0 3 030 177.3 1 1

Since the long-term model tracks recorded observa-tions reasonably well, it is used to simulate behaviourof the system over 100 years. After the 30 years ofrecorded rainfall, random values for annual rainfall andthe season of rainfall are generated using the estimatedlong-term mean and variability. The results show thatalthough animal numbers vary widely between years,there is no long-term decline in productivity of the sys-tem (Fig. 5). Thus, the system is sustainable but notstable (Marten, 1988). The model predicts that underthe simulated system of livestock management, the per-centage cover of WP decreases slightly during the first30 years, but thereafter fluctuates around a mean valueadequate to maintain the productivity of the system(Fig. 5). Apparently, this is because low rainfall leadsto the production of small amounts of edible foragewhich affects animal growth and reproductive and sur-vival rates within one year. However low rainfall and

severe defoliation has only a small effect on plant pop-ulations in the subsequent year.

4.2. Bulk runs

If single replicates use randomly generated rainfalldata (amount, season, summer bonus) they may be re-peated many (typically 1000) times to estimate aver-ages of various outputs. Such repeated replicates arereferred to as “bulk runs”.

Bulk runs may be used to arrive at the initial vegeta-tion scenarios listed inTable 7, as follows. Initial valuesof the three perennial guilds are changed by trial-and-error until the average annual values (during the 100-year replicates) and average final values (at the end ofthe 100-year replicates) after a bulk run are the same asthe initial values. This does not mean that guild coveris constant at these values throughout each replicate,but rather that these are the values one might expect inthe long term.

The pristine initial scenario then corresponds to thesituation where there are no livestock, the standard sce-nario where each replicate starts with 1600 non-kidsand livestock numbers are allowed to vary freely, andthe degraded scenario where the number of goats isfixed at 5000.

One of the objectives of this model is to examinepossible management interventions. We report herebriefly on one such management option—limiting(capping) the stocking rate. There are two obviousm nualos goatp er inF ont is tom d ata izem glek capt thea mi-t tob therea doei fewerd nt ofm

easures of the productivity of a rangeland: anfftake (defined above) and annual milk yield.Fig. 6hows the effect on these measures of capping theopulation in varying degrees (each bar and markig. 6represents a bulk run with that particular cap

he goat population). If the management objectiveaximize offtake, then the goats should be cappebout 2000. However, if the objective is to maximilk yield (where in this case only does with sin

ids are milked), it does not seem necessary tohe population. When the population is capped,nimals tend to be in better condition with conco

antly higher birth rates; higher offtake is thereforee expected. However, as the cap increases andre more does with single kids, the milk yield per

s reduced. Capping is an advantage here sinceoes have to be milked to produce the same amouilk, so reducing labour requirements.


Fig. 5. Model predictions of stock numbers and guild cover (over 100 years) with recorded stock numbers (over 30 years).

Fig. 6. Output (annual goat offtake and milk yield) of model bulk runs for different caps on the goat population.

222B.D.H

ahnetal./E

colog

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odellin

g183(2005)211–230

Table 9Percentage changes to averages over 1000 replicates for the standard initial guild scenario with variable animals (A) in response to: rainfall (B) increased (C) decreased by 20%;sell-by date (D) increased (E) decreased by 20%; (F) rainfall constant but seasonality random; rainfall random but always (G) early, (H) even, (I) late; rainfall and seasonality randombut summer bonus (J) never, (K) always; rainfall constant with no summer bonus but always (L) early, (M) even, (N) late; and (O) rainfall constant with summer bonus but alwayseven

A B C D E F G H I J K L M N O

WP annual 14.1 17.0 −55.3 −0.7 3.5 −36.9 −45.4 −23.4 92.2 −9.9 44.7 −79.4 −78.0 70.2 −19.9ME annual 6.8 83.8 −79.4 0.0 1.5 −16.2 −70.6 −8.8 133.8 −2.9 11.8 −82.4 −73.5 158.8 −72.1GA annual 12.1 41.3 −73.6 2.5 −2.5 39.7 5.0 1.7 −37.2 8.3 −36.4 −31.4 −15.7 46.3 −16.5AG annual 9.7 4.1 −20.6 −1.0 1.0 −15.5 −37.1 −1.0 14.4 −7.2 35.1 −53.6 −10.3 −15.5 10.3

Total perennial coverannual

33 39.7 −67.0 0.3 1.2 −4.5 −32.4 −11.5 53.3 −1.8 8.2 −62.4 −54.5 79.4 −29.4

WP final 14.1 17.0 −80.1 −1.4 6.4 −49.6 −59.6 −32.6 129.8 −12.8 60.3 −100.0 −100.0 85.8 −27.7ME Final 6.8 104.4 −88.2 1.5 4.4 −32.4 −83.8 −10.3 216.2 −2.9 27.9 −100.0 −100.0 219.1 −100.0GA final 12 50.0 −87.5 5.8 −5.0 78.3 23.3 12.5 −94.2 18.3 −65.8 −60.0 −30.0 50.8 −30.8

Total perennial coverfinal

32.9 47.1 −84.2 2.1 2.1 0.6 −34.3 −11.6 66.0 0.3 7.9 −85.4 −74.5 100.6 −44.1

Rainfall 211.3 20.0 −20.0 0.2 0.2 0.3 0.1 0.4 0.3 0.1 0.1 0.3 0.3 0.3 0.3Rainfall CV 38.1 −0.3 0.0 −0.5 0.0 −100.0 0.0 −0.3 −0.3 0.3 −0.5 −100.0 −100.0 −100.0 −100.0Goat offtake annual 603 40.1 −63.0 −2.7 3.8 14.4 −0.3 −14.9 −86.1 −10.3 44.9 −40.0 −56.7 −82.4 59.4Milk yield annual 30018 40.5 −63.0 5.1 −7.0 21.7 −7.5 −13.7 −85.8 −10.3 45.3 −34.8 −51.7 −81.4 76.1Goals annual 1280 40.2 −61.9 6.2 −8.4 25.0 −9.5 −13.4 −81.7 −10.2 44.9 −30.2 −47.0 −77.3 81.8Goats cv 52.1 −3.1 84.3 5.2 −11.5 −47.2 42.0 4.4 174.1 4.0 −3.5 113.4 113.6 148.8 −79.3Goats replicate min 267 42.3 −85.8 0.4 15.0 237.5 −46.8 −15.4 −99.6 −13.5 52.8 −98.1 −98.1 −99.6 398.5Goats replicate max 2928 35.8 −39.1 11.8 −17.4 −6.8 22.4 −14.9 −49.8 −8.3 43.0 20.7 −21.0 −48.1 −4.1Goats final 1268 38.4 −86.0 6.6 −7.0 0.2 −36.1 −25.2 −97.7 −14.0 61.9 99.6 −99.6 −99.9 70.4Goat offtake final 597 37.9 −87.4 −3.5 5.2 −9.2 −29.3 −26.3 −99.3 −14.1 61.3 −100.0 −100.0 −100.0 47.6Probability goat

mortality >75%0.08% 3.6 9790.4 119.3 −49.4 −100.0 963.9 68.7 −24.1 27.7 −9.6 25201.2 46888.0−100.0 −100.0

Kids per dose 1.13 0.0 −4.4 −0.9 0.9 −8.0 3.5 −1.8 49.6 0.0 0.0 −31.0 −46.9 45.1 −9.7Years to goat max 51.2 6.2 −82.0 2.9 1.2 −66.8 −54.9 −33.6 −96.5 −10.4 29.7 −82.4 −80.5 −96.1 −70.7Probability goat max

before 20 years (%)20.6 −34.5 330.6 −2.9 −0.5 278.2 217.5 131.6 385.4 37.9 −82.0 385.4 385.4 385.4 385.4

Probability of goatextinction (%)

0.00 0.00 3.98 0.00 0 10.00 19.00 0.00 0.00

The last row (probability of goat extinction) shows absolute

0.00 0.00 0.23 0.03 0.40 0.00 0.0

values not percentage changes since this quantity is zero in the standard run (A).


4.3. Model responses to parameter changes

Conventional sensitivity analysis examines the re-sponses of a model to changes in each parameter inturn. However, it is often of more interest to considerparameter changes as reflecting realistic differences inthe field. These in turn drive complex changes in themodel response that cannot easily be picked up in thefield.

Table 9shows how model output with livestock al-lowed to vary (standard initial guild scenario) respondsto various changes. The responses are in terms of per-centage changes from a “standard” bulk run before anysuch changes have been made (column A), averagedas before over 1000 replicates. (The small change inrainfall CV – coefficient of variation – shown in mostcolumns is a result of rainfall being generated ran-domly.)

Columns B and C ofTable 9reflect the effects ofmean rainfall increasing and decreasing by 20% (butnevertheless randomized, with randomized seasonalityand summer bonus as usual). In column B, WP andME increase as one would expect, but there is a muchgreater increase in GA. This could be a consequence ofthe 40% increase in goat numbers which has a relativelydeleterious effect on WP and ME. Reduced rainfall hasthe expected effect (column C). In particular, goat num-bers fluctuate far more (their CV is up by 84%), there isnow a non-zero probability of a livestock wipeout, andthe probability of livestock mortality exceeding 75%h

thes bil-i byo opu-l tionr her(

f2 merb iffi-c ourG ver,l allerC G,w

ayse ith

the amount of rainfall still randomized). The effect ofrain being always early is as expected. It favours onlyGA, livestock numbers are more variable and mortalityis up. The effect of rain always late (column I) tendsto be the opposite of rain always fixed (column D): itfavours WP and ME at the expense of GA, yet livestocknumbers are down on average, with lower mortality, butmore variable.

Column K depicts the effect of always having a sum-mer rainfall bonus. WP, ME and annuals are up, live-stock numbers are up, mortality is down.

Probably the most interesting changes are those de-picted in column M, where the annual rainfall is fixed,it is always evenly distributed, and there is never a sum-mer bonus, i.e. there are no stochastic effects in the cli-mate. The result is devastating: all guilds and livestocknumbers are down, mortality is very high. Columns Aand M represent two climatic extremes: highly variablerainfall with a CV of 38% and stochastic seasonalityon the one hand (column A), with fixed rainfall andseasonality on the other hand (column M). This com-parison seems to suggest that it is largely the climaticvariability which leads to a “sustainable” rangeland (asdepicted in column A: initial, annual average and fi-nal average guild covers roughly the same; livestocknumbers allowed to vary freely, very low probabilityof livestock mortality exceeding 75%).

5. Discussion and conclusions

terme tocki us-i del( db e ass rtheri ingw re-s m-b ser-v nt on.T( -ct sons

as increased by nearly 10,000%.Columns D and E show the effects of changing

ell-by date of old does. In column D, the probaty of livestock mortality exceeding 75% increasesver 100%—a consequence of an older livestock p

ation. By contrast, the on average younger populaeflected in column E has a lower CV, with a higlower) average replicate minimum (maximum).

In column F, the rainfall isfixedat the average o12 mm each year; however, seasonality and sumonus is still randomized. The effect here is more dult to explain. The constant rainfall appears to favA at the expense of all the other guilds. Howe

ivestock numbers are up somewhat but with a smV. Constant rainfall implies a constant supply of Aith consequently fewer years of high mortality.Columns G–I show the effects of rain being alw

arly, always evenly distributed, or always late (w

This paper describes a comprehensive long-cosystem model of the climate–vegetation–lives

nteractions in a communal semi-arid rangeland,ng output from a short-term production system moRichardson et al., 2000). The model is validatey comparison with recorded data and experienchown above and provides a sound basis for fu

nvestigation. Preliminary runs of the model (startith values of 20, 12 and 10 for WP, ME and GA,pectively) support the view that when livestock nuers vary in a manner consistent with recorded obations (Fig. 5) herbivory has little long-term impact ohe guild composition and productivity of vegetatihis supports the concepts proposed byShackleton1993)andScoones (1995)but is contrary to the conlusion reached byIllius and O’Connor (1999)af-er analysis of published data. There are two rea


for this. First, plants which have reserves of inediblebiomass are able to survive under conditions when live-stock starve to death (Sullivan and Rohde, 2002). Forthe most important forage guild in the model, WP, 85%of the above ground biomass is inedible. Second, modeloutput shows that when years of adequate rainfall fol-low a drought that leads to a severe reduction in the an-imal population, the cover of productive species (WPand ME) recovers much faster than livestock numbers(Fig. 5).

The two-level approach in this paper of linking thelong-term ecosystem model described here with theshort-term production system model ofRichardson etal. (2000)addresses the major problem of processeswithin the system operating on different time scales.Cooper and Huffaker (1997)use slow-manifold theoryto link the fast (annual) dynamics of a herbivore–plantrelationship to the slow (decade-long) dynamics ofplant species competition. That method is not suitablefor representation of a Succulent Karoo system as itis designed to find equilibrium levels and successionalthresholds. It can only accommodate competition be-tween two plant types in the long-term and the pro-duction of forage by two plant types in the short-term.Forage is consumed by growing animals and produc-tion is measured as weight gain. The effects of climatevariability cannot be accommodated and neither thereproductive rates nor mortality of livestock are con-sidered. The two-level approach used here, however,provides a flexible framework for the modelling of sucha mo-d ntsa in ane rmp taryi en-e

.,2 tailw fallr ts oft sesw h isn ofa int -s falli tion

system model (seeAppendix A). The effect of season-ality is seen inFig. 4(b) when the model is run underthe evenly distributed rainfall scenario; in this case themodel output does not track recorded data nearly aswell.

In a communal pastoral system the reality is thatlivestock numbers vary widely between years and thisvariation affects the vegetation. An important featureof the present model is the way in which it realisticallysimulates changes in animal numbers over time in re-sponse to variations in forage availability and doe bodycondition. The GRAZPLAN decision support modelfor grazing enterprises in temperate Southern Australia(Donnelly et al., 1997; Freer et al., 1997; Moore etal., 1997) models sheep reproductive rates mechanisti-cally as a function of body condition and time of year,but not in terms of available forage. The spreadsheetmodel of cattle production in communal grazing areasof Zimbabwe (Campbell et al., 2000) was developed toevaluate the economics of various stocking rate strate-gies ranging from opportunistic (high stocking rates butsubject to crashes especially during droughts) to a con-stant low stocking rate. That model does not simulateany interaction between livestock and vegetation, nordoes it simulate livestock population dynamics: spe-cific calving, offtake and mortality rates are assumedfor each of the different strategies. The ecological eco-nomic simulation model of a grazing system in theNama Karoo (Beukes et al., 2002) does not specifi-cally simulate livestock mortality or reproductive ratesa ola of ac heldcs ionsa d di-g mu-l

ofv thern tialt tesm ions.T theN et hei The

system. The long-term ecosystem model accomates the population dynamics of four guilds of pland several age and physiological classes of goatsnvironment of highly variable rainfall. The short-teroduction model simulates the selection and volun

ntake of different guilds and the partition of dietaryrgy between various productive processes.

The production system model (Richardson et al000) simulates the behaviour of the system in deith a time step of 0.01 days and uses daily rain

ecords as input. These features enable the effeche temporal distribution of rainfall on all procesithin the system to be modeled, something whicot possible in models with a simulation time stepyear (Beukes et al., 2002) or even four months as

he IMAGES model (Hacker et al., 1991). This emphaises the importance of including seasonality of rainn the regression equations derived from the produc

lthough the effects of variation in rainfall on wond meat production are modelled. It is a modelommercial system where livestock numbers areonstant. In the IMAGES model (Hacker et al., 1991)tocking rate is determined by management decisnd simulated mortality rate depends on predicteestible dry matter intake. That model does not si

ate livestock breeding.Changes in the species (guild) composition

egetation in the Succulent Karoo, especiallyeplacement of palatable woody species by toxicGale-ia, influence livestock production. So, it is essen

hat any model of the production system simulaechanisms that cause changes in guild populathe model of grazing livestock and vegetation inama Karoo (Beukes et al., 2002) does not simulat

he effects of variation in annual rainfall nor tmpact of grazing on plant species composition.


objective of that model is to investigate the effects ofincreasing the number of paddocks in a given area onlivestock production and on profitability with the aimof achieving “non-selective grazing” and defoliationof all species to the same extent. This strategy maybe inappropriate for the Succulent Karoo where toxicGaleniacompetes with more palatable species.

The IMAGES model of a semi-arid grazing systemsimulates the responses of shrubs and ephemerals (an-nuals) to temporal variations in rainfall and to grazing(Hacker et al., 1991). Like the model described here,IMAGES also simulates the partial independence offorage production from the population of edible shrubs.In both models, the amount of edible forage producedper hectare by shrubs is a function of current rainfalland the number of shrubs per hectare. The latter vari-able changes over a longer time scale than does the den-sity of edible forage. However in the IMAGES modelshrubs are represented as one global population anddifferent types of shrubs are not considered; this ap-proach is not adequate for models of the Succulent Ka-roo where desirable WP compete with toxicGalenia.

The detailed mechanistic sub-model of milk pro-duction in the short-term production system model(Richardson et al., 2000) enables that model to predicton a daily basis the effects of energy intake and numberof kids suckled combined with optional partial hand-milking on the performance of both does and their off-spring. These effects are carried over into the long-termecosystem model. However, in GRAZPLAN (Freer etat c-c ord-i eiry

ys-t nal)l tioni tiono ) isa duc-t ri-c rr am-i calfa imu-l t in-v AN

model (Donnelly et al., 1997; Freer et al., 1997; Mooreet al., 1997) simulates the phenological development ofgrasses and forbs, while animal protein metabolism isrepresented specifically. One problem with both thesemodels is the amount of soil and vegetation informationrequired about specific areas to run the model. Thesedata are not readily available for communal rangelandsin arid regions such as Namaqualand.

The present model ignores spatial variation in veg-etation, when in reality grazing gradients occur aroundwater points and stockposts (night kraals) (Riginos andHoffman, 2003). A model of the arid chenopod shrub-lands of central Australia takes account of both spatialand temporal variability (Sparrow et al., 1997). In thatmodel one species of shrub and grass compete for spaceand water but the shrubs are not browsed as cattle arethe only livestock represented; herbage dynamics aredependent on soil moisture, stocking rate and the dis-tance from permanent water. Livestock breeding is notsimulated.

The present model is used in a subsequent paper(Richardson et al., 2004) to examine some of the con-cepts of the “new rangeland science” (Behnke andScoones, 1993), viz. that semi-arid rangelands operateunder non-equilibrium rather than equilibrium condi-tions and that pastoralists should adopt an opportunisticstrategy rather than a single conservative stocking rate.

Work is in progress on further papers which will usethe model to address the following key questions:

• andll un-nal,s?

• duc-thattesnge-eli-

• otaldi-

A

lat-i arch

l., 1997) Wood’s equation (Wood, 1967,1979) is usedo predict daily milk yield which is then adjusted aording to energy intake or body condition and accng to whether the dams are milked or suckling thoung.

There are two very detailed models of grazing sems that simulate commercial (rather than commuivestock production and in regions where vegetas more productive than in Namaqualand. Simulaf production and utilisation of rangelands (SPURcomprehensive mechanistic model of cattle pro

ion from a variety of grass types on North Amean rangelands (Foy et al., 1999). It can predict wateunoff, soil losses, soil carbon and nitrogen dyncs, plant production, forage selection, and cow,nd steer liveweights. However, SPUR does not s

ate the population dynamics of woody species thaade American rangelands. Similarly, the GRAZPL

How does the proportion of succulent, palatableunpalatable shrubs change in response to rainfader a range of stocking rates reflecting commucommercial and conservation land use practiceHow do management interventions (e.g. stock retion, supplementary feeding, changing breeds)lead to improved birth rates, lower mortality raand increased production per animal affect raland condition, offtake and hence people’s livhoods?How will climate change (especially a change in trainfall and rainfall seasonality) affect range contion and livestock production in the future?

cknowledgements

We thank Tony Starfield for assistance in formung aspects of the model, and the National Rese


Foundation and the University of Cape Town for fi-nancial assistance. Thanks also to Nicky Allsopp andothers who have participated in the Paulshoek project.The Mazda Wildlife Vehicle fund is thanked for theuse of a courtesy vehicle. This work was funded inpart by the European Commission under INCO-DC:International Cooperation with Development Coun-tries (2000–2004), Contract No. ERBIC18CT970162.However, the European Commission does not acceptresponsibility for any information provided or viewsexpressed.

Appendix A

SeeTable 10for an explanation of symbols used inall equations.

Plant regrowth factor after defoliation

Repeated defoliation of 0.70 of the edible materialof Karoo bushes reduces regrowth by 0.5 (Tainton andDanckwerts, 1999). This is too severe for our condi-tions where the defoliation is the amount removed overa year. For each perennial guild we therefore fit a curvewith the regrowth factorr set at 0.5 when the defoliationis 0.8:

r = 1.021234− 0.0441142e(5.211013d),

w hatg

isa

A

i re-l nd

TE

S

AAA ith

A

Table 10 (Continued)

Symbol Explanation

AGd(k) kg AG eaten per breeding doe (and her kids) withkkidsBy Birth rate of yearlingsBt(k) Birth rate of two-year-olds presently withk kidsBd(k) Birth rate of mature does presently withk kidsCy Conception rate of yearlingsCt(k) Conception rate of two-year-olds presently withk kidsCd(k) Conception rate of mature does presently withk kidsgh Goat units per hectareGAu kg edible GA available per goat unitGAy0 kg GA eaten per barren yearlingGAy(k) kg GA eaten per breeding yearling (and her kids) with

k kidsGAd0 kg GA eaten per barren doeGAd(k) kg GA eaten per breeding doe (and her kids) withkkidsh 1 or 0 according to whether kid’s mother is hand-milked

or notMEu kg edible ME available per goat unitMEy0 kg ME eaten per barren yearlingMEy(k) kg ME eaten per breeding yearling (and her kids) with

k kidsMEd0 kg ME eaten per barren doeMEd(k) kg ME eaten per breeding doe (and her kids) withkkidsMkd(k) kg milk consumed by kid of mature doe withk kidsMky(k) kg milk consumed by kid of yearling withk kidsMhy(k) kg milk from yearling with k kids for human

consumptionMhd(k) kg milk from doe withk kids for human consumptions Rainfall season (1, early; 2, even; 3, late)Skd(k) Survival rate of kid of mature doe withk kidsSky(k) Survival rate of kid of yearling withk kidsSy0 Survival rate of barren yearlingSy(k) Survival rate of yearling withk kidsSd0 Survival rate of barren doeSd(k) Survival rate of breeding withk kidsug Number of goat unitswky(k) Weight of kid of yearling withk kids (from previous

year)wkd(k) Weight of kid of mature doe withk kids (from previous

year)wd(k) Weight of mature doe withk kids (from previous year)wk Weight of kid (from previous year)w′

y0 Weight of barren yearling at end of seasonw′

y(k) Weight of breeding yearling withkkids at end of seasonw′

d0 Weight of barren doe at end of seasonw′

d(k) Weight of breeding doe withk kids at end of seasonWPe kg edible WPWPy0 kg WP eaten per barren yearlingWPy(k) kg WP eaten per breeding yearling (and her kids) with

k kidsWPd0 kg WP eaten per barren doeWPd(k) kg WP eaten per breeding doe (and her kids) withkkidsWPT Total kg WP eatenWPde Dead WP available (next year) per goat unitWPu Edible WP available per goat unit

hered is the proportion of the edible mass of tuild eaten the previous year.

The minimum value ofr is set at 0.2 since therelways some regrowth.

dditional effect of rainfall on edible matter

The edible biomass of each guild shown inTable 2s a potential representing unlimited rainfall. Theationship between growth of the edible fraction a

able 10xplanation of symbols used in the text

ymbol Explanation

Gu kg edible AG available per goat unitGy0 kg AG eaten per barren yearlingGy(k) kg AG eaten per breeding yearling (and her kids) w

k kidsGd0 kg AG eaten per barren doe


rainfall is assumed to be sigmoidal. Edible biomass ismultiplied by a factor

Er = ac

{ab + (cb − ab)e(−0.012431r)}(1/b),

wherea= 0.079396,b= 0.902389,c= 1.013134 andr is the annual rainfall. The maximum value ofEr is set to 1.The factorEr is based on the Richards equation (France and Thornley, 1984).

Conception and birth rates

Cy = 1.002547− 11.37819 exp(−0.096391wk) + 0.000024WPu,

By = 1.657087− 5.435499 exp(−0.048008wk) + 0.000039WPu,

Ct(k) = 1.002547− 11.37819 exp(−0.096391wy(k)) + 0.000024WPu (k indicates number of kids)

Bt(k) = 1.657087− 5.435499 exp(−0.048008wy(k)) + 0.000039WPu,

Cd(k) = 1.002547− 11.37819 exp(−0.096391wd(k)) + 0.000024WPu,

Bd(k) = 1.657087− 5.435499 exp(−0.048008wd(k)) + 0.000039WPu.

Milk consumption, survival rates and weights of kids

Mkd(k) = 84.949− 87.5579 exp(−0.004659WPu) − 9.5587s + 1.2393wd(k) − 29.2374k − 15.2168h,

Skd(k) = 1.168332− 5.649015 exp(−0.037813Mkd(k)) − 0.023425s,

wkd(k) = 26.5687− 28.6581 exp(−0.004741WPu) − 2.9964s + 0.2023wd(k) − 4.7567k − 2.1914h,

Mky(k) = 84.949− 87.5579 exp(−0.004659WPu) − 9.5587s + 1.2393wk − 29.2374k − 15.2168h,

Sky(k) = 1.168332− 5.649015 exp(−0. 037813Mky(k)) − 0.023425s,

wky(k) = 26.5687− 28.6581 exp(−0.004741WPu) − 2.9964s + 0.2023wk − 4.7567k − 2.1914h.

Amounts of each guild eaten

WPd0 = 685.747− 368.026 exp(−0.002577WPu) − 61.19 log(AGu) − 9.112s + 11.253 log(WPde),

MEd0 = 86.7877+ 24.4948 log(MEu) − 13.3526 log(WPu) − 20.229 log(AGu),

GAd0 = 12.2276+ 1.61903 log(GAu) − 2.9369 log(WPu),

AGd0 = 752.973− 273.472 exp(−0.002798AGu) − 23.874 log(WPu) − 61.1821 log(WPde) − 5.1305s,

WPd(k) = 746.186− 414.895 exp(−0.002192WPu) − 80.518 log(AGu) − 7.9285s

+ 3.1601 log(WPde) + 17.4311k,


MEd(k) = 122.702+ 31.9996 log(MEu) − 20.5183 log(WPu) − 25.5127 log(AGu) + 4.7098k,

GAd(k) = 16.3583+ 2.1366 log(GAu) − 3.9456 log(WPu),

AGd(k) = 973.465− 357.334 exp(−0.002503AGu) − 30.1075 log(WPu) − 83.156 log(WPde) + 9.9304k,

WPy0 = 685.747− 368.026 exp(−0.002577WPu) − 61.19 log(AGu) + 9.112s + 11.253 log(WPde),

MEy0 = 86.7877+ 24.4948 log(MEu) − 13.3526 log(WPu) − 20.229 log(AGu),

GAy0 = 12.2276+ 1.61903 log(GAu) − 2.9369 log(WPu),

AGy0 = 752.973− 273.472 exp(−0.002798AGu) − 23.874 log(WPu) − 62.1821 log(WPde) − 5.1305s,

WPy(k) = 746.186− 414.895 exp(−0.002192WPu) − 80.518 log(AGu) − 7.9285s

+ 3.1601 log(WPde) + 17.4311k,

MEy(k) = 122.702+ 31.9996 log(MEu) − 20.5183 log(WPu) − 25.5127 log(AGu) + 4.7098k,

GAy(k) = 16.3583+ 2.1366 log(GAu) − 3.9456 log(WPu),

AGy(k) = 973.465− 357.334 exp(−0.002503AGu) − 30.1075 log(WPu) − 83.156 log(WPde) + 9.9304k.

Milk production for human use

Mhd(k) = 32.9921− 45.5718 exp(−0.006505WPu) − 3.2679s + 0.4248wd(k) − 9.382k,

Mhy(k) = 32.9921− 45.5718 exp(−0.006505WPu) − 3.2679s + 0.4248wk − 9.382k.

W

w

w

w

w

S

S

S

S

eights of does and yearlings

′d0 = 53.299− 59.058 exp(−0.005431WPu) − 5.9395s + 0.3162wd(0),

′y0 = 53.299− 59.058 exp(−0.005431WPu) − 5.9395s + 0.3162wk,

′d(k) = 53.943− 46.382 exp(−0.003667WPu) + 0.1083wd(k) − 2.1664k − 4.7898s,

′y(k) = 53.943− 46.382 exp(−0.003667WPu) + 0.1083wk − 2.1664k − 4.7898s.

urvival rates of does and yearlings

d0 = 0.5581− 48479.89 exp(−0.309838w′d0) + 0.0113534wd(0),

d(k) = 0.961523− 605580.2 exp(−0.415356w′d(k)),

y(k) = 0.5581− 48479.89 exp(−0.309838w′y(k)) + 0.0113534wk.


Appendix B. Supplementary data

Supplementary data associated with this articlecan be found, in the online version, atdoi:10.1016/j.ecolmodel.2004.07.028.

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