# World 2 equations, variables, and parameters

In this page we list the equations, the variables and the parameters of the World2 model as described in Chapter 3 and Appendix B of the book World dynamics (1973). Even if it is not said explicitily in the book, the World2 model consists of six systems containing several subsystems.

## Equations

System#DefinitionReferenceNotes
Population1$\frac{d\mathtt{p}}{dt} = \mathtt{br} - \mathtt{dr}$page 33see initialisation of $\mathtt{p}$
Population2$\mathtt{br} = \mathtt{p}\times\mathit{clip}(\mathtt{brn}, \mathtt{brn1},\mathit{t},\mathtt{swt1})\times\mathtt{brfm}\times\mathtt{brmm}\times\mathtt{brcm}\times\mathtt{brpm}$page 34see values of $\mathtt{brn}$, $\mathtt{brn1}$, and $\mathtt{swt1}$
Population3$\mathtt{brmm}=\mathit{interpolate}\left(\mathtt{msl}\right)$page 35see table $\mathtt{brmm}$
Capital investment4$\mathtt{msl}=\frac{\mathtt{ecir}}{\mathtt{ecirn}}$page 36see value of $\mathtt{ecirn}$
Capital investment5$\mathtt{ecir}=\frac{\mathtt{cir}\times(1-\mathtt{ciaf})\times\mathtt{nrem}}{1-\mathtt{ciafn}}$page 37see value of $\mathtt{ciafn}$
Natural resources6$\mathtt{nrem}=\mathit{interpolate}\left(\mathtt{nrfr}\right)$page 38see table $\mathtt{nrem}$
Natural resources7$\mathtt{nrfr}=\frac{\mathtt{nr}}{\mathtt{nri}}$page 38see initialisation of $\mathtt{nr}$
Natural resources8$\frac{d\mathtt{nr}}{dt} = - \mathtt{nrur}$page 39see initialisation of $\mathtt{nr}$
Natural resources9$\mathtt{nrur} = \mathtt{p}\times\mathit{clip}(\mathtt{nrun}, \mathtt{nrun1},\mathit{t},\mathtt{swt2})\times\mathtt{nrmm}$page 39see values of $\mathtt{nrun}$, $\mathtt{nrun1}$, and $\mathtt{swt2}$
Population10$\mathtt{dr} = \mathtt{p}\times\mathit{clip}(\mathtt{drn}, \mathtt{drn1},\mathit{t},\mathtt{swt3})\times\mathtt{drfm}\times\mathtt{drmm}\times\mathtt{drcm}\times\mathtt{drpm}$page 40see values of $\mathtt{drn}$, $\mathtt{drn1}$, and $\mathtt{swt3}$
Population11$\mathtt{drmm}=\mathit{interpolate}\left(\mathtt{msl}\right)$page 41see table $\mathtt{drmm}$
Population12$\mathtt{drpm}=\mathit{interpolate}\left(\mathtt{polr}\right)$page 42see table $\mathtt{drpm}$
Population13$\mathtt{drfm}=\mathit{interpolate}\left(\mathtt{fr}\right)$page 43see table $\mathtt{drfm}$
Population14$\mathtt{drcm}=\mathit{interpolate}\left(\mathtt{cr}\right)$page 44see table $\mathtt{drcm}$
Population15$\mathtt{cr}=\frac{\mathtt{p}}{\mathtt{la}\times\mathtt{pdn}}$page 44see values of $\mathtt{la}$ and $\mathtt{pdn}$
Population16$\mathtt{brcm}=\mathit{interpolate}\left(\mathtt{cr}\right)$page 45see table $\mathtt{brcm}$
Population17$\mathtt{brfm}=\mathit{interpolate}\left(\mathtt{fr}\right)$page 46see table $\mathtt{brfm}$
Population18$\mathtt{brpm}=\mathit{interpolate}\left(\mathtt{polr}\right)$page 47see table $\mathtt{brpm}$
Agricultural investment19$\mathtt{fr} = \frac{\mathtt{fpci}\times\mathtt{fcm}\times\mathtt{fpm}\times\mathit{clip}(\mathtt{fc}, \mathtt{fc1},\mathit{t},\mathtt{swt7})}{\mathtt{fn}}$page 47see values of $\mathtt{fc}$, $\mathtt{fc1}$, $\mathtt{fn}$, and $\mathtt{swt7}$
Agricultural investment20$\mathtt{fcm}=\mathit{interpolate}\left(\mathtt{cr}\right)$page 48see table $\mathtt{fcm}$
Agricultural investment21$\mathtt{fpci}=\mathit{interpolate}\left(\mathtt{cira}\right)$page 49see table $\mathtt{fpci}$
Capital investment22$\mathtt{cira}=\frac{\mathtt{cir}\times\mathtt{ciaf}}{\mathtt{ciafn}}$page 50see value of $\mathtt{ciafn}$
Capital investment23$\mathtt{cir}=\frac{\mathtt{ci}}{\mathtt{p}}$page 50
Capital investment24$\frac{d\mathtt{ci}}{dt} = \mathtt{cig}-\mathtt{cid}$page 50see initialisation of $\mathtt{ci}$
Capital investment25$\mathtt{cig} = \mathtt{p}\times\mathtt{brfm}\times\mathit{clip}(\mathtt{cign}, \mathtt{cign1},\mathit{t},\mathtt{swt4})$page 51see values of $\mathtt{cign}$, $\mathtt{cign1}$, and $\mathtt{swt4}$
Capital investment26$\mathtt{cim}=\mathit{interpolate}\left(\mathtt{msl}\right)$page 52see table $\mathtt{cim}$
Capital investment27$\mathtt{cid} = \mathtt{ci}\times\mathit{clip}(\mathtt{cidn}, \mathtt{cidn1},\mathit{t},\mathtt{swt5})$page 53see values of $\mathtt{cidn}$, $\mathtt{cidn1}$, and $\mathtt{swt5}$
Agriculture investment28$\mathtt{fpm}=\mathit{interpolate}\left(\mathtt{polr}\right)$page 54see table $\mathtt{fpm}$
Pollution29$\mathtt{polr} = \frac{\mathtt{pol}}{\mathtt{pols}}$page 54see value of $\mathtt{pols}$
Pollution30$\frac{d\mathtt{pol}}{dt} = \mathtt{polg}-\mathtt{pola}$page 54see initialisation of $\mathtt{pol}$
Pollution31$\mathtt{polg} = \mathtt{p}\times\mathit{clip}(\mathtt{poln}, \mathtt{poln1},\mathit{t},\mathtt{swt6})\times\mathtt{polcm}$page 55see values of $\mathtt{poln}$, $\mathtt{poln1}$, and $\mathtt{swt6}$
Pollution32$\mathtt{polcm}=\mathit{interpolate}\left(\mathtt{cir}\right)$page 56see table $\mathtt{polcm}$
Pollution33$\mathtt{pola}=\frac{\mathtt{pol}}{\mathtt{polat}}$page 57see table $\mathtt{polcm}$
Pollution34$\mathtt{polat}=\mathit{interpolate}\left(\mathtt{polr}\right)$page 58see table $\mathtt{polat}$
Agriculture investment35$\frac{d\mathtt{ciaf}}{dt} = \frac{\mathtt{cfifr}\times\mathtt{ciqr}-\mathtt{ciaf}}{\mathtt{ciaft}}$page 59see initialisation of $\mathtt{ciaf}$ and value of $\mathtt{ciaft}$
Agriculture investment36$\mathtt{cfifr}=\mathit{interpolate}\left(\mathtt{fr}\right)$page 60see table $\mathtt{cfifr}$
Quality life37$\mathtt{ql} = \mathtt{qls}\times\mathtt{qlm}\times\mathtt{qlc}\times\mathtt{qlf}\times\mathtt{qlp}$page 60see value of $\mathtt{qls}$
Quality life38$\mathtt{qlm}=\mathit{interpolate}\left(\mathtt{msl}\right)$page 61see table $\mathtt{qlm}$
Quality life39$\mathtt{qlc}=\mathit{interpolate}\left(\mathtt{cr}\right)$page 62see table $\mathtt{qlc}$
Quality life40$\mathtt{qlf}=\mathit{interpolate}\left(\mathtt{fr}\right)$page 63see table $\mathtt{qlf}$
Quality life41$\mathtt{qlp}=\mathit{interpolate}\left(\mathtt{polr}\right)$page 64see table $\mathtt{qlp}$
Natural resources42$\mathtt{nrmm}=\mathit{interpolate}\left(\mathtt{msl}\right)$page 64see table $\mathtt{nrmm}$
Agriculture investment43$\mathtt{ciqr}=\mathit{interpolate}\left(\frac{\mathtt{qlm}}{\mathtt{qlf}}\right)$page 65see table $\mathtt{ciqr}$

## Variable and (ODE) subsystem correspondance

DescriptionNameMain subsystemOther subsystems
Populationppopulationbirth_rate, capital_investment, capital_investment_generation, death_rate, natural_resources_usage_rate, pollution_generation
Birth ratebrbirth_ratepopulation
Birth rate from crowding multiplierbrcmbirth_rate
Birth rate from food multiplierbrfmbirth_rate
Birth rate from material multiplierbrmmbirth_rate
Birth rate from pollution multiplierbrpmbirth_rate
Capital investmentcicapital_investmentcapital_investment_discard
Capital fraction indicated by food ratiocfifragriculture_investment
Capital investment in agriculture fractionciafagriculture_investmentcapital_investment
Capital investment discardcidcapital_investment_discardcapital_investment
Capital investment generationcigcapital_investment_generationcapital_investment
Capital investment multipliercimcapital_investment_generation
Capital investment from quality ratiociqragriculture_investment
Capital investment ratiocircapital_investmentpollution_generation
Capital investment ratio in agricultureciracapital_investmentagriculture_investment
Crowding ratiocrpopulationagriculture_investment, birth_rate, death_rate, quality_life
Death ratedrdeath_ratepopulation
Death rate from crowding multiplierdrcmdeath_rate
Death rate from food multiplierdrfmdeath_rate
Death rate from material multiplierdrmmdeath_rate
Death rate from pollution multiplierdrpmdeath_rate
Effective capital investment ratioecircapital_investment
Food from crowding multiplierfcmagriculture_investment
Food potential from capital investmentfpciagriculture_investment
Food from pollution multiplierfpmagriculture_investment
Food ratiofragriculture_investmentbirth_rate, death_rate, quality_life
Material standard of livingmslcapital_investmentbirth_rate, capital_investment_generation, death_rate, natural_resources_usage_rate, quality_life
Natural resourcesnrnatural_resources
Natural resource extraction multipliernremnatural_resourcescapital_investment
Natural resource fraction remainingnrfrnatural_resources
Natural resources from material multipliernrmmnatural_resources_usage_rate
Natural resource usage ratenrurnatural_resources_usage_ratenatural_resources
Pollutionpolpollutionpollution_absorption
Pollution absorptionpolapollution_absorptionpollution
Pollution absorption timepolatpollution_absorption
Pollution from capital multiplierpolcmpollution_generation
Pollution generationpolgpollution_generationpollution
Pollution ratiopolrpollutionagriculture_investment, birth_rate, death_rate, pollution_absorption, quality_life
Quality of lifeqlquality_life
Quality of life from crowdingqlcquality_life
Quality of life from foodqlfquality_lifeagriculture_investment
Quality of life from materialqlmquality_lifeagriculture_investment
Quality of life from pollutionqlpquality_life

## Variable initializations

DescriptionInitializationReference
Capital investment$\mathtt{ci}(0)=0.4e9$page 50
Agriculture investment$\mathtt{ciaf}(0)=0.2$page 59
Natural resources$\mathtt{nr}(0)=900e9$page 39
Population$\mathtt{p}(0)=1.65e9$page
Pollution$\mathtt{pol}(0)=0.2e9$page 54

## Parameters

DescriptionNameValueSystem
Birth rate normal$\mathtt{brn}$$0.04Population Birth rate normal no. 1\mathtt{brn1}$$0.04$Population
Capital investment in agriculture fraction normal$\mathtt{ciafn}$$0.3Capital investment Capital investment in agriculture fraction adjustment time\mathtt{ciaft}$$15$Agriculture investment
Capital investment discard normal$\mathtt{cidn}$$0.025Capital investment Capital investment discard normal no. 1\mathtt{cidn1}$$0.025$Capital investment
Capital investment generation normal$\mathtt{cign}$$0.05Capital investment Capital investment generation normal no. 1\mathtt{cign1}$$0.05$Capital investment
Death rate normal$\mathtt{drn}$$0.028Population Death rate normal no. 1\mathtt{drn1}$$0.028$Population
Effective capital investment ratio normal$\mathtt{ecirn}$$1Capital investment Food coefficient\mathtt{fc}$$1$Agriculture investment
Food coefficient no. 1$\mathtt{fc1}$$1Agriculture investment Food normal\mathtt{fn}$$1$Agriculture investment
Land area$\mathtt{la}$$135e6Population Natural resource usage normal\mathtt{nrun}$$1$Natural resources
Natural resource usage normal no. 1$\mathtt{nrun1}$$1Natural resources Population density normal\mathtt{pdn}$$26.5$Population
Pollution normal$\mathtt{poln}$$1Pollution Pollution normal no. 1\mathtt{poln1}$$1$Pollution
Pollution standard$\mathtt{pols}$$3.6e9Pollution Quality of life standard\mathtt{qls}$$1$Quality life
Switch time no. 1 for brn$\mathtt{swt1}$$1970Population Switch time no. 2 for nrun\mathtt{swt2}$$1970$Natural resources
Switch time no. 3 for nrun$\mathtt{swt3}$$1970Population Switch time no. 4 for cign\mathtt{swt4}$$1970$Capital investment
Switch time no. 5 for cidn$\mathtt{swt5}$$1970Capital investment Switch time no. 6 for poln\mathtt{swt6}$$1970$Pollution