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Ecological Modelling 351 (2017) 96–108

Contents lists available at ScienceDirect

Ecological Modelling

j ourna l h omepa ge: www.elsev ier .com/ locate /eco lmodel

ime-delayed biodiversity feedbacks and the sustainability ofocial-ecological systems

.-S. Lafuite ∗, M. Loreauentre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, Moulis, France

r t i c l e i n f o

rticle history:eceived 25 November 2016eceived in revised form 20 February 2017ccepted 23 February 2017

eywords:iodiversitycological economicscosystem servicesxtinction debtntegrative sustainability thresholdocial-ecological system

a b s t r a c t

The sustainability of coupled social-ecological systems (SESs) hinges on their long-term ecologicaldynamics. Land conversion generates extinction and functioning debts, i.e. a time-delayed loss of speciesand associated ecosystem services. Sustainability theory, however, has not so far considered the long-term consequences of these ecological debts on SESs. We investigate this question using a dynamicalmodel that couples human demography, technological change and biodiversity. Human populationgrowth drives land conversion, which in turn reduces biodiversity-dependent ecosystem services toagricultural production (ecological feedback). Technological change brings about a demographic transi-tion leading to a population equilibrium. When the ecological feedback is delayed in time, some SESsexperience population overshoots followed by large reductions in biodiversity, human population sizeand well-being, which we call environmental crises. Using a sustainability criterion that captures the vul-nerability of an SES to such crises, we show that some of the characteristics common to modern SESs (e.g.high production efficiency and labor intensity, concave-down ecological relationships) are detrimentalto their long-term sustainability. Maintaining sustainability thus requires strong counteracting forces,such as the demographic transition and land-use management. To this end, we provide integrative sus-tainability thresholds for land conversion, biodiversity loss and human population size - each threshold

being related to the others through the economic, technological, demographic and ecological parametersof the SES. Numerical simulations show that remaining within these sustainable boundaries preventsenvironmental crises from occurring. By capturing the long-term ecological and socio-economic driversof SESs, our theoretical approach proposes a new way to define integrative conservation objectives thatensure the long-term sustainability of our planet.

© 2017 Elsevier B.V. All rights reserved.

. Introduction

Current trends in human population growth (Cohen, 2003;erland et al., 2014) and environmental degradation (Vitousekt al., 1997) raise concerns about the long-term sustainability ofodern societies, i.e. their capacity to meet their needs “with-

ut compromising the ability of future generations to meet theirwn needs” (Brundtland et al., 1987). Many of the ecosystemervices supporting human systems are underpinned by biodiver-ity (Cardinale et al., 2012), and current species extinction rateshreaten the Earth’s capacity to keep providing these essential ser-

ices in the long run (Pereira et al., 2010; Ehrlich and Ehrlich,013). The long-term ecological feedback of ecosystem services onuman societies has been largely ignored by neo-classical economic

∗ Corresponding author.E-mail address: lafuite.as@gmail.com (A.-S. Lafuite).

ttp://dx.doi.org/10.1016/j.ecolmodel.2017.02.022304-3800/© 2017 Elsevier B.V. All rights reserved.

theory, mainly due to the focus on short-term feedbacks, andthe assumption that ecosystem processes can be substituted forby human capital (e.g. tools and knowledge) and labor, therebyreleasing ecological checks on human population and economicgrowth (Boserup, 1965; Dasgupta and Heal, 1974). In particular,tremendous increases in agricultural productivity resulted in morethan 100% rise in aggregate food supply over the last century(Schmidhuber and Tubiello, 2007).

However, the substitution of human capital for naturalresources, also called the “technology treadmill” (Pezzey, 1992), iscurrently facing important limitations. One such limitation is landscarcity, as the remaining arable land reserve might be exhaustedby 2050 (Lambin and Meyfroidt, 2011). Moreover, recent pro-jections suggest a slowdown in the growth rate of agricultural

Total Factor Productivity (TFP), which measures the effect of tech-nological inputs on total output growth relative other inputs(Kumar et al., 2008). Technological improvements may not com-pensate for arable land scarcity (Zeigler and Steensland, 2015),

gical M

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A.-S. Lafuite, M. Loreau / Ecolo

hus questioning the potential for continued TFP growth in theuture (Gordon, 2012; Shackleton, 2013). Another limitation comesrom the loss of many biodiversity-dependent ecosystem serviceshich play a direct or indirect role in agricultural production, such

s soil formation (Barrios, 2007), nutrient cycling, water retention,iological control of pests (Martin et al., 2013) and crop pollinationGallai et al., 2009; Garibaldi et al., 2011). Substituting ecologicalrocesses with energy and agrochemicals has mixed environmen-al impacts, with unintended consequences such as water useisturbance, soil degradation, and chemical runoff. These effectsre responsible for a slowdown in agricultural yield growth sincehe mid-1980s (Pingali, 2012) and have adverse consequences oniodiversity, human health (Foley et al., 2005) and the stability ofcosystem processes (Loreau and de Mazancourt, 2013).

Moreover, biodiversity does not respond instantaneously toand-use changes. Habitat fragmentation (Haddad et al., 2015)ncreases the relaxation time of population dynamics (Ovaskainennd Hanski, 2002) - i.e. how fast a species responds to environmen-al degradation. As a consequence, species extinctions are delayedn time, which generates an extinction debt (Tilman et al., 1994)nd a biodiversity-dependent ecosystem service debt (Isbell et al.,015). These ecological debts may persist for more than a cen-ury, and increase as species get closer to extinction (Hanski andvaskainen, 2002). The accumulation of these ecological debts mayave long-term effects on modern human societies.

Such time-delayed ecological feedbacks have been neglected byhe most influential population projection models (Meadows et al.,972, 2004; Turner, 2014). Yet, environmental degradation canave catastrophic consequences for human societies even withoutny delayed effect (Diamond, 2005; Ponting, 1991). A well-knownxample is Easter Island, this Polynesian island in which civilizationollapsed during the 18th century due to overpopulation, exten-ive deforestation and overexploitation of its natural resourcesBrander, 2007). In order to investigate the mechanisms behindhis collapse, Brander and Taylor (1998) modeled the growth ofhe human population as endogenously driven by the availabil-ty of natural resources, the depletion of which was governedy economic constraints. Their model was essentially a Lotka-olterra predator-prey model, which is familiar to ecologists, withn economic interpretation. It showed that one of Easter Island’scological characteristics - the particularly low renewal rate of itsorests - may be responsible for the famine cycles which broughtorward the collapse of this civilization.

Given the unprecedented rates of current biodiversity andcosystem service loss (Pereira et al., 2010), accounting for theirong-term feedback on modern human societies appears crucial. Inn attempt to delimit safe thresholds for humanity, a 10% loss inocal biodiversity was defined as one of the core “planetary bound-ries” which, once transgressed, might drive the Earth system into

new, less desirable state for humans (Scholes and Biggs, 2005;teffen et al., 2015). However, in practice, the biodiversity thresholdbove which ecosystem processes are significantly affected variesmong ecosystems (Hooper et al., 2012; Mace et al., 2014), ands correlated to other thresholds such as land-use change (Oliver,296). The definition of integrative thresholds thus requires thate consider the interaction between the economic and ecological

omponents of systems.In order to explore the long-term consequences of ecological

ebts for human societies, we build upon Brander and Taylor’sramework, but we allow the human population to produce itswn resources through land conversion. In our approach, terrestrialatural habitats provide essential ecosystem services to the agri-

ultural lands - which are assumed to be unsuitable to biodiversity.iodiversity and human population dynamics are coupled throughhe time-delayed effect of biodiversity-dependent ecosystem ser-ices on agricultural production (ecological feedback). It may be

odelling 351 (2017) 96–108 97

viewed as a minimal social-ecological system (SES) model that cou-ples basic insights from market economics and spatial ecology.From market economics, we derive human per capita consump-tions and the rate of land conversion. From spatial ecology, we usea classical species-area relationship (SAR) to capture the depend-ence of species diversity on the remaining area of natural habitat,and account for ecological debts through the relaxation rate of com-munities following habitat loss.

We investigate the behavior of the system at equilibrium ana-lytically, and then numerically evaluate the trajectories to theequilibrium. We show that the transient dynamics of an SESdepends on its ecological, economic, demographic and technologi-cal parameters. Some SESs experience large population overshootsfollowed by reductions in biodiversity, human population size andwell-being - which we call “environmental crises”. We then analyt-ically derive an integrative sustainability criterion that captures thevulnerability of an SES to such crises. This criterion allows assessingthe effects of some parameters on the long-term sustainability ofthe SES, and deriving integrative land conversion and biodiversitythresholds.

2. Methods

2.1. Coupling human and ecological dynamics

We model the long-term dynamics of three variables: thehuman population (H), technological efficiency (T) and biodiversity(B).⎧⎪⎨⎪⎩

H = �(B, T) H

T = � T(1 − T/Tm)

B = −� [B − S(H)]

(1)

The human population endogenously grows at a rate �(B, T),which is explicitly defined as a function of the per capita agricul-tural and industrial consumptions in Section 2.1.1. Technologicalefficiency is assumed to follow logistic growth at an exogenousrate �, until a maximum efficiency Tm is reached (Section 2.1.2).We use an economic general equilibrium framework to deriveper capita human consumptions at market equilibrium, i.e. whenproduction supply equals the demand of the human population(Section 2.1.3). Using these consumptions, we derive a proportionalrelationship between land conversion and the size of the humanpopulation (Section 2.1.4). Land conversion affects biodiversitythrough a change in the long-term species richness S(H). Currentbiodiversity B reaches its long-term level S(H) at a relaxation rate �(Section 2.2). Biodiversity-dependent ecosystem services then feedback on agricultural production and affect the per capita agriculturalconsumption and the human growth rate, �(B, T). Model structureis summarized in Fig. 1.

2.1.1. Human demographyThe interaction between human population, technology and

income has been mainly studied by endogenous growth theory,which distinguishes three phases of economic development (Galorand Weil, 2000; Kogel and Prskawetz, 2001): (1) a Malthusianregime with low rates of technological change and high rates ofpopulation growth preventing per capita income to rise; (2) a Post-Malthusian Regime, where technological progress rises and allowsboth population and income to grow, and (3) a Modern Growthregime characterized by reduced population growth and sustainedincome growth (Peretto and Valente, 2015). Transition to this third

regime results from a demographic transition which reverses thepositive relationship between income and population growth.

In order to consider the basic linkages between human demog-raphy and economics, the growth rate of the human population is

98 A.-S. Lafuite, M. Loreau / Ecological M

Fig. 1. Coupling between human and ecological dynamics. Black boxes: produc-tion sectors; grey boxes: dynamical variables; white box: auxiliary economic model;dashed lines: production inputs (labor Li , land Ai and technology T); solid lines:per capita outputs (yi); grey line: biodiversity-dependent ecosystem services (eco-lcA

ay

�

watdctntN2wls

2

Mn(niy(agoc

cpltef

ogical feedback, fS); black dotted line: effect of land conversion on biodiversity;ircle: total land divided into converted land (A1 + A2) and natural habitat (A3), where1 + A2 + A3 = 1.

ssumed to depend on the per capita consumptions of agricultural1 and industrial goods y2 (Anderies, 2003):

= �max(1 − exp(ymin1 − y1))exp(−b2y2) (2)

here �max is the maximum human population growth rate,nd ymin

1 is the minimum per capita agricultural goods consump-ion, such that human population size increases if y1 > ymin

1 , andecreases if y1 < ymin

1 . b2 is the sensitivity of � to industrial goodsonsumption, and thus captures the strength of the demographicransition. A higher agricultural goods consumption increases theet human growth rate while a higher industrial goods consump-ion eventually limits the net growth rate of the human population.ote that both y1 and y2 vary with the states of the system (Section.1.3). The system reaches its long-term equilibrium (y1 = ymin

1 )hen further technological change and habitat conversion no

onger increase total agricultural production, i.e. no longer compen-ate for the negative ecological feedback on agricultural production.

.1.2. Technological changeTechnological change is central to explain the transition from a

althusian to a modern human population growth regime. Tech-ology is often captured through the Total Factor Productivity termTFP) of a production function, which accounts for effects of tech-ological inputs on total output growth relative to the other inputs,

.e. labor and land in our model. Accelerating TFP growth in recentears partially compensated for the slowing down in input growthespecially land) and allowed total output growth to maintain itselfround 2% per year (Fuglie, 2008). However, recent reviews sug-est that agricultural TFP growth is slowing down in a numberf countries (Kumar et al., 2008) and that this trend is likely toontinue (Gordon, 2012; Shackleton, 2013).

In our model, technological change increases production effi-iency in the agricultural and industrial sectors, leading to higherroductions for a given level of inputs (i.e. converted land and

abor). We assume a logistic growth of production efficiencyowards a maximum efficiency denoted by Tm, where � is the exog-nous rate of technological change (system (1)). We explore otherorms of technological change in Appendix A.1.

odelling 351 (2017) 96–108

2.1.3. Instantaneous market equilibriumWe assume a closed market where total labor is given by the size

of the human population. Consumption and production levels arederived by solving for a general market equilibrium, where prices ofthe production inputs and consumption goods vary endogenously.As the economic dynamics are much faster than the demographicand ecological dynamics, the market is assumed to reach an equi-librium between supply and demand instantaneously.

On the demand side, the human population is assumed to behomogeneous, i.e. composed of H identical agents. Per capita con-sumptions of agricultural and industrial goods (y1 and y2, resp.) arederived from the maximization of a utility function, U(y1, y2), whichis a common economic measure of the satisfaction experienced byconsumers (Appendix A.2.1):

U(y1, y2) = y�1y1−�

2 (3)

where y1 is per capita agricultural demand, y2 is per capita industrialdemand, and � is the preference for agricultural goods.

On the supply side, a total quantity Yi of goods is produced bysector i, using two production inputs, labor Li, and converted landAi (Appendix A.2.2):

Y1 = TL˛11 A1−˛1

1 fS(B) Y2 = TL˛22 A1−˛2

2 (4)

where production efficiency is captured by the variable T, ˛i is therelative use of labor compared to land in sector i, and the functionfS(B) measures provisioning of biodiversity-dependent ecosystemservices to agricultural production (ecological feedback). SuchCobb-Douglas production functions with constant returns to scaleare a common assumption of growth models and allow for thesubstitution of land by labor. Potential ecological feedbacks onindustrial production were ignored.

When demand equals supply, the per capita consumptions are(Appendix A.2.3):

y1(B, T) = �1 fS(B) T/Tm y2(T) = �2 T/Tm (5)

where �1 and �2 are explicitly defined as functions of the parame-ters of the system in Table 1.

The per capita consumption of agricultural goods y1 is thussubject to the trade-off between the negative effect of decreasingbiodiversity-dependent ecosystem services, and the positive effectof technological change. Conversely, the per capita consumptionof industrial goods y2 monotonously increases with technologicalefficiency - as does the strength of the demographic transition.

2.1.4. Land conversion dynamicsRising agricultural and industrial productions require the con-

version and maintenance of land surfaces A1 and A2 respectively,at a cost of � units of labor per unit of converted area. Land conver-sion reduces the area of natural habitat A3, where A1 + A2 + A3 = 1.Let us denote by A the converted land, i.e. A = A1 + A2. At the marketequilibrium, the relationship between A and the human populationsize H is (Appendix A.2.4):

H = �A (6)

where � represents the density of the human population on con-verted land, and is explicitly defined as a function of the economicparameters in Table 1. The converted area A decreases with the landoperating cost �, since high operating costs reduce the incentive toconvert natural habitat. Proportional relationships between human

population sizes and converted surfaces are commonly observedin local and regional developing economies, where subsistenceagriculture remains strong and the transition to a modern growthregime is not achieved yet (Meyer and Turner, 1992).

A.-S. Lafuite, M. Loreau / Ecological M

Table 1Definition and default values of the parameters and dynamical variables.

Dynamical variables Initial value

H Human population 10−3

T Technology 10−3

B Biodiversity 1

Parameters Default values

Economic parameters� Agents preference for agricultural goods 0.5˛1 Labor intensity in the agricultural sector 0.3˛2 Labor intensity in the industrial sector 0.9

Technological parametersTm Maximum technological efficiency varies� Rate of technological change 0.5� Land operating cost 0.35

Demographic parameters�max Maximum growth rate 1ymin

1 Minimum per capita agricultural consumption 0.5b2 Sensitivity to industrial goods’ consumption 0.2

Ecological parameters Concavity of the BES relationship 1z Concavity of the SAR 0.3� Relaxation rate 0.6

Aggregate parameters

� �/(1 − ˛1� − ˛2(1 − �))

�1 �Tm˛˛11

((1 − ˛1)/�

)1−˛1

�2 (1 − �)Tm˛˛22

((1 − ˛2)/�

)1−˛2

′ 4zymin1 e−b2�2

′((

min) 1

z

)

2s

dtecta

2

anoSlcsr

S

I0dhdt

21

allow the human population to persist in the long term, and (2) a

�1/y1 − 1

� � − �max

.2. Spatio-temporal dynamics of biodiversity and ecosystemervices

We need two types of relationships, which capture (1) theependence of biodiversity upon the area of natural habitat A3 andhe size of the human population (S(H)), and (2) the dependence ofcosystem services upon biodiversity (fS(B)). Since species richnessan be related to the area and spatial characteristics of natural habi-ats (Mac Arthur and Wilson, 1967), we choose species richness as

biodiversity measure in our system.

.2.1. Species-area relationship (SAR)SARs are commonly used tools in ecological conservation to

ssess the effect of natural habitat destruction on species rich-ess (Pereira et al., 2014). A common way to describe the decreasef species richness with habitat loss is to use a power function

= c Az3, where c and z represent the intercept and the slope in

og-log scale. By choosing a unitary intercept (c = 1) and under theonstraint A1 + A2 + A3 = 1, the number of species S(H) that can beupported in the long run on an area A3 of natural habitat can beewritten:

(H) ={

(1 − H/�)z for H/� ≤ 1

0 for H/� > 1(7)

n terrestrial systems, the slope z typically ranges from 0.1 to.4 (Connor and McCoy, 1979; McGuiness, 1984), with valuesepending on ecosystem characteristics and species response toabitat loss. In our model, the long-term species richness S(H) thusecreases when the human population grows, and equals zero if

he human population exceeds its maximum viable size �.

We assume, following empirical (Ferraz et al., 2007; Wearn et al.,012) and theoretical (Mac Arthur and Wilson, 1967; Diamond,972) expectations, that the rate of community relaxation to

odelling 351 (2017) 96–108 99

this long-term richness is proportional to the difference betweencurrent richness, B, and long-term richness S(H) (system (1)), wherethe relaxation rate is � (Diamond, 1972).

2.2.2. Biodiversity-ecosystem service relationship (BES)Species richness has a positive effect on the level of many regu-

lating services (Cardinale et al., 2012), among which the pollinationand pest regulation services are particularly important to agricul-tural production - since the production of over 75% of the world’smost important crops and 35% of the food produced is dependentupon animal pollination (Klein et al., 2007).

The relationship between biodiversity and ecosystem servicescan be captured by a power function (Kremen, 2005; O’Connor et al.,2017):

fS(B) = B (8)

where < 1, since the shape of the function fS is mostly concave-down in terrestrial systems (Mora et al., 2014).

2.3. Model summary

System (1) can be rewritten as:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

H = �maxH(

1 − exp(ymin1 − �1BT/Tm)

)exp(−b2�2T/Tm)

T = �T(1 − T/Tm)

B ={

−� [B − (1 − H/�)z] for H < �

−� B for H > �

(9)

The aggregate parameters �1, �2, and � result from the eco-nomic derivations of sections 2.1.3, 2.1.4 and Appendix A.2. Allparameters and aggregate parameters are explicitly defined inTable 1.

In the short term, technological change and human populationgrowth drive land conversion and increase human consumptionand well-being. In the long term, the time-delayed biodiversity lossreduces the supply of biodiversity-dependent ecosystem servicesto agricultural production (ecological feedback). The ecologicalrelaxation rate � generates a time lag between human and eco-logical dynamics, mediated through the ecological feedback. In thenext section, we investigate the consequences of this time lag onthe long-term sustainability of an SES.

3. Results and discussion

3.1. Analysis of the dynamical system

For a given set of parameters and initial conditions, the dynam-ics of our SES is driven by the interaction between the dynamicalvariables summarized in eq. (9). In section 3.1.1, we characterize thesteady states that can potentially be reached by the SES in the longterm, and the necessary condition for their stability. We then sim-ulate in section 3.1.2 the transient dynamics of the SES over time,for various time lags between the ecological and human dynamics(�).

3.1.1. Steady states and stability conditionA steady state is reached by the dynamical variables when the

system is at equilibrium, i.e. when H = T = B = 0 (Appendix A.3.1).There are two potential steady states (H, T, B) for our system: (1)an undesirable equilibrium (0, Tm, 1), when the parameters do not

desirable equilibrium (H*, Tm, B*):

H∗ = �(

1 −(

ymin1 /�1

) 1z

)B∗ =

(ymin

1 /�1) 1

(10)

100 A.-S. Lafuite, M. Loreau / Ecological Modelling 351 (2017) 96–108

0 200 400 600 800 10000

0.5

1

Hum

an p

opul

atio

n

Time

A

0 200 400 600 800 10000

0.5

1

0 200 400 600 800 10000

0.5

1

Time

B

0 200 400 600 800 10000

0.5

1

Bio

dive

rsity

0 200 400 600 800 10000

0.5

1

Hum

an p

opul

atio

n

Time

C

0 200 400 600 800 10000

0.5

1

Bio

dive

rsity

0.6 0.7 0.8 0.9 10

0.5

1

Hum

an p

opul

atio

n

Biodiversity

D

H*

(1)

(3)

(2)

H*

B*

B*

Modern Growth Regime

Post−MalthusianGrowth Regime

Fig. 2. Transient human population and biodiversity dynamics with varying relaxation coefficients (�). Parameters are given in Table 1, with Tm = 1. A: negligibler imes ff 0); blac d; do

Fa(

�

Ct1l(in

rtr

3

bth

rett

btrrc

elaxation time (� = 10); B: high relaxation time (� = 0.02); C: differential relaxation tor biodiversity and human population. Grey curve: negligible relaxation time (� = 1urve), 2 (dashed curve) and 3 (dotted curve) until the equilibrium (B*, H*) is reache

or a given set of parameters, only one of these equilibria is stablend reached by the SES (Appendix A.3.2). The desirable equilibriumH*, Tm, B*) is stable if condition (11) is met:

1 > ymin1 (11)

ondition (11) captures the capacity of an SES to persist in the longerm, given its technological and economic characteristics (Table). The higher �1, the higher the initial human growth rate and the

ower the biodiversity at equilibrium B*. When condition (11) isnot) met, the SES reaches the desirable (undesirable) equilibriumn the long term. According to eq. (10), natural habitat (1 − H/�) isever entirely converted at equilibrium, since H/� < 1.

Condition (11) only guarantees that the desirable equilibrium iseached in the long term, but tells nothing about the trajectorieso the equilibrium, and especially about the effect of the relaxationate �.

.1.2. Transient dynamics for varying relaxation rates �In order to test for the effect of a time-delayed ecological feed-

ack on the transient dynamics of an SES, we numerically evaluatehe trajectories to the equilibrium and increase the lag between theuman and ecological dynamics by decreasing �.

When the relaxation time is negligible (1/� → 0), biodiversityesponds instantaneously to habitat conversion (B = S(H)) and thecological feedback on agricultural production is instantaneous. Inhis case, Fig. 2.A shows that transient trajectories converge mono-onically to the viable equilibrium (H*, B*).

Introducing a time-delay between the dynamics of humans andiodiversity by decreasing � leads to damped oscillations during

he transient dynamics to the equilibrium (Fig. 2B). Moreover, if theelaxation rates of species extinction and recovery are distinct (e.g.ecovery takes longer than extinction), these damped oscillationsan lead to the collapse of the human population (Fig. 2C).

or biodiversity loss (�− = 0.02) and recovery (�+ = �−/20); D: phase plane trajectoriesck curve: high relaxation time (� = 0.02) leading to the repetition of phases 1 (thick

uble-arrow: maximum amplitude of the transient crisis.

These damped oscillations result from the repetition of threesuccessive phases (Fig. 2D):

(1) Human population growth and biodiversity decline (� ≥ 0).During the initial growth phase, the human population reachesits equilibrium size H*. But at this point, the biodiversity debt isnot paid off yet, i.e. biodiversity and ecosystem services are stillin excess (B > B*). As a consequence, the per capita consumptionis not at equilibrium (y1 > ymin

1 ), and human population keepsgrowing until y1 = ymin

1 .(2) Environmental crisis and population decline (� ≤ 0). When

biodiversity eventually reaches its equilibrium value (B = B*),both human population growth and habitat conversion stop(y1 = ymin

1 ). However, the human population now exceeds itsequilibrium size (H > H*). This population overshoot results inan over-conversion of natural habitat which generates an addi-tional biodiversity debt. While this debt is paid off, humangrowth rate becomes negative (y1 < ymin

1 ) and human popu-lation declines.

(3) Human population and biodiversity recovery (� ≥ 0). As thehuman population declines, the converted surface decreasesand allows for the recovery of biodiversity. When biodiversity isnot in deficit anymore (B = B*), the human population can reachits equilibrium size, H*.

3.2. Sustainability sensitivity analysis

3.2.1. Sustainability conditionsTransient environmental crises are undesirable from a sustaina-

bility perspective, since these collapses result from a decrease inper capita agricultural consumption, and thus in human utility (eq.(3)) - which we use as a proxy for human well-being. Let us definean SES as sustainable when its transient dynamics monotonously

gical M

ceto

T

w(hadwc

�

wgTtcg

�w�i

FTtctw

A.-S. Lafuite, M. Loreau / Ecolo

onverge to the long-term equilibrium (i.e. no crisis) and the systemxperiences a net growth in human utility. Condition (12) ensureshat technological change is sufficient to compensate for the effectf biodiversity loss on human utility:

m/T(0) > (�1/ymin1 )

�(12)

here T(0) is the initial technological efficiency. When condition12) is met, human utility at equilibrium is higher than the initialuman utility (U* > U(0)), but environmental crises may still lead to

decrease in human utility and population size during the transientynamics. Using dynamical system properties (see Appendix A.3.2),e derive a general condition for the absence of environmental

rises:

> �max (13)

here is explicitly defined as a function of the economic, demo-raphic, technological and ecological parameters of the SES inable 1. Condition (13) implies that a given SES is not vulnerableo environmental crises if its ecological dynamics (�) is fast enoughompared to the growth of its human population (≈�max), for aiven set of parameters ().

Let us rewrite condition (13) as a sustainability criterion,

= � − �max. From the expression of in Table 1, it is straightfor-ard to deduce the effect of some parameters on �. In particular,

decreases with the preference for agricultural goods, �, sincet increases the global demand for agricultural goods. � increases

0 5 10 15 200

0.5

1

1.5

Eq.

hum

an p

op. (

H* )

A

0 5 10 15 200

0.5

1

1.5

0 5 10 15 20−6

−4

−2

0

2

Sus

t. cr

iterio

n (Δ

)

D

0 5 10 15 200

5

10

Eq.

util

ity (

U* )

Max. techn. efficiency (Tm

)

G

0 5 10 15 200

5

10

0 0.5 0

0.5

1B

0 0.5

0 0.5 −2

−1.5

−1

−0.5

0

0.5

1

E

0 0.5 0

0.5

1

1.5

Labor inten

H

0 0.5

ig. 3. Effect of varying parameter values on the steady states, the sustainability critem = 1.8 (B-E-H) and Tm = 1.2 (C-F-I). A to C: Effect of Tm , ˛1, and on the human populationransition is weak (b2 = 0.2). Grey areas represent the maximum amplitude of the transieriterion (� = � − �max), when demographic transition is weak (solid curve, b2 = 0.2) ando sustainable trajectories. G to I: Effect of Tm , ˛1, and on per capita human well-beinhen demographic transition is weak (b2 = 0.2). Grey areas represent the maximum amp

odelling 351 (2017) 96–108 101

with the land operating cost �, since high operating costs reducethe incentive to convert natural habitat (see Appendix A.4 formore details). The strength of the demographic transition (b2) alsoincreases the sustainability criterion �, by reducing the net growthrate of the human population.

In the next sections, we explore the effect of the other parame-ters through numerical simulations. We use the default parametersof Table 1 such that conditions (11) and (12) are met, i.e. the SES canreach its desirable equilibrium, and human well-being at equilib-rium is higher than the initial well-being. A given SES is sustainable(unsustainable) if its parameters satisfy � > 0 (� < 0) and condition(12). Any increase (decrease) in � has a positive (negative) effecton long-term sustainability. We then assess the effect of some char-acteristics of SESs on their steady states (H*, B*) and sustainability(�) by varying a single parameter at a time: the maximum techno-logical efficiency (Tm), the agricultural labor intensity (˛1) and theconcavity of the BES relationship (). For each of these parameters,we also plot the effect of the strength of the demographic transitionb2.

3.2.2. Effect of technological changeIn our model, production efficiency increases with technologi-

cal change until a maximum technological efficiency Tm. Increasing

Tm results in larger human population sizes and less biodiversity atequilibrium (Fig. 3A). When the demographic transition is weak(b2 = 0.2), the effect of technological change on sustainability isnegative (Fig. 3D), and the amplitude of the transient crises rises

110

0.5

1

1

1

sity ( α1)

10

0.5

1

1.5

0 0.5 1 1.5 20

0.5

1

C

0 0.5 1 1.5 20

0.5

1

Eq.

bio

dive

rsity

(B* )

0 0.5 1 1.5 2−10

−5

0

5F

0 0.5 1 1.5 20

0.5

BES concavity ( Ω)

I

0 0.5 1 1.5 20

0.5

Tot

al u

tility

(H

* .U* )

rion and human well-being at equilibrium. Parameters are given in Table 1, with size (black curve) and biodiversity (grey curve) at equilibrium, when demographicnt crises as defined in Fig.2.D. D to F: Effect of Tm , ˛1 and on the sustainability

strong (dashed curve, b2 = 2). Parameter values for which � is positive correspondg (U*, black curve) and total human well-being (H*. U*, grey curve) at equilibrium,litude of the transient variations in well-being.

1 gical M

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consider two alternative land-use scenarios: (1) no restriction onland conversion, and (2) conservation of an area of natural habitat

Fwn

02 A.-S. Lafuite, M. Loreau / Ecolo

ith technological change (Fig. 3A and D). On the other hand, tech-ological change has a positive effect on per capita and total humantility at equilibrium (Fig. 3G), since it increases industrial con-umption (y2) and the size of the human population at equilibrium*.

However, for a stronger demographic transition (b2 = 2), theffect of technological change on sustainability switches from nega-ive to positive at Tm = 4 (dashed curve in Fig. 3D). Further increasingm allows the industrial consumption y2 to reduce the net humanrowth rate to the point where the system does not experiencenvironmental crises (� > 0 for Tm > 10). Note that b2 only affectshe transient dynamics of the system, and thus does not preventigh levels of biodiversity loss when technological efficiency rises.

.2.3. Effect of labor intensityAs labor intensity ˛1 captures the relative use of labor com-

ared to land in the agricultural sector, a moderate increase in laborntensity (˛1 < 0.7) reduces land conversion requirements and ben-fits both biodiversity and sustainability (Fig. 3B and E). However,urther increasing labor intensity (˛1 > 0.7) increases agriculturalroductivity to the point where it lowers agricultural prices and

ncreases the demand for natural habitat conversion. This reboundffect thus allows the human population and total human util-ty to rise again, while biodiversity, sustainability and per capitauman utility decrease (Fig. 3B, E and H). A stronger demographicransition positively affects the sustainability criterion for all laborntensities, and thus partially mitigates the negative rebound effectdashed curve, Fig. 3E). Note that the shape of the relationshipetween sustainability and labor intensity depends on the landperating cost, �. If � = 1, the bell-shaped curve is centered on1 = 0.5. If � > 1 (� < 1, resp.), the sustainability criterion is max-

mum at ˛1 < 0.5 (˛1 > 0.5, resp.). Indeed, higher operating costseduce the incentive to convert natural habitat and increase sus-ainability (Appendix A.4), thus reducing the sustainability-optimalabor intensity.

.2.4. Effect of the concavity of ecological relationshipsThe concavity of the ecological relationships, i.e. the SAR and BES

elationships, is captured by their parameters z and , respectively.ince both parameters have similar effects on �, we only presenthe results for . Commonly observed concave-down BES relation-

hips ( < 1) lead to highly populated and biodiversity-poor SESst equilibrium (Fig. 3C). These SESs are characterized by a higherotal human utility (Fig. 3I) and a lower sustainability (Fig. 3F) thanESs with concave-up BES relationships. A stronger demographic

0.

1.

0 20 40 60 80 1000

0.5

1

1.5

2

Eq.

hum

an p

op. (

H* )

A

Time0 20 40 60 80 100

0

0.5

1

1.5

2

ig. 4. Land-use scenarios. For both scenarios, parameters are given in Table 1, with Tm =ith no land-use management. B: Implementation of the sustainable land conversionatural habitat 1 − AS is set aside.

odelling 351 (2017) 96–108

transition increases sustainability for both concave-down and -upBES relationships (dashed curve, Fig. 3F).

Modern SESs may thus be particularly vulnerable to the currentrise in technological efficiency and reduction in labor intensity -unless the demographic transition is strong enough to counteracttheir negative effects on long-term sustainability. Sustainable SESsare characterized by higher levels of biodiversity, lower humanpopulation sizes, industrial consumption and consumption util-ity at equilibrium, compared to unsustainable SESs. In section 3.3,we derive sustainability thresholds for land conversion, biodiver-sity and human population size, and explore their efficiency inpreserving the long-term sustainability of SESs through numericalsimulations.

3.3. Application to land-use management

3.3.1. Integrative sustainability threshold for land conversionOur sustainability condition � > 0 can be rewritten as A < AS

(Appendix A.3.3), where AS is the sustainable land conversionthreshold:

AS = 1 −(

′�max

� + ′�max

)(14)

′ is explicitly defined as a function of the economic, technologicaland ecological parameters of the SES in Table 1. AS represents themaximum area of natural habitat that a given SES can convert with-out becoming unsustainable, and depends on the economic andecological characteristics of the SES. In particular, AS decreases asthe ecological relaxation rate � decreases. As a result, the larger theecological relaxation time (i.e. the lower �), the more natural habi-tat a SES needs to preserve in order to remain sustainable. Using thisland conversion threshold AS, it is also possible to derive a sustain-able biodiversity threshold BS = (1 − AS)z, and a human populationthreshold HS = �AS.

3.3.2. Land-use scenariosIn this section, we explore the efficiency of the land conversion

threshold (Eq.(14)) in preventing environmental crises. To do so, we

1 − AS, e.g. through the creation of a protected area. In order to stopland conversion at the sustainable threshold AS without (directly)limiting the dynamics of the human population, we have to defineconverted land A as a fourth dynamical variable:

0 20 40 60 80 1000

5

1

5

2B

Time0 20 40 60 80 100

0

0.5

1

1.5

2

Eq.

bio

dive

rsity

(B* )

5. A: No restriction on land conversion; dynamics of an unsustainable SES (� < 0) threshold; dynamics of the same SES (� < 0) when the sustainable threshold of

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˙ =

{H/� for A ≤ AS

0 else(15)

ig. 4 shows that, in an unsustainable SES (i.e. � < 0), the firstcenario generates transient environmental crises (Fig. 4A), whilehe second scenario prevents environmental crises from occurringFig. 4B). A precautionary approach to natural habitat conservationppears necessary to prevent biodiversity and human populationrom exceeding their own sustainable thresholds, thus resulting innvironmental crises in the long run.

. Conclusions

We explore the long-term dynamics of coupled SESs byodelling the main reciprocal feedbacks between human popu-

ation and biodiversity, and show that the temporal trajectoriesf the human population and biodiversity are not necessarilyonotonous. In particular, for large ecological relaxation times,e observe the emergence of transient environmental crises, i.e.

arge reductions in biodiversity, human population size and well-eing. More complex population projection models calibrated withata for the world population, industrialization, pollution, food pro-uction and non-renewable resources obtained similar “overshootnd collapse” scenarios (Meadows et al., 1972). Recent updatesven suggest that the business-as-usual trajectory of our societ-es may be approaching such a global collapse (Meadows et al.,004; Turner, 2014). However, neither these population projectionodels nor neo-classical economic models consider any long-

erm ecological feedback of biodiversity on agricultural production,hich may worsen these crises scenarios.

The present work thus sheds new light upon the importancef accounting for ecological time lags when studying the sustaina-ility of SESs and defining management policies. Accumulating datan extinction debts (Wearn et al., 2012) and on the relationshipetween biodiversity and the provisioning of ecosystem servicesO’Connor et al., 2017) make it possible to include such time-elayed ecological feedbacks in more complex simulation models.he ability of our toy-model to perform population projections isimited, since many of our economic and ecological parameters doot vary endogenously. For example, the ecological relaxation rate

is known to decrease with habitat loss and fragmentation (Hanskind Ovaskainen, 2002), and population density on converted land

rose between 1993 and 2009 (Venter et al., 2016), since the rate ofand degradation (9%) was lower than that of the human populationnd economic growth (resp. +23% and +153%). We calibrated ourodel using mean values for these parameters during the period

960-2010, and chose other parameters in order to reproduce thebserved human population and TFP growth (Appendix A.5). Pop-lation projections suggest strong effects of ecological time lagsfter 2100, which are worsened if population density keeps rising.hese results urge for the inclusion of long-term ecological feed-acks in simulation models of population projections, as is alreadyone in models that couple the economy and climate (Nordhaus,993, 2010).

Although detailed policy-relevant population scenarios wouldequire a simulation model of greater complexity, our toy-modelllows assessing the qualitative effects of some parameters of SESsn their long-term sustainability. It provides us with an integra-ive sustainability criterion that captures the vulnerability of anES to environmental crises, given its economic, demographic,echnological and ecological characteristics. Using this criterion,

e show that some of the characteristics common to modern

ESs, such as high technological efficiency and low labor inten-ity, are detrimental to their long-term sustainability. Even thoughand-use efficiency reduces land conversion in the short term,

odelling 351 (2017) 96–108 103

it also increases the time lag between the dynamics of humansand biodiversity and the vulnerability of the SES to environmen-tal crises. Indeed, a more efficient use of land resources releasesecological checks on human population growth, which eventuallyincreases both land conversion and the extinction debt. Moreover,the commonly observed concave-down SARs and BES relationshipsexacerbate this effect, since a given amount of habitat conversiontranslates into a lower loss of biodiversity and ecosystem serviceswith concave-down than with concave-up relationships.

We show that counteracting forces such as the demographictransition and natural habitat conservation can mitigate environ-mental crises - provided that the ecological relaxation rates andthe sensitivity of the human growth rate to industrial consump-tion are high enough. Indeed, the demographic transition is oftenpresented as the solution to accelerate economic development andreduce environmental impact in developing countries (Bongaarts,2009; Brander, 2007). However, recent projections cast doubt uponits stabilizing potential (Bradshaw and Brook, 2014). Our resultsalso show that, in order to efficiently counteract time-delayed bio-diversity feedbacks, natural habitat conservation should take placewhen biodiversity is still abundant. Alternatively, habitat restora-tion can also help mitigating crises, and century-long extinctiondebts may be seen as windows of conservation opportunity (Wearnet al., 2012). However, our results suggest that we should not rely onhabitat restoration, since restoration delays are often longer thanextinction debts (Naaf and Kolk, 2015; Maron et al., 2012), whichworsens environmental crises.

The lack of such considerations about ecological time lags incurrent conservation policies calls for a precautionary approachto natural habitat conservation. To this end, we provide an inte-grative land conversion threshold that captures the long-termecological dynamics of species confronted with the destruction oftheir habitat, given the economic, technological and demographicparameters of an SES. The smaller the ecological relaxation rate(� → 0), the more natural habitat should be preserved in order toavoid environmental crises. Since relaxation rates decrease withhabitat loss and fragmentation (Hanski and Ovaskainen, 2002), thesustainable amount of natural habitat may increase when more andmore natural habitat is lost.

Regarding rates of current biodiversity loss (Newbold et al.,2016), it is thus crucial to determine how much natural habitat weneed to preserve in the long run. Recent attempts to define globalconservation thresholds for biodiversity and land use (Steffen et al.,2015) neglect the interaction between the social and ecologicalcomponents of SESs. For instance, as biodiversity moves closer toits own potential threshold, it reduces the land-use change thresh-old (Mace et al., 2014). There is also a lack of context-dependencyfor the application of these thresholds at local scales, as SESs withdifferent sets of economic, technological and ecological characteris-tics present different thresholds. Our theoretical model of a coupledSES proposes a way to move beyond these limitations. Our integra-tive thresholds for land conversion and biodiversity loss are notdefined independently from each other, but related through thesystem’s parameters. As a consequence, it is possible to predict howa change in one of the thresholds affects the other, and to definecontext-dependent land-use policies that prevent environmentalcrises from occurring.

Despite its simplicity, our model may be seen either as a thoughtexperiment about the sustainability of our planet, or as a rep-resentation of a local SES - which could be connected to otherSESs by considering trade, human migrations and species dispersal.Improvements to the model include adding property rights and

trade in order to gain realism in land-use change dynamics (Lambinet al., 2001), and internalizing the value of ecosystem services viapayments for ecosystem services and taxes on land conversion(Costanza et al., 2014). Finally, our results emphasize the critical

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eed for a better assessment of ecological time lags and how theyeed back on human systems. Reducing uncertainties on ecologicalime lags effects is crucial to be able to define integrative thresholdsnd foster sustainability of our planet, which may be seen as a globalocial-ecological system (Ehrlich and Ehrlich, 2013). The pursuit ofuch objectives requires taking into account the main internal feed-acks and specificities of coupled social-ecological systems, whichynamical system modelling allows.

cknowledgements

This work was supported by the TULIP Laboratory of Excel-ence (ANR-10-LABX-41) and the Midi-Pyrénées Region. We wouldike to thank two anonymous reviewers for their careful readingf our manuscript and their insightful comments and suggestions.e also thank Franc ois Salanié, Claire de Mazancourt, Bart Haege-an, Audrey Valls, Robin Delsol, David Shanafelt, Charles Perrings

nd Ann Kinzig for valuable discussions and helpful comments onarlier versions of the manuscript.

ppendix A. Appendices

.1. Endogenous or exponential technological progress

Technological efficiency T represents the Total Factor Produc-ivity (TFP) of the industrial and agricultural sectors. Followingecent concerns regarding the potential for continued TFP growthn the future (Gordon, 2012; Shackleton, 2013), we choose to modelechnological change as a logistic function, such that technologicalfficiency grows at an exogenous rate �, until it reaches a maximumechnological efficiency (Tm).

˙ = �T(1 − T/Tm) (A.1)

his function allows us to retain analytical tractability, and fornstance, derive analytical sustainability criteria. We could alsoave modeled technological change as endogenous, e.g. driven byhe rate of human population growth:

˙ = �(�)T(1 − T/Tm) (A.2)

here �(�) is a function of the human growth rate, for instance(�) = �(B, T). The behaviour of the system is not qualitativelyffected by this assumption. However, the system becomes ana-ytically intractable.

0 10 20 30 40 50 600

0.5

1

Eq.

hum

an p

op. (

H* )

Max. tech. efficiency (Tm

)

A

0 10 20 30 40 50 600

0.5

1 1

ig. A5. Effect of endogenous (A) and exponential (B) technological change on long-tehe transient and long-term dynamics of the system, when the maximum technological effirises. Parameters are given in Table 1, with = 2 and b2 = 0.5.; B: effect of an infinite techen the labor intensity ˛1 varies. Parameters are given in Table 1, with = 2, b2 = 0.1 an

odelling 351 (2017) 96–108

Another possibility is to assume an exponential technologi-cal progress, i.e. technological efficiency keeps growing instead ofreaching a maximum technological efficiency Tm:

T = �T (A.3)

In this case, the system has no global equilibrium (H*, T*, B*) and noanalytical results. However, as technological change increases thestrength of the demographic transition, it reduces the growth rateof the human population (�(T, B) → 0) along with land conversion,so that the human population and biodiversity reach a stationarystate in the long run.

In both cases (endogenous or exponential technologicalchange), however, our results would not be qualitatively affected.Indeed, with endogenous technological change and a strong demo-graphic transition (i.e. high value of the sensitivity of the humangrowth rate to industrial consumption b2), the effect of themaximum technological efficiency Tm on the vulnerability to envi-ronmental crises switches from negative to positive and reducesthe amplitude of the transient environmental crises (Fig. A5A).This corresponds to the dashed curve in fig.3.D (b2 = 2). Similarly,with an exponential technological change, the effect of vary-ing labor intensity ˛1 is non-linear (Fig. A5B), as it first reducesenvironmental crises before increasing human population sizeagain, through an economic rebound effect. This corresponds toFig. 3B and E.

A.2. Economic derivations

In this section, the aim is to derive the per capita agriculturaland industrial consumptions at the market equilibrium, i.e. whenthe production supply equals the demand of the human popula-tion. The human population is assumed to be composed of identicalagents, with preferences �. Each agent allocates part of his revenuew to buy agricultural goods, and the rest to buy industrial goods,which generates an aggregate demand for the human populationH (Section A.2.1). On the supply side, firms produce agriculturaland industrial goods given the costs of production inputs, i.e. landand labor, which gives the aggregate supply of the production sec-

tors (Section A.2.2). Section A.2.3 solves for the general equilibriummodel, and derives the per capita consumptions at the market equi-librium. Section A.2.4 then derives the rate of land conversion at theeconomic market equilibrium.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

0

Labor intensity (α1)

B

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

Eq.

bio

dive

rsity

(B* )

rm sustainability. A: effect of an endogenous technological change (� = �(B, T)) onciency Tm varies. Grey areas represent the amplitude of the transient environmentalhnological change (T = �) on the transient and long-term dynamics of the system,d � = 0.1.

gical Modelling 351 (2017) 96–108 105

A

iwayptT

L

F⎧⎪⎨⎪⎩Ast

p

A

uafiaop

Y

wcc⎧⎪⎨⎪⎩A

p

A

p{

(

A

a

L

A

(L(

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

Eq.

hum

an p

op. (

H* )

A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

Eq.

bio

dive

rsity

(B* )

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−10

−8

−6

−4

−2

0

2

Sus

t. cr

iterio

n (Δ

)

B

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.5

0.55

0.6

0.65

0.7

Land conversion cost ( κ)

Eq.

util

ity (

U* )

C

Fig. A6. Effect of the land conversion cost (�). A: Effect of varying � on the viableequilibria (H*, B*), with default parameters of Table 1 and Tm = 1.8. The grey arearepresents the amplitude of the transient crises. B: Effect of � on the sustainabilitycriterion (�). Values of � for which � is positive correspond to sustainable trajec-

A.-S. Lafuite, M. Loreau / Ecolo

.2.1. Consumer optimizationOverall, each agent supplies one unit of labor, so that his revenue

s the wage w. Let U(y1, y2) be the individual consumer well-being,here y1 and y2 are per capita consumption rates of agricultural

nd industrial goods. Agents maximize their well-being U(y1, y2) =�1y1−�

2 under the revenue constraint p1y1 + p2y2 ≤ w, where p1 and2 are the prices of agricultural and industrial goods respectively, whe per capita income, and � the preference for agricultural goods.o solve this maximization problem, we define the Lagrangian:

≡ U(y1, y2) − �[p1y1 + p2y2 − w]

irst order conditions are:

∂L/∂y1 = �U(y1, y2)y1

− �p1 = 0

∂L/∂y2 = (1 − �)U(y1, y2)y2

− �p2 = 0(A.4)

dding both conditions yields U(y1, y2)/� = p1y1 + p2y2 = w, andolving (A.4) for y1 and y2, and substituting for U(y1, y2)/� yieldshe aggregate demands:

1YD1 = �wH p2YD

2 = (1 − �)wH (A.5)

.2.2. Firms optimizationFirms in sectors 1 and 2 produce a quantity Yi (i = 1, 2) of output,

sing labor Li and land Ai. A units of land and H units of labor arevailable. Agricultural firms occupy A1 units of land, and industrialrms occupy A2 units of land, such that A = A1 + A2. This comes atn operating cost (including clearance and maintenance) of � unitsf labor per unit of land, so that each firm maximizes a profit �i =iYi − wLi − �wAi. The production functions Yi are:

1 = T fS(B) L1˛1 A1

1−˛1 Y2 = T L2˛2 A2

1−˛2 (A.6)

here fS(B) is the ecological feedback, T captures production effi-iency and ˛i is labor intensity in sector i. Therefore, first orderonditions are:

∂�i/∂Li = ˛ipiYi

Li− w = 0

∂�i/∂Ai = (1 − ˛i)piYi

Ai− �w = 0

(A.7)

dding both lines of (A.7) gives the total supply in sector i:

iYSi = w(Li + �Ai) (A.8)

.2.3. Economic general equilibriumMarkets’ clearing for the agricultural and industrial sectors

iYDi

= piYSi

(eq. (A.5) and (A.8)) yields:

�H = L1 + �A1

(1 − �)H = L2 + �A2

(A.9)

Input factors in each sector verify Li/˛i = �Ai/(1 − ˛i) (see eq.A.7)), so that the equilibrium land allocations are:

1 = H�(1 − ˛1)/� A2 = H(1 − �)(1 − ˛2)/� (A.10)

nd the equilibrium labor allocations are:

1 = H�˛1 L2 = H(1 − �)˛2 (A.11)

.2.4. Land conversion

The allocation of total labor H between agricultural production

L1), industrial production (L2) and land conversion (�A) writes1 + L2 + �A = H. Replacing L1 and L2 by their optimal allocationsEq.(A.11)), we deduce the relationship A = H/� between the human

tories. C: Effect of � on human well-being at equilibrium. The grey area representsthe amplitude of the transient variations in well-being.

population size H and the converted land A, where the density ofthe human population per unit of converted land is:

� = �

1 − ˛1� − ˛2(1 − �)(A.12)

A.2.5. Per capita consumptionsUsing the equilibrium allocations of labor Li and land Ai, the per

capita agricultural and industrial consumptions Y/H (eq.(A.6)) canbe rewritten:

y1 = �1BT/Tm y2 = �2T/Tm (A.13)

where

�1 = �Tm˛˛11

(1 − ˛1

�

)1−˛1

�2 = (1 − �)Tm˛˛22

(1 − ˛2

�

)1−˛2

(A.14)

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06 A.-S. Lafuite, M. Loreau / Ecolo

.3. Dynamical system analysis

.3.1. Equilibria and stability conditionThe Jacobian matrix at the viable equilibrium (H*, Tm, B*) writes:

∗ =

⎛⎜⎝

0 J1B−1∗ J1B∗

−�z

�

(1 − H∗

�

)z−1−� 0

0 0 −�

⎞⎟⎠

1 = �max�1 exp(−b2y2)Hast

The eigenvalues �i (i = 1 : 3) of the system are solutions ofet(J* − �I) = 0, where

et(J∗ − �I) = −(� + �)

(�(� + �) + ��max

4

)(A.15)

nd

= 4z

�

(1 − H∗

�

)z−1 J1B−1∗

�max(A.16)

The first eigenvalue is �1 =− �, and the two others are solutionf the characteristic equation �2 + �� + ��max/4 =0, which discrim-

nant is:

= �(� − �max) (A.17)

2 = (−� −√

D)/2 �3 = (−� +√

D)/2 (A.18)

1960 2060 2160 2260 23600

5

10

15

20

25

30

35

40

Years

Hum

an p

op. s

ize

(bill

ions

)

Bio

dive

rsity

t1/2 = 35 yrs

t1/2 = 23 yrs

1960 2060 2160 2260 23600

20

40

60

80

Years

Hum

an p

op. s

ize

(bill

ions

)

φ=30

φ=15

ig. A7. Model calibration and population projections. Top panels: Projected popula = 15 corresponding to the mean population density between 1960 and 2010. Half-lives

o extinction. A half-life of 35 years corresponds to a relaxation rate � = 0.02, and a half-lopulation and biodiversity trajectories for varying human population densities (�), wieforested land between 1960 and 2010. Parameter values: �max = 0.05; b2 = 2.29; Tm =

odelling 351 (2017) 96–108

The viable equilibrium is stable if all the eigenvalues have nega-tive real parts, i.e.

√D < �, which gives a stability condition for the

viable equilibrium (H*, Tm, B*):

> 0 ⇔ H∗ < � ⇔ �1 > ymin1 (A.19)

A.3.2. Sustainability conditionFor parameters such that the viability condition �1 > ymin

1is met, there are two possible transient dynamics dependingon the eigenvalues of the system: (1) for real eigenvalues, amonotonous convergence to equilibrium, and (2) for complex con-jugate eigenvalues, damped oscillations. The former case stands fora sustainable system, while the later one stands for an environmen-tal crisis. The eigenvalues are real if and only if the discriminantof the characteristic equation is positive, so that a sustainabilitycondition for our system is:

� > �max (A.20)

A sustainability criterion for our system is � = � − �max, where� > 0 stands for sustainable trajectories.

A.3.3. Sustainability thresholdsUsing the relationship between the biodiversity and human

population size at equilibrium H* = �A*, (eq.(A.16)) can be rewrit-ten as a function of the equilibrium converted area A*. The

sustainability criterion � then rewrites:

A∗ ≤ 1 −(

′�max

� + ′�max

)

1960 2060 2160 2260 23600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Years

t1/2

= 35 yrs

t1/2

= 23 yrs

1960 2060 2160 2260 23600

0.2

0.4

0.6

0.8

1

Years

Bio

dive

rsity

φ=15

φ=30

tion and biodiversity trajectories for varying ecological relaxation rates (�), with t1/2 = ln(2)/� measure the time it takes to loose half of the species that are doomedife of 23 years corresponds to a relaxation rate � = 0.03. Bottom panels: Projectedth � = 0.02. � = 15 corresponds to the mean density of human population on total

0.15; �1 = 60; �2 = 5; ymin1 = 1.7; � = 0.05. Dashed lines: long-term equilibria.

gical M

wOitoa

H

A

ataa(tbstsms

A

mcd≈l2pt

lttacllsp

sa2iha

lu

R

A

B

B

B

A.-S. Lafuite, M. Loreau / Ecolo

here ′ = 4zymin1 e−b2�2 , and � is the ecological relaxation rate.

ver a sustainable trajectory (� > 0), converted land never exceedsts equilibrium level, i.e. A ≤ A*. Condition (A.3.3) is thus equivalento A ≤ AS, where AS is the sustainable land conversion threshold ofur system. Similarly, the sustainability criterion can be written as

human or a biodiversity threshold as follows:

∗ ≤ �AS B∗ ≥ (1 − AS)z

.4. Effect of land operating costs on sustainability

The exploitation of converted land comes with operating costst each period of time, which include the initial cost of natural habi-at conversion, and the cost of land maintenance. Operating costs �re in units of human labor per unit of converted land. High oper-ting costs are beneficial to the long-term sustainability of the SESFig. A6.B), since they reduce the incentive to convert natural habi-at - in a similar manner to taxes on converted land. Therefore,iodiversity at equilibrium increases with � (Fig. A6A) while con-umption utility decreases (Fig. A6C). The effect of � on the size ofhe human population at equilibrium H* is non-linear (Fig. A6A),ince low values reduce the population density �, and high valuesake the system unviable (�1 → ymin

1 ), i.e. natural habitat conver-ion becomes to expensive.

.5. Model calibration

The model was calibrated on the period 1960-2010, using esti-ates for some parameters, e.g. z = 0.27 (Wearn et al., 2012), and

hoosing aggregate and demographic parameters in order to repro-uce the observed changes in human growth rate (≈2% in 1960,1.2% in 2010), human population size (3 billions in 1960, 6.9 bil-

ions in 2010) and TFP growth (nearly tripled) between 1960 and010. Trajectories lead to a population size around 15 billions peo-le in 2100, slightly higher than the medium fertility predictions ofhe United Nations (11.2 billions).

In order to explore the effect of relaxation rates on the popu-ation projections, we choose two relaxation rates correspondingo half-lives of 23 and 35 years. Top panels of Fig. A7 show thatrajectories are unaffected by ecological time-lags between 1960nd 2100, but diverge greatly afterwards. Note that the half-liveshosen are much shorter than the usual estimate of 57 years inocal communities (Wearn et al., 2012). The longer the half-life, thearger the growth of the human population and the resulting cri-is, since the long-term equilibrium is unchanged (with constantarameters, and especially constant population density).

Bottom panels show the effect of an increase in population den-ity on the long-term trajectories, with a 35 years half-life. Oncegain, trajectories are affected by population density only after100, the larger density leading to higher human population sizes

n the long term. Moreover, larger population densities result in aigher amplitude of the crisis, thus worsening the effect of relax-tion rates.

These results suggest that current trends of increasing popu-ation density and ecological relaxation times may lead to highlynsustainable pathways in the long run.

eferences

nderies, J.M., 2003. Economic development, demographics, and renewableresources: a dynamical systems approach. Environ. Develop. Econ. 8 (02),219–246.

arrios, E., 2007. Soil biota, ecosystem services and land productivity. Ecol. Econ.

64 (2), 269–285.

ongaarts, J., 2009. Human population growth and the demographic transition.Philos. Trans. R Soc. Lond. B: Biol. Sci. 364 (1532), 2985–2990.

oserup, E., 1965. The conditions of agricultural growth. Population Stud. 20 (1),1–108.

odelling 351 (2017) 96–108 107

Bradshaw, C.J.A., Brook, B.W., 2014. Human population reduction is not a quick fixfor environmental problems. Proc. Natl. Acad. Sci. 111 (46), 16610–16615.

Brander, J.A., 2007. Viewpoint: Sustainability: Malthus revisited? Can. J. Econ. 40(1), 1–38.

Brander, J.A., Taylor, M.S., 1998. The simple economics of Easter Island: ARicardo-Malthus Model of Renewable Resource Use. Am. Econ. Rev. 88 (1),119–138.

Brundtland, G., Khalid, M., Agnelli, S., Al-Athel, S., Chidzero, B., Fadika, L., Hauff, V.,Lang, I., Shijun, M., Morino de Botero, M., Singh, M., Okita, S., et al., 1987. OurCommon Future (The Brundtland Report). Oxford University Press.

Cardinale, B.J., Duffy, J.E., Gonzalez, A., Hooper, D.U., Perrings, C., Venail, P.,Narwani, A., Mace, G.M., Tilman, D., Wardle, D.A., Kinzig, A.P., Daily, G.C.,Loreau, M., Grace, J.B., Larigauderie, A., Srivastava, D.S., Naeem, S., 2012.Biodiversity loss and its impact on humanity. Nature 486 (7401), 59–67.

Cohen, J.E., 2003. Human population: the next half century. Science 302 (5648),1172–1175.

Connor, E., McCoy, E., 1979. The statistics and biology of the species-arearelationship. Am. Naturalist 113, 791–833.

Costanza, R., de Groot, R., Sutton, P., van der Ploeg, S., Anderson, S.J., Kubiszewski,I., Farber, S., Turner, R.K., 2014. Changes in the global value of ecosystemservices. Global Environ. Change 26 (1), 152–158.

Dasgupta, P., Heal, G., 1974. The Optimal Depletion of Exhaustible Resources. Rev.Econ. Stud. 41 (5), 3.

Diamond, J., 1972. Biogeographic kinetics: estimation of relaxation times foravifaunas of southwest pacific islands. Proc. Natl. Acad. Sci. USA 69 (11),3199–3203.

Diamond, J., 2005. Collapse: How Societies choose to fail or survive. Penguin.Ehrlich, P.R., Ehrlich, a.H., 2013. Can a collapse of global civilization be avoided?

Proc. R. Soc. Lond. B: Biol. Sci. 280 (1754), 20122845.Ferraz, G., Nichols, J., Hines, J., Stouffer, P., Bierregaard, J., Lovejoy, T., 2007. A

large-scale deforestation experiment: Effects of patch area and isolation onAmazon birds. Science 315 (5809), 238–241.

Foley, J.A., Defries, R., Asner, G.P., Barford, C., Bonan, G., Carpenter, S.R., Chapin, F.S.,Coe, M.T., Daily, G.C., Gibbs, H.K., Helkowski, J.H., Holloway, T., Howard, E.A.,Kucharik, C.J., Monfreda, C., Patz, J.A., Prentice, I.C., Ramankutty, N., Snyder,P.K., 2005. Global consequences of land use. Science (New York, N.Y.) 309(5734), 570–574.

Fuglie, K., 2008. Is a slowdown in agricultural productivity growth contributing tothe rise in commodity prices? Agric. Econ. 39, 431–441.

Gallai, N., Salles, J., Settele, J., Vaissiére, B., 2009. Economic valuation of thevulnerability of world agriculture confronted with pollinator decline. Ecol.Econ. 68 (3), 810–821.

Galor, O., Weil, D., 2000. Population, technology, and growth: from malthusianstagnation to the demographic transition and beyond. Am. Econ. Rev. 90 (4),806–828.

Garibaldi, L., Steffan-Dewenter, I., Kremen, C., Morales, J., Bommarco, R.,Cunningham, S., Carvalheiro, L., Chacoff, N., Dudenhffer, J., Greenleaf, S.,Holzschuh, A., Isaacs, R., Krewenka, K., Mandelik, Y., Mayfield, M., Morandin, L.,Potts, S., Ricketts, T., Szentgyrgyi, H., Viana, B., Westphal, C., Winfree, R., Klein,A., 2011. Stability of pollination services decreases with isolation from naturalareas despite honey bee visits. Ecol. Lett. 14 (10), 1062–1072.

Gerland, P., Raftery, a.E., Ev Ikova, H., Li, N., Gu, D., Spoorenberg, T., Alkema, L.,Fosdick, B.K., Chunn, J., Lalic, N., Bay, G., Buettner, T., Heilig, G.K., Wilmoth, J.,2014. World population stabilization unlikely this century. Science 346 (6206),234–237.

Gordon, R., 2012. Is U. S. Economic Growth Over? Faltering Innovation Confrontsthe Six Headwinds., pp. 18315.

Haddad, N.M., Brudvig, L.a., Clobert, J., Davies, K.F., Gonzalez, a., Holt, R.D., Lovejoy,T.E., Sexton, J.O., Austin, M.P., Collins, C.D., Cook, W.M., Damschen, E.I., Ewers,R.M., Foster, B.L., Jenkins, C.N., King, a.J., Laurance, W.F., Levey, D.J., Margules,C.R., Melbourne, B.a., Nicholls, a.O., Orrock, J.L., Song, D.-X., Townshend, J.R.,2015. Habitat fragmentation and its lasting impact on Earth’s ecosystems. Sci.Adv. 1 (2), e1500052–e1500052.

Hanski, I., Ovaskainen, O., 2002. Extinction debt at extinction threshold. Conserv.Biol. 16 (3), 666–673.

Hooper, D., Adair, E., Cardinale, B., Byrnes, J., Hungate, B., Matulich, K., Gonzalez, A.,Duffy, J., Gamfeldt, L., O’Connor, M., 2012. A global synthesis revealsbiodiversity loss as a major driver of ecosystem change. Nature 486, 105–108.

Isbell, F., Tilman, D., Polasky, S., Loreau, M., 2015. The biodiversity-dependentecosystem service debt. Ecol. Lett. 18 (2), 119–134.

Klein, A.-M., E.B., Cane, J.H., Steffan-Dewenter, I., Cunningham, S.A., Kremen, C.,Tscharntke, T., 2007. Importance of pollinators in changing landscapes forworld crops. Proc. R. Soc. Lond. B: Biol. Sci. 274 (1608), 303–313.

Kogel, T., Prskawetz, A., 2001. Agricultural Productivity Growth and Escape fromthe Malthusian Trap. J. Econ. Growth 6, 337–357.

Kremen, C., 2005. Managing ecosystem services: what do we need to know abouttheir ecology? Ecol. Lett. 8 (5), 468–479.

Kumar, P., Mittal, S., Hossain, M., 2008. Agricultural Growth Accounting and TotalFactor Productivity in South Asia: A Review and Policy Implications. Agric.Econ. Res. Rev. 21, 145–172.

Lambin, E.F., Meyfroidt, P., 2011. Global land use change, economic globalization,

and the looming land scarcity. Proc. Natl. Acad. Sci. U. S. A. 108 (9), 3465–3472.

Lambin, E.F., Turner, B.L., Geist, H.J., Agbola, S.B., Angelsen, A., Bruce, J.W., Coomes,O.T., Dirzo, R., Fischer, G., Folke, C., George, P.S., Homewood, K., Imbernon, J.,Leemans, R., Li, X., Moran, E.F., Mortimore, M., Ramakrishnan, P.S., Richards,J.F., Skånes, H., Steffen, W., Stone, G.D., Svedin, U., Veldkamp, T.A., Vogel, C., Xu,

1 gical M

L

M

M

M

M

M

M

M

M

M

N

N

N

N

O

08 A.-S. Lafuite, M. Loreau / Ecolo

J., 2001. The causes of land-use and land-cover change: Moving beyond themyths. Global Environ. Change 11 (4), 261–269.

oreau, M., de Mazancourt, C., 2013. Biodiversity and ecosystem stability: Asynthesis of underlying mechanisms. Ecol. Lett. 16 (s1), 106–115.

ac Arthur, R., Wilson, E., 1967. The theory of island biogeography. Monographs inPopulation Biology.

ace, G., Reyers, B., Alkemade, R., Biggs, R., Chapin III, F., Cornell, S., Diaz, S.,Jennings, S., Leadley, P., Mumby, P., Purvis, A., Scholes, R., Seddon, A., Solan, M.,Steffen, W., Woodward, G., 2014. Approaches to defining a planetary boundaryfor biodiversity. Global Environ. Change 28, 289–297.

aron, M., Hobbs, R.J., Moilanen, A., Matthews, J.W., Christie, K., 2012. Faustianbargains? Restoration realities in the context of biodiversity offset policies.Biol. Conserv. 155, 141–148.

artin, E.A., Reineking, B., Seo, B., Steffan-Dewenter, I., 2013. Natural enemyinteractions constrain pest control in complex agricultural landscapes. Proc.Natl. Acad. Sci. 110 (14), 5534–5539.

cGuiness, K., 1984. Equations and explanations in the study of species-areacurves. Biol. Rev. 59, 423–440.

eadows, D.H., Meadows, D.L., Randers, 2004. Limits to growth: the 30 yearupdate. EarthScan. London, Sterling, VA.

eadows, D.H., Meadows, D.L., Randers, J., Behrens III, W.W., 1972. The limits togrowth. Universe Books, New York, US.

eyer, W.B., Turner, B.L., 1992. Human Population Growth and GlobalLand-Use/Cover Change. Annu. Rev. Ecol. Syst. 23 (1), 39–61.

ora, C., Danovaro, R., Loreau, M., 2014. Alternative hypotheses to explain whybiodiversity-ecosystem functioning relationships are concave-up in somenatural ecosystems but concave-down in manipulative experiments. Scientificreports. Nature 4, 5427.

aaf, T., Kolk, J., 2015. Colonization credit of post-agricultural forest patches in NEGermany remains 130-230years after reforestation. Biol. Conserv. 182,155–163.

ewbold, T., Hudson, L.N., Arnell, A.P., Contu, S., De Palma, A., Ferrier, S., Hill, S.L.L.,Hoskins, A.J., Lysenko, I., Phillips, H.R.P., Burton, V.J., Chng, C.W.T., Emerson, S.,Gao, D., Pask-Hale, G., Hutton, J., Jung, M., Sanchez-Ortiz, K., Simmons, B.I.,Whitmee, S., Zhang, H., Scharlemann, J.P.W., Purvis, A., 2016. Has land usepushed terrestrial biodiversity beyond the planetary boundary? A globalassessment. Science 353 (6296), 288–291.

ordhaus, W., 1993. Optimal Greenhouse-gas Reductions and Tax Policy in the“DICE” model. Am. Econ. Rev. 83 (2), 313–317.

ordhaus, W., 2010. Economic aspects of global warming in a post-Copenhagen

environment. Proc. Natl. Acad. Sci. 107 (26), 11721–11726.

’Connor, M.I., Gonzalez, A., Byrnes, J., Cardinale, B., Duffy, J., Gamfeldt, L., Griffin, J.,Hooper, D., Hungate, B., Paquette, A., Thompson, P., Dee, L., Dolan, K., 2017. Ageneral biodiversity-function relationship is mediated by trophic level. Oikos126 (1), 18–31.

odelling 351 (2017) 96–108

Oliver, T., 2016. How much biodiversity loss is too much. Nature 353 (6296),220–221.

Ovaskainen, O., Hanski, I., 2002. Transient dynamics in metapopulation response toperturbation. Theor. Popul. Biol. 61 (3), 285–295.

Pereira, H., Ziv, G., Miranda, M., 2014. Countryside Species-Area Relationship as aValid Alternative to the Matrix-Calibrated Species-Area Model. Conserv. Biol.28 (3), 874–876.

Pereira, H.M., Leadley, P.W., Proenc a, V., Alkemade, R., Scharlemann, J.P.W.,Fernandez-Manjarrés, J.F., Araújo, M.B., Balvanera, P., Biggs, R., Cheung, W.W.L.,Chini, L., Cooper, H.D., Gilman, E.L., Guénette, S., Hurtt, G.C., Huntington, H.P.,Mace, G.M., Oberdorff, T., Revenga, C., Rodrigues, P., Scholes, R.J., Sumaila, U.R.,Walpole, M., 2010. Scenarios for global biodiversity in the 21st century.Science 330 (6010), 1496–1501.

Peretto, P., Valente, S., 2015. Growth on a finite planet: resources, technology andpopulation in the long run. J. Econ. Growth 20 (3), 305–331.

Pezzey, J., 1992. Sustainability: an interdisciplinary guide. Environ. Values 1,321–362.

Pingali, P.L., 2012. Green Revolution: Impacts, limits, and the path ahead. Proc.Natl. Acad. Sci. 109 (31), 12302–12308.

Ponting, C., 1991. A green History of the World: The Environment & the Collapse ofGreat Civilizations. No. 35. Penguin.

Schmidhuber, J., Tubiello, F.N., 2007. Global food security under climate change.Proc. Natl. Acad. Sci., 19703–19708.

Scholes, R., Biggs, R., 2005. A biodiversity intactness index. Nature 434, 45–49.Shackleton, R., 2013. Total factor productivity growth in historical perspective,

Working Paper 2013-01.Steffen, W., Richardson, K., Rockstrom, J., Cornell, S.E., Fetzer, I., Bennett, E.M.,

Biggs, R., Carpenter, S.R., de Vries, W., de Wit, C.A., Folke, C., Gerten, D., Heinke,J., Mace, G.M., Persson, L.M., Ramanathan, V., Reyers, B., Sorlin, S., 2015.Planetary boundaries: Guiding human development on a changing planet.Science 347 (6223), 1259855–1259870.

Tilman, D., May, R.M., Lehman, C.L., Nowak, M.A., 1994. Habitat destruction and theextinction debt. Nature 371 (6492), 65–66.

Turner, G., 2014. Is Global Collapse Imminent?Venter, O., Sanderson, E., Magrach, A., Allan, J., Beher, J., Jones, K., Possingham, H.,

Laurance, W., Wood, P., Fekete, B., Levy, M., Watson, J., 2016. Sixteen years ofchange in the global terrestrial human footprint and implications forbiodiversity conservation. Nature Commun. 7 (12558.).

Vitousek, P.M., Mooney, H.a., Lubchenco, J., Melillo, J.M., 1997. Human Dominationof Earth’ s Ecosystems. Science 277, 494–499.

Wearn, O.R., Reuman, D.C., Ewers, R.M., 2012. Extinction Debt and Windows ofConservation Opportunity in the Brazilian Amazon. Science 337 (6091),228–232.

Zeigler, M., Steensland, A., 2015. Global Agricultural Productivity Report (GAPReport). Global Harvest Initiative.