Non-Linear Dynamics of Population and Natural Resources: The Emergence of Different Patterns of Development Simone D’Alessandro∗† May 30, 2006
Abstract This paper studies the long-term dynamics between the exploitation of natural resources and population growth. Brander and Taylor (1998) started a sequence of papers which sought to deduce from historical and archaeological studies some stylized links between ecological and economic systems. In this strand of literature all the services of the natural environment are aggregated in a ecological complex which is characterized by a simple logistic dynamics. Given such assumptions these models show a unique long-term steady state. The aim of this paper is to obtain a more general framework that could account for the heterogeneity of environmental development paths followed by past societies. Two new assumptions are introduced: i) the disaggregation of the ecological complex into two different resources; ii) irreversibility - namely, an inexorable tendency to exhaustion when the renewable resource stock is below a certain threshold. Analysis of the dynamic properties of the system shows a multiplicity of steady states which makes it possible to consider the effects of technical progress, cultural and climate changes on the resilience of the existing development path.
JEL classification: Q20; N50; C61 Keywords: Renewable Resource; Population Growth; Multiple Equilibria; Critical Depensation; Feast-Famine Cycles; Resilience .
∗
Simone D’Alessandro: Department of Economics, University of Siena, Piazza San Francesco 7, Siena, Italy. E-mail address:
[email protected] † I am grateful to Tommaso Luzzati, Ennio Bilancini and two anonymous referees of Ecological Economics for very helpful comments on a previous version of the paper.
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1
Introduction
In the course of history, several societies have experienced paths of rapid growth followed by dramatic slumps. Most of the times, feast-famine cycles have been caused by two interconnected causal factors: uncontrolled population growth and the overexploitation of basic renewable resources (Diamond, 2005; Page, 2005; Ponting, 1991). Instead, there are examples of traditional societies which managed – and continue to manage – natural resources sustainably, maintaining a nearly constant level of population and wealth for an extremely long period of time (Ostrom, 1990). In order to understand the great heterogeneity of environmental paths of development, anthropologists, archaeologists and historical economists compare the socioeconomic organizations of different societies. It is however important to recognize that also the resilience of the natural environment plays a crucial role in determining the failure or success of societies in the sustainable exploitation of basic renewable resources; hence, endogeneity can be traced both ways between institutions and ecological systems. For this reason ecological-economic models can improve our ability to understand the long-term dynamics of our societies. Brander and Taylor (1998) presented a simple predator-prey model of renewable resource use which simulates the history of the Easter Island Civilization, describing the presence of feast-famine cycles. They showed that the overexploitation of natural resources caused a sharp reduction in the human population. Several other authors have developed this model taking into account additional aspects such as institutions, property rights and technical progress (Anderies, 1998; Dalton and Coats, 2000; Dalton et al., 2005; Pezzey and Anderies, 2003; Reuveny and Decker, 2000). All these models aggregate the services supplied by the environment in a unique complex renewable resource – as Brander and Taylor (1998, p.122) did – without distinguishing the dynamical properties of different resources. This simplification implies that the society always approaches a long-term steady state with positive population and positive stock of nat2
ural resource. In other words, it is not possible to have an equilibrium with positive population and the exhaustion of a part of the complex renewable stock which alters the organization of the society. This, for instance, contradicts the evidence from the history of Easter Island where the exhaustion of palm forest induced a collapse of the society.1 This collapse irreversibly changed the organization of Easter Island Civilization, while the simulations of the mentioned models would predict the recovery of the system in the long run. This example proves that such models are not able to describe the historical emergence of extremely different ways of managing renewable resources. This paper aims to modify the structure of the basic model in order to enable it to take into account the heterogeneity of historical paths of development. More precisely, this work explores the implications of two new assumptions: i) the presence of two natural resources and ii) a critical depensation curve which describes the dynamics of the renewable resource stock.2 The presence of two natural resources allows for a better specification of the different features of different resources. In order to stress this property of the natural environment, I assume that one resource is renewable and one is inexhaustible. This implies that, for certain values of parameters, the society can exhaust the renewable resource stock without extinguishing itself.3 Critical depensation allows for explicit investigation of irreversibility in the dynamics of the renewable natural resource. When the resource stock falls below a certain threshold the regeneration rate becomes negative and the natural resource stock is likely to become extinct. The dynamics of the resource stock shows three equilibria, two of which are stable: that implying extinction of the resource and that of carrying-capacity 1 When the first European explorers arrived on the island, they found that local inhabitants had extinguished the palm forests, hence the population was dramatically diminished but still constituted a viable society. 2 Critical depensation dynamics, as it is explained later, means that there exists a level of the stock of a renewable resource such that below that level the stock cannot be sustained and the rate of regeneration becomes negative. This feature is broadly referred to as the Allee effect (Allee, 1938). 3 See D’Alessandro (2006) for further implications of the introduction of this single assumption.
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level – the maximum stock that the limited environment can support. The third equilibrium is unstable and it is the critical level for the renewable resource, the threshold to extinction. Until recently, this specific dynamics was considered of limited importance, but it has recently been asserted that there is a widespread evidence for critical depensation (see for instance Liermann and Hilborn, 2001), which can also arise indirectly as a consequence of changes in the habitat caused by human activity.4 These two more general assumptions imply that the non-linear dynamics of population and natural resources allows for the emergence of multiple steady states. Consequently, small differences on both sides – the ecological aspects (e.g. intrinsic regeneration rate) and the socio-economic ones (e.g. preferences and technology) – may entail a marked divergence in the development path, leading the social-ecological system to approach either the sustainable or the unsustainable equilibrium.5 Moreover, since the time scale under consideration is extremely long it is interesting to investigate whether changes in technology, culture and climate may alter the resilience of the system.6 Hence, in the last part of the paper, I evaluate the effects of these changes on the basins of attraction of the system, investigating their consequences on the precariousness of the development path. The paper is organized as follows: sections II and III present respectively the static model and its dynamical component; section IV shows the qualitative properties of the dynamic system; section V performs some exercises in comparative dynamics, evaluating the effects of changes in technology, culture and climate on the resilience of the system; finally section VI presents some concluding remarks. 4
A typical example of induced Allee effect is the extinction of Passenger Pigeon, see Conrad (2005). In this paper, sustainable equilibrium is defined as the equilibrium of the dynamical system which allows the renewable resource to maintain a positive stock in the long run. Although this definition of sustainability can be consider oversimplified, it alludes to the strong influence of the exhaustion of a basic renewable resource on the organization of a society – take for instance the case of Easter Island Civilization. 6 Resilience is defined as “[...] the capacity of a system to absorb disturbance and reorganize while undergoing change so as to still retain essentially the same function, structure, identity, and feedbacks” (Walker et al., 2004). 5
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2
Analytical Model: Economic Structure
This section elaborates the analytical relations of a stylized economy in order to derive the equilibrium for any stock of renewable resource and for any level of population. The double causal link between these two variables and their dynamics are analyzed in the next section. The productive structure of the economy consists of two sectors which exploit two different natural resources – a renewable resource, called forest7 and an inexhaustible one, land – in order to produce two subsistence goods: wood 8 and corn. In any period, forest is exploited according to Schaefer production function (Schaefer, 1957)
H = αSLH ,
(1)
where H is the harvest produced – i.e. the quantity of wood – , S is the current stock of forest, LH is the number of workers employed in resource harvesting and α is a positive constant – i.e. a technological parameter. The index of time t is suppressed for convenience. There is a fixed amount of land and it is used in the agricultural sector in order to produce corn, according to the following production function
C = λLδC ,
(2)
where C is the quantity of corn produced, LC is the number of workers employed in the production of corn; the parameter δ ∈ (0, 1) implies that this technology shows 7
Of course, the choice of the name of the renewable resource does not change anything in this theoretical framework. I could also call it fish or wild animals. Forest was chosen in order to maintain the link with the history of Easter Island. 8 Actually wood is not a consumption good, rather a production factor. Indeed, in traditional societies, wood is used to make canoes and catch fish, for firewood, and the forest is also a nesting place for birds. It would also be possible to consider wood an intermediate commodity and include an additional final sector for the production of another subsistence good – e.g. fish. However, this change would only complicate the model, adding very little.
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decreasing returns to scale, and λ > 0 is a constant index of land fertility. The production function of corn shows decreasing returns to scale. This formalization is common in economics after the formulation of the Ricardian theory of land. The same theory does not apply to forest. Given a certain stock of forest it would be possible in principle to double the harvest of wood by doubling the labour force – hence the elasticity of labour with respect to wood in equation (1) is equal to 1. However, when wood is exploited, the stock of forest declines and this effect diminishes the productivity of labour.9 Let pH be the price of wood in terms of corn. I assume that both natural resources are open access and then each worker gets, as income, the amount produced in one period. This income will be called hereafter wage. Furthermore, assuming identical workers and free mobility between sectors, workers get the same wage w. Then, in each sector:
C = wLC ,
(3)
pH H = wLH .
(4)
It is useful to express the wage and the price of wood in the following way. From equations (2) and (3) and from (1) and (4) I obtain
w = λLδ−1 , C
pH =
w . αS
(5)
(6)
Each worker consumes corn and wood on the basis of a parameter of preferences β which lies in the interval (0, 1). The parameter β represent the quota of income spent on wood. Therefore the individual demands are equal to 9
Consider for instance the increase in transportation costs when the stock of forest is reduced.
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h = βw/pH
c = (1 − β)w ,
(7)
where h and c are respectively the individual demand of wood and corn.10 If consumers have the same preferences, total demand is equal to the individual demand multiplied by the level of population L, that is
C D = (1 − β)wL ,
HD =
βwL , pH
(8)
(9)
where the superscript D denotes the demand. In equilibrium, it must holds C = C D and H = H D . Therefore, from equations (2), (5) and (8), the equilibrium number of workers employed in the corn sector is obtained as a function of the total population,
L∗C = (1 − β)L ,
(10)
and, from (1), (6) and (9), the equilibrium production of wood is obtained as a function of the stock of renewable resource and the level of population,
H ∗ = αβSL .
(11)
Finally, from (10) and (5), the equilibrium level of wage is given by
w∗ = λ(1 − β)δ−1 Lδ−1 . 10
(12)
This formalization of preferences is consistent with the maximization of the Cobb-Douglas utility function u = hβ c1−β , subject to the boundary pH h + c = w. This specification may seem oversimplified since it does not take into account the fact that the fraction of income spent on wood β should decline when the natural resource stock is dramatically reduced. However, If it is assumed endogenous preferences according to a positive relation between the level of the renewable stock and β, for example by using ¯ where K ¯ is the carrying capacity, βK¯ > 0, F ′ (S/K) ¯ > 0, F ′′ (S/K) ¯ < 0 and the form β(S) = βK¯ F (S/K) F (0) = 0; the qualitative dynamic analysis does not change. Hence, I avoid considering this additional feature.
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Consequently, the equilibrium relations for the other variables of the model are trivially obtained – i.e. C ∗ , L∗H , p∗H . It is also useful to express the individual demands at equilibrium,
h∗ = αβS,
(13)
c∗ = λ(1 − β)δ Lδ−1 .
(14)
Finally, I assume that, in each period given any couple {L, S}, the economy is in equilibrium, since the dynamics of population and renewable resource is much slower than that of the economy.11 The variation in time of these two variables describe the development path of the complex ecological-economics system and also its long-term position.
3
Analytical Model: Equations of Motion
˙ The change in the stock of the natural resource at time t, S(t), is equal to the function G(S(t)) – which represents the natural dynamics of the stock in a virgin state – minus the human harvest H(t). In the literature, the natural rate of growth has been described by the simple logistic function G(S(t)) = ρ(1 − S(t)/K)S(t), where ρ is the intrinsic regeneration rate and K is the carrying capacity. This compensatory growth function12 is commonly employed to analyze the growth of a renewable natural resource stock in a limited environment. As I pointed out in the introduction, I do not use this function. In order to consider irreversibility in the dynamics of the renewable resource, I introduce a so-called critical depensation growth function.13 The main feature of this function is that the rate of 11 The time unit of the dynamical system is quite long – e.g. ten years in Brander and Taylor (1998). Hence the assumption that the economy instantaneously reaches the equilibrium is not restrictive. 12 A growth function is defined as compensatory when the rate of growth is always a decreasing function of the resource stock. 13 The term depensation illustrates the fact that the rate of growth can be a positive function of the renewable stock in some intervals. The adjective critical means that in addition the rate of growth is negative for small levels of the stock.
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growth becomes negative below a certain threshold, driving the natural resource stock to exhaustion. Several ecological studies stress that the fertility rate of many species declines when their number is excessively reduced, and there is evidence that some plants show the same dynamics.14 The use of the critical depensation curve aims to emphasize that human activities may damage the regeneration rate itself of renewable resources through a modification of the environment.15 This dynamics can be represented by the cubic function G1 (S(t)) = ρ(S(t)/K − 1)(1 − S(t)/K)S(t), where K is the carrying capacity and K is the unstable equilibrium below which the natural resource stock tends to the extinction, even without human exploitation. As I pointed out above, the total dynamics is obtained by deducting the harvest of wood H(t), from G1 (S(t)). Hence, dropping the time variable for convenience
S˙ = ρ(S/K − 1)(1 − S/K)S − H .
(15)
Given the equilibrium production of wood in equation (11), equation (15) can be written as
S˙ = [ρ(S/K − 1)(1 − S/K) − αβL]S .
(16)
The other equation of motion is given by the dynamics of population L. Like Brander and Taylor (1998) and the subsequent literature, I consider a Malthusian population dynamics, but unlike their models, here the consumption of both goods influences the fertility rate.16 According to Ricardo, the engine of population growth is the consumption of a certain quantity of corn beyond the quantity embodied in the natural wage level. In the particular case of one only wage good, there is a strict correspondence between the quantity consumed and the level of wage. With two wage goods, however, 14
Liermann and Hilborn (2001) reported some examples of the Allee effect on plants; see also Groom (1998); Kunin (1992); Lamont et al. (1993). 15 See for instance Conrad (2005); Albers (1996). 16 see also Prskawetz et al. (1994) for some additional concerns on the importance of Malthusian forces in developing countries.
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this correspondence does not hold. In order to maintain the same intuition, I define the goods in term of calories and assume that the population grows if and only if the quantity of calories consumed by workers is greater than a certain natural caloric level. In this interpretation the natural level of calories is expressed by that required by a worker to survive. Corn and wood have a different caloric power which do not need to be represented by their relative prices. For simplicity, I assume that the variation in time of the population will be equal to the difference between the calories consumed per capita and the per capita natural level of calories, multiplied by the number of workers, hence
L˙ = (γc + φh − σ)L .
(17)
where γ and φ are respectively the caloric unit of corn and wood, and σ is the per capita natural caloric level. For convenience, equation 17 can be written as
¯ −σ L˙ = γ(c + φh ¯ )L ,
(18)
where φ¯ := φ/γ and σ ¯ := σ/γ are expressed in terms of mass units of corn. Substituting the equilibrium level of individual demands, equations (13) and (14), it holds
¯ L˙ = γ(λ(1 − β)δ Lδ−1 + φαβS −σ ¯ )L .17
(19)
Note that the fixed quantity of land, the decreasing returns to scale in agriculture and the limitedness of the stock of the renewable resource mean that the population cannot persistently increase.
4
Interactions and Regimes
The equations of motion (16) and (19) form the system of simultaneous differential equations 17
Unlike Brander and Taylor (1998, p.124), I have the additional non-linear term λ(1 − β)δ Lδ−1 .
10
S˙ = [ρ(S/K − 1)(1 − S/K) − αβL]S ; L˙ = γ(λ(1 − β)δ Lδ−1 + φαβS ¯ −σ ¯ )L .
Steady State Analysis.
(20)
System (20) is in a steady state when S˙ and L˙ are simul-
taneously equal to zero. There are at least four equilibria of the dynamical system: 1. L∗ = 0,
S ∗ = 0;
2. L∗ = 0,
S ∗ = K;
3. L∗ = 0,
S ∗ = K; δ
)(1/(1−δ)) , 4. L∗ = ( λ(1−β) σ ¯
S ∗ = 0.
The first three equilibria are irrelevant for the purpose of this paper, as the dynamic system can never reach L = 0 if the initial value of population is positive, i.e. if L(0) > 0 (see appendix A). Other equilibria are possible. They are given by the interceptions of the two isoclines in the positive quadrant, namely the positive solutions {S ∗ , L∗ } of the following system: (
ρ(S/K − 1)(1 − S/K) − αβL = 0 ; ¯ λ(1 − β)δ Lδ−1 + φαβS −σ ¯=0.
(21)
Let N be the isocline of the natural resource given by the first equation of system (21); and P the isocline of the population given by the second one. Note that the N-isocline is a ∩-shaped parabola in the plane S, L; while the P-isocline is an increasing curve. Given the shapes of the two isoclines, depending on the parameter values, there may by two or zero internal steady states. Dynamics.
In order to simplify the analysis of stability, I investigate two different
cases.
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Case 1. When system (21) has no positive solutions, the dynamics of the system is shown in figure 1.18 In this case, the equilibrium E = {L∗ , S ∗ } - corresponding to the fourth steady state - is a stable node and it is also globally stable (see Appendix A); the resource stock will be exhausted in the long run and the population will approach δ
the value L∗ = ( λ(1−β) )(1/(1−δ)) .19 The arrows in the phase diagram of figure 1 show σ ¯ the global dynamics: when the system is on the left of the N-isocline both population and natural resource stock are increasing; after a while when the trajectory crosses N, the natural resource stock starts decreasing. After that point the natural resource stock can never increase, while the population continues growing. Once the P-isocline is crossed, the trajectory will asymptotically approach E remaining below that line. A possible trajectory is depicted in figure 1 by curve t. Notice that when the resource stock goes below the threshold level K – the lower intercept of N-isocline in figure 1 – the natural resource stock cannot be restored. Below that threshold no price level and no institutional change are capable of sustaining a positive stock of the natural resource in the long run.20 Case 2. When there exist positive solutions for system (21), that is there are three relevant equilibria, the behaviour of the dynamical system is much more complex. Notice δ
that the fourth equilibrium E = (( λ(1−β) )(1/(1−δ)) , 0) is always locally stable. The study σ ¯ of local stability for the internal equilibria is developed in the Appendix A. This analysis shows that the higher the level of natural stock at equilibrium, the more likely it is that this equilibrium is locally stable. Moreover, if one of the two internal equilibria lies in the upper branch of the parabola – i.e. the N-isocline – that equilibrium is locally stable 18
When possible, the values of the parameters are the same as those of Brander and Taylor (1998). However, given the theoretical approach followed in this paper, these values just aim to be reasonable. 19 Assuming β = 0 when S = 0, the population will approach the equilibrium L∗β=0 = (λ/¯ σ)(1/(1−δ)) along the horizontal axis L. Since the system does not reach E in finite time, I maintain the hypothesis that β is constant. This assumption would be inconsistent if the system lay in E: in that case β would be zero and the population would move along the horizontal axis until it reached the new equilibrium with a higher level of population. As remarked in note 10, endogenous preferences would take this element into account but the qualitative results of the model would not be substantially affected. 20 I am excluding that the natural resource stock can be regenerated, that is H < 0.
12
Figure 1: No internal equilibria. Parameters: α = 0.0001, λ = 20, δ = 0.7, β = 0.3, ¯ = 1.8. P and N are the ρ = 0.025, K = 700, K = 12000, γ = 0.1, φ¯ = 3, σ population and the resource isoclines; t is the trajectory given by L(0) = 200 and S(0) = 12000; E is the global stable equilibrium.
(see Appendix A). When both equilibria are in the lower branch of the parabola, local stability strictly depends on the values of parameters.21 Given this variety of scenarios, it is better to summarize the possible dynamics in two subcases: 1. The fourth equilibrium is the only locally stable one. In this case the system approaches asymptotically this steady state (as in case 1 ) or converges in a limit cycle around the upper internal equilibrium (see figure 2). 2. Also the upper internal equilibrium is locally stable. In this case the initial conditions will determine to which equilibrium the system moves, and the size of two basins of attraction depends on the values of parameters (see figure 3, where the threshold ts separates the two basins of attraction of the two locally stable 21
Appendix A performs a graphical analysis of the stability of internal equilibria by drawing two curves representing the trace and the determinant of the Jacobian of the linearization of system (21) in the plane L, S.
13
Figure 2: Limit Cycle. Parameters: α = 0.0001, λ = 12.95, δ = 0.7, β = 0.3, ¯ = 1.4. P and N are the ρ = 0.025, K = 700, K = 12000, γ = 0.1, φ¯ = 3, σ population and the resource isoclines; t is the trajectory given by L(0) = 200 and S(0) = 12000; ts is the threshold between the basin of attraction of the limit cycle and the one of equilibrium E.
equilibria E and E1). In Case 1, the parameter values lead to a unique global stable steady state characterized by the exhaustion of the renewable resource. This is possible when for instance the regeneration rate of the renewable resource is very slow with respect to the growth rate of the human population. In Case 2 instead, the dynamic system shows multiple steady states and a great heterogeneity of the stability properties of the equilibria. Small differences in technology, culture – preferences and reproductive choices – or climate may imply a strong divergence in the long-term development path. However, assuming a stationary environment – i.e. no change in any variable of the model – the trajectory followed by the social-ecological system is deterministic. Nevertheless, since the time scale under consideration is extremely long, it is important to evaluate what happens to a path of development under changes in parameter values. 14
Figure 3: Locally stable internal equilibrium. Parameters: α = 0.0001, λ = 18.5, δ = 0.7, ¯ = 1.8. P and N are the β = 0.3, ρ = 0.025, K = 700, K = 12000, γ = 0.1, φ¯ = 3, σ population and the resource isoclines; t is the trajectory given by L(0) = 200 and S(0) = 12000; ts is the threshold between the basin of attraction of equilibrium E1 and the one of E.
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Comparative Dynamics
For the sake of clarity, I assume that the social-ecological system shows the qualitative properties of the system in figure 3 and that the initial conditions are such as to lead the society to approach the upper internal equilibrium. At a certain point in time there is a shock which affects the value of a parameter. This change alters the basin of attraction of the internal equilibrium, modifying its resilience and therefore the path of convergence.22 Technical Progress.
I now proceed discussing the implications of an improvement
in the three technological parameters: α, λ and δ. The first represents the technology used in harvesting the renewable resource. An increase in α affects the dynamics of 22
Since there are many interpretations of the concept of resilience in social-ecological systems, I follow the terminology used by Walker et al. (2004).
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both the stock of the renewable resource and the level of population – see equation (20). Given a fixed level of population, the exploitation of the forest increases in α shifting the N-isocline to the left. At the same time, given a fixed level of the resource, the additional availability of wood increases the equilibrium level of population, shifting the P-isocline to the right. Figure 4 shows the shift due to the increase in α in the isoclines – from N to N1 and from P to P1. This implies that the resilience of the upper internal equilibrium is reduced. Hence, the improvement in the wood production technology causes a reduction in the latitude of the basin of attraction with a consequent increase of precariousness of the path of development. If the technology is significantly improved, then the trajectory followed by the society may also enter the basin of attraction of the unsustainable equilibrium. Figure 4 shows the case in which the society would follow a sustainable development path towards the long-term position E, but the improvement in technology means that the system enters the basin of attraction of equilibrium E1, which is characterized by the exhaustion of the renewable stock. The other two parameters determine the technology of the agricultural sector: λ is an index of land fertility, while δ exemplifies the decreasing returns to scale of the intensive use of land. Changes in both parameters affect only the dynamics of the population. Given a fixed amount of the renewable resource, the population equilibrium level increases, shifting the P-isocline to the right. This change has the same aggregate effect as an increase in α, reducing the resilience of the internal equilibrium. Cultural Changes.
The culture of the society is represented in this stylized model by
preferences and reproductive decisions. Changes in preferences are captured by changes in the parameter β. An increase of β entails a stronger preference for wood: given a fixed level of population, the harvest of wood H increases implying a shift of N-isocline to the left. A higher β also means that the fraction of population employed in the wood sector increases. Thus, for high levels of the stock of forest, food availability increases, which means that the population equilibrium level increases as well, while for low levels 16
Figure 4: Technical Progress (α). Parameters: λ = 18.5, δ = 0.7, β = 0.3, ρ = 0.025, K = 700, K = 12000, γ = 0.1, φ¯ = 3, σ ¯ = 1.8. At the starting time α = 0.00009 the isoclines are P and N and the system follows trajectory t (L(0) = 200 and S(0) = 12000). After a while, α changes to α1 = 0.00011, then P1, N1 are the new isoclines and t1 the new trajectory.
of the natural stock the population equilibrium level must decrease. Hence, the Pisocline rotates clockwise. Although the effects of an increase in β on the stability of the sustainable equilibrium cannot analytically be determined, simulations suggest that its resilience should decrease. Nevertheless, even changes in preferences may provoke – under certain conditions on parameters – a switch from one basin of attraction to the other. Along the path of development, the human population may also change its reproductive decisions. In particular, changes in a society’s habits may well increase the subsistence level of consumption summarized in this stylized model by σ ¯ . An increase in σ ¯ means that the P-isocline shifts to the left while the N-isocline is not affected. This change leads to an increase in the resilience of the sustainable equilibrium since the population increases more slowly. Therefore, the extension of the basin of attraction of
17
the sustainable equilibrium increases. Climate Change.
Finally, variations in the ecological sphere can involve the intrinsic
regeneration rate ρ, the minimum viable population K and the carrying capacity K and have a straightforward implication. Indeed, these factors influence only the isocline of the renewable resource: they have a negative impact on the resilience of the sustainable equilibrium if either ρ decreases, K increases or K decreases, while they have the opposite effect otherwise.
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Concluding Remarks
In the literature on the dynamic interactions between population and renewable resources, up until now the formalization of the natural environment has been oversimplified. In this paper I added two elements to it. First, I considered two natural resources instead of one; and, secondly, I introduced irreversibility in the renewable resource dynamics. These two new assumptions allow for the emergence of multiple steady states in the non-linear dynamical system. This contributes to explain the divergences in the development path of different societies. Moreover, The model allows analysis of the behaviour of the system in comparative dynamic exercises. Given the long time scale under consideration, I showed that small changes in technology, culture and climate modify the resilience of the system and can provoke irreversible switches from one basin of attraction to the other. Unlike the stream of literature originated by Brander and Taylor (1998), modifications of parameters not only alter the path of convergence to the steady state – and then the size of feast-famine cycles – but can also lead to a significant and irreversible change in the long-term position itself. In conclusion, examples from the past show that several societies followed paths of development which exhausted some basic renewable resources with a consequent dramatic deterioration of their wellbeing. The present analysis attempted to provide a 18
qualitative description of two possible mechanisms towards exhaustion: i) the initial conditions of the social-ecological system may not allow for sustainable development; ii) along the development path there may be some negative changes, which means a shift of the system into the basin of attraction of the unsustainable equilibrium. There is scope for further research to improve the model, allowing for additional details from historical and archaeological studies in order to make it suitable for empirical applications.
A
Appendix
In this appendix I study the local stability properties of all critical points. The method of linearization cannot be used in the equilibria with L∗ = 0 (i.e. equilibria one, two and three) because the derivative of the RHS of equation (19) is not defined for L = 0. However, starting from any strictly positive value of δ ¯ – where L ¯ := ( λ(1−β) )(1/(1−δ)) L, population cannot approach the L = 0 axis. Indeed for any L ∈ (0, L) σ ¯ is the horizontal intercept of P-isocline – L˙ > 0. Therefore, if the level of population at the initial time ¯ for any level of natural resource stock, the trajectory of the system will move towards is less than L, the right. The direction of the population shift will change only when the system reaches the isocline of ¯ Therefore, the first population and the level of population cannot decrease below the positive level L. three steady states are not interesting for the purposes of this paper. In order to study the local stability of the other equilibria I linearize through a Taylor series expansion the dynamical system around the equilibria. The steady states can be represented in the form E = (S ∗ , L∗ ), and can therefore be analyzed linearizing the vector e = (eS , eL ) = (S − S ∗ , L − L∗ ) around e = 0, that is around E. Then, it holds e˙ = Je + R(S, L),
(22)
where J is the Jacobian matrix of first order partial derivatives with respect to S and L evaluated at (S∗, L∗) that is
2 6 J=4
3ρ − KK S ∗2 + 2 (K+K)ρ S ∗ − ρ − αβL∗ KK ∗ ¯ γ φαβL
−αβS ∗ δ
∗(δ−1)
γ[δλ(1 − β) L
∗ ¯ + φαβS −σ ¯]
3 7 5.
(23)
The general solution of the linear system e˙ = Je is e = ξ1 ν1 eµ1 t + ξ2 ν2 eµ2 t , where ξ1 and ξ2 are integration constants, µ1 and µ2 are the eigenvalues of matrix J, and ν1 and ν2 are the corresponding eigenvectors. In order to prove the local stability of the equilibria I must determine the sign of eigenvalues. Notice that the eigenvalues of (23) are:
i p 1h trJ ± (trJ)2 − 4detJ , (24) 2 where trJ denotes the trace of the Jacobian J and detJ its determinant. δ Before differentiating case 1 and case 2, I prove that E = (( λ(1−β) )(1/(1−δ)) , 0) is always locally σ ¯ stable. Substituting E in the Jacobian (23) it holds µ1,2 =
2 6 J4 = 4
δ
−ρ − αβ( λ(1−β) )(1/(1−δ)) σ ¯ λ(1−β)δ (1/(1−δ)) ¯ γ φαβ( ) σ ¯
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0 γσ ¯ (δ − 1)
3 75 .
(25)
Since δ < 1, the trace of matrix (25) is negative and the determinant is positive. Hence, both the eigenvalues of (24) are negative for any considered value of parameters. It is also easy to prove that δ
the discriminant of the matrix is positive. Therefore, equilibrium E = (( λ(1−β) )(1/(1−δ)) , 0) is locally σ ¯ stable and the trajectories approach it monotonically. In case1 – that is, when there are no intercepts between the two isoclines – if I exclude L = 0 from δ the initial conditions, equilibrium E = (( λ(1−β) )(1/(1−δ)) , 0) is the only one which can be reached by the σ ¯ dynamical system and therefore it is globally stable. The arrows of the phase diagram in figure 1 show this result. Instead, in case 2 there are two internal equilibria, whose local stability generally depends on the values of parameters. Since the two internal equilibria – when they exist – are the solution of system (21), I can manipulate the Jacobian (23) in order to obtain a clearer expression. I define a1 the first element of the main diagonal of a Jacobian, and b2 the second element of the main diagonal. Subtracting the first equation of (21) from the term a1 of (23), and the second equation multiplied by γ from the term b2, it holds
2 6 J5,6 = 4
2ρ S ∗2 + − KK
(K+K)ρ ∗ S KK
∗ ¯ γ φαβL
−αβS ∗
3 7 5,
(26)
(δ − 1)γλ(1 − β)δ L∗ (δ−1)
where J5,6 denotes the Jacobian of the internal equilibria five and six. Since b2 of 26 is negative, a1 ≤ 0, that is S ∗ ≥ (K + K)/2, is a sufficient condition to have trJ < 0. In this case detJ > 0, therefore both eigenvalues are negative and the equilibrium is locally stable. Condition S ∗ ≥ (K + K)/2 holds when the population isocline crosses the superior branch of the natural resource isocline. The analysis of the discriminant of the characteristic equation shows that, depending on parameters, the convergence can be monotonic of cyclical. When both the equilibria lie in the inferior branch of the natural resource isocline, their local stability strictly depends on the values of parameters. To obtain some conditions on these values I consider a different manipulation of Jacobian (23). Subtracting the first equation of (21) multiplied by three from a1, and subtracting the second equation multiplied by δγ from b2, I get
2 6 J5,6 = 4
− (K+K)ρ S ∗ + 2ρ + 2αβL∗ KK
−αβS ∗
∗ ¯ γ φαβL
∗ ¯ γ(1 − δ)(φαβS −σ ¯)
3 75 .
(27)
Hence, the trace is negative if and only if trJ = −
(K + K)ρ ¯ S + 2ρ + 2αβL + γ(1 − δ)(φαβS −σ ¯ ) < 0. KK
(28)
The line −
(K + K)ρ ¯ S + 2ρ + 2αβL + γ(1 − δ)(φαβS −σ ¯) = 0 KK
(29)
can be drawn in plan [L, S]. When the internal equilibrium (L∗ , S ∗ ) is on the left of this line, trJ < 0. The expression of the determinant is
detJ
(K + K)ρ ¯ ¯ 2 β 2 SL = S + 2ρ + 2αβL][γ(1 − δ)(φαβS −σ ¯ )] + γ φα KK
=
[−
=
−aS 2 + bS − c,
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(30)
Figure 5: Local stability (trace and determinant). Parameters: α = 0.0001, λ = 17, δ = 0.7, ¯ = 1.8. T r(J) and β = 0.4, ρ = 0.025, K = 800, K = 12000, γ = 0.05, φ¯ = 2, σ Det(J) are the trace and the determinant of the Jacobian (26).
where: a
:=
b
:=
c
:=
(K + K)ρ ¯ γ(1 − δ)φαβ KK (K + K) ¯ ¯ 2β2L ργ(1 − δ)¯ σ + γ(1 − δ)φαβ(2ρ + 2αβL) + γ φα KK (2ρ + 2αβL)γ(1 − δ)¯ σ.
Since a, b, and c are positive, the determinant of (23) is positive if and only if 1/2a(b −
p
b2 − 4ac) < S < 1/2a(b +
where the two solutions are functions of L. Equations S1 (L)
=
1/2a(b −
S2 (L)
=
1/2a(b +
p
b2 − 4ac),
p b2 − 4ac) p2
(32)
b − 4ac) ∗
(31)
(33) ∗
can be drawn in the plan [L, S]. When the internal equilibrium (S , L ) is in between these two curves, detJ > 0. When the two conditions trJ < 0 and detJ > 0 are satisfied, the internal equilibrium under consideration is locally stable. In Figure 5, line T r(J) represents trJ = 0 of Jacobian (26), while curve Det(J) represents function (32). Curve (33) is not represented in this figure because for the range of parameters under consideration the value of S is always greater than K, so it is never binding. Therefore, the area of the graph where the internal equilibria are locally stable is that on the right of line T r and above curve Det. In this case, equilibria E and E1 are locally stable, while the lower internal equilibrium is locally unstable, so the characteristics of the global dynamics of the system are those of subcase (ii) of case2. When, instead, both internal equilibria are locally unstable the system is that described in subcase (i).
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