The models Directory UMM :Data Elmu:jurnal:E:Ecological Economics:Vol35.Issue3.Dec2000:

Even with simple models, the analysis is not easy. The two model behaviors a and b are separated by what is called a Hopf bifurcation which occur when, as a parameter is varied, the steady state changes from being locally stable to unstable, and a periodic orbit develops around the steady state. For simple systems, standard ana- lytic methods of dynamical systems theory can be applied reasonably easily Kuznetsov, 1995. For systems with more complex expressions, such analysis becomes increasingly difficult and may be impractical. The main tool I employ here is a numerical technique known as pseudo arclength continuation available in the software package Auto Doedel, 1981. By combining known ana- lytical results with powerful computational al- gorithms, models that are analytically tractable, but for which the analysis is impractical or cum- bersome, can be more efficiently, thoroughly, and accurately analyzed. The analysis amounts to starting at a known fixed point of the system and tracking its stability as a parameter is varied in very small steps. By locating points where the stability of the fixed point changes, we can detect local bifurcations and use these to divide the parameter space as mentioned above. This powerful tool for analyz- ing dynamical systems is freely available. Inter- ested readers should visit http:www.iam.ubc. caguidesxppaut for more details and download information. For more details on the application of the method for ecological models see van Coller, 1997.

4. The models

Presented here is a summary of the detailed models of Anderies 1998a and Brander and Tay- lor 1998 to illustrate the modeling approach and to serve as a basis for comparison for my exten- sion of the Easter Island model presented in Sec- tion 5. 4 . 1 . The Tsembaga The Tsembaga occupy a rugged mountainous region in the Simbai and Jimi River Valleys of New Guinea, along with several other Maring speaking groups with whom they engage in some material and personnel exchanges through mar- riages and ritual activity. These groups each oc- cupy semi-fixed territories that intersperse in times of plenty and become more rigidly separated in times of hardship. Outside these interactions, the Tsembaga act as a unit in ritual performance, material relations with the environment, and in warfare. The Tsembaga rely on a simple swidden slash- and-burn agricultural system as a means of sub- sistence. At the time of Rappaport’s 1968 field work they occupied about 830 ha, 364 of which were cultivable. The Tsembaga also practice ani- mal husbandry the most prominent domesticated animal being pigs, but derive little energetic value from this activity. Pork probably serves as a concentrated source of protein for particular seg- ments of the population as they are rarely eaten other than on ceremonial occasions, and several taboos surround its consumption that seem to direct it to women and children who need it most. The key point to keep in mind as the model is developed is that pigs are not an important food source for the Tsembaga in meeting their basic nutritional requirements. As such, the effect of pig consumption on the population growth rate is negligible compared to staple foods of plant origin. Much of the activity of the Tsembaga is related to the observance of rituals tied up with spirits of the low ground and the red spirits. The spirits of the low ground are associated with fertility and growth while the red spirits, which occupy the high forest, forbid the felling of trees. The ritual activity that is the focus here is the Kaiko, a year-long pig festival. The Kaiko serves to end a 5 – 25 year-long ritual cycle that is coupled with pig husbandry and warfare. It is this ritual cycle that Rappaport hypothesized acted as self-regula- tory mechanism for the Tsembaga population, preventing the degradation of their ecosystem. The three main ingredients of the ritual cycle, pig husbandry, the Kaiko itself, and the subse- quent warfare, are intricately interwoven with the political relationships between the Tsembaga and the neighboring groups. The Tsembaga maintain perpetual hostilities with some groups and are allied with other groups without whose support they will not go to war. There are two important aspects of pig husbandry: raising pigs requires more energy than is derived from their consump- tion, and pigs are the main source of conflict between neighboring groups because they invade gardens. From this perspective the keeping of pigs is completely nonsensical. However, the effort required to raise pigs is a strong information source about pressure on the ecosystem. The greater the pig population, the greater the chance an accidental invasion of others gardens will oc- cur. Each time a garden is invaded, there is a chance that the person whose garden was invaded will kill the owner of the invading pig. Records are kept of such deaths which must be avenged during the next ritually sanctioned bout of war- fare. From this perspective, pigs provide a meter of ecological and human population pressure and help measure the right amount of human popula- tion reduction to maintain ecosystem integrity. The Kaiko, when all but a few of the host group’s pig herd are slaughtered, helps facilitate material transfers with other groups, allows the host group to assess the support of its allies, and resets the pig population. The ritual cycle as the homeostatic mechanism proposed by Rappaport operates as follows: hu- man and pig populations grow until the work required to raise pigs is too great. A Kaiko is called and most of the pig herd is slaughtered. The Tsembaga are released from taboos prohibit- ing conflict with neighbors. Warfare begins with a series of minor ‘nothing fights’ where casualties are unlikely, then escalates to the ‘true fight’ where axes are the weapons of choice and casu- alties are much more likely. Periods of active hostilities seldom end in decisive victories but rather when both sides have agreed on ‘enough killing’ related to blood revenge from past injus- tices. The ritual cycle then begins anew with both the pig and human populations reduced to hope- fully levels that will not cause ecological degradation. Of course, Rappaport’s hypothesis that the rit- ual cycle could stabilize the system was chal- lenged. Rappaport’s detailed ethnographic and ecological information invited several efforts at formal modeling aimed at testing his hypothesis. Several very detailed simulation models were de- veloped that suggested that the ritual cycle could produce a stable equilibrium for the Tsembaga ecosystem, but this result was very sensitive to parameter choices Shantzis and Behrens, 1973. Later work using more realistic parameter choices concluded that the ritual cycle could not stabilize the Tsembaga model ecosystem Foin and Davis, 1984. The stylized model I developed focused on this question. The form of the functions RS, HS, L, GH, L in Eqs. 1a and 1b are based on Rappaport’s detailed ethnographic and ecolog- ical information, and some basic ecological con- siderations in tropical forests. Based on the qualitative aspects of the regeneration of tropical forests after fire see Anderies, 1998a for a de- tailed discussion, the resource dynamics are char- acterized by logistic regeneration, i.e. RS = rS1 − S 2 where r is the intrinsic regeneration rate of the resource. This is the simplest way to capture the forest regeneration process that is qualitatively similar to more complex models. Based on the nature of Tsembaga agriculture, I assumed HS, L food production function took a Cobb – Douglas form in labor and the resource stock: HS, L = kcL g S 1 − g 3 where c is the proportion of the labor force, Lt, devoting 1-person-year of energy 2000 h at 350 kcalh to agriculture, and k is a conversion fac- tor. The term cL embodies the work to cut down the forest for a garden, subsequently burn it, plant, and then finally harvest the plant biomass. The term S embodies the ‘forest condition’, i.e. it aggregates the effect of the biomass of forest cut to provide nutrients, soil structure, etc. in one term. Eq. 3 implies that the population devotes a constant proportion of its available labor to agri- culture. Rappaport’s rough numbers concerning the productivity of labor in Tsembaga agriculture enable one to make a reasonable guess for the parameters k, and g. Finally, based on very basic Fig. 3. Limit cycles that develop as the system becomes unstable. The inner cycle is for the case where the work level is constant at 0.14. The outer cycle represents the case where the work level is set by demand. assumptions about fertility and nutrition, I as- sumed that GH, L = g − g 1 exp g 2 H L 4 where g \ 0 is the nutrition independent compo- nent of the population growth rate, g 1 \ 0 is the maximum negative growth caused by malnutri- tion, and g 2 B 0 measures the sensitivity of the growth rate to nutrition. Eq. 4 says that the effects of malnutrition that suppress growth rates fall off exponentially as per capita intake of biore- sources increases. Again I refer interested readers to Anderies 1998a for more detail. The parameter of interest here is c which mea- sures how hard the population works. Here, a representative agent framework in which individu- als are identical is tacitly assumed. By performing a bifurcation analysis, it can be shown how in- creasing c could move the model system from case a to case b in Fig. 2 Anderies, 1998a. Thus, if the population did not work too hard, the system would be stable and there would be no need for collective action in the form of the ritual cycle. It turns out that for the value of c estimated by Rappaport at the time of his ethnographic study, c = 0.09, and for a wide range of physically mean- ingful values for the other parameters, the model exhibits a stable equilibrium population density of 0.6 persons per hectare a total of around 200 individuals in agreement with Rappaport’s data. This would suggest that the ritual cycle does not play an important role in mediating the impact of the Tsembaga on their ecosystem. The problem is that the Tsembaga adjust c to meet the nutritional demands of themselves and their pigs. To illus- trate, I added a very simple equation to describe the dynamic change of c over time: dc dt = l d − HL 5 where d is a constant minimum per capita food demand. My approach in this case is obviously quite different from a standard neo-classical eco- nomic model where each Tsembaga is assumed to be maximizing utility, and there is a well-defined market with an agricultural sector that almost instantaneously determines the efficient labor allo- cation, c. I am, rather, assuming that Tsembaga family units have imperfect knowledge of their system and adjust labor up or down according to their nutritional status. They react at a speed of l which is assumed to be much faster than ecologi- cal dynamics. These assumptions about behavior may seem a bit simplistic. However, when we compare this to the more complex economic model used by Brander and Taylor for Easter Island, we will see that there is very little differ- ence in the end. The Tsembaga do not collectively determine c at the population level; it is determined at the level of the family unit or individuals within the family unit. As such, Eq. 5 must be interpreted as representing individual decisions. If per capita intake is below the fixed minimum demand, the work level increases, and vice versa. By assuming a population of identical representative agents, d is constant across the population, per capita in- take is HL, and all agents make identical deci- sions. This unrealistic structure is a major limitation of this approach and completely ig- nores important heterogeneity across agents that would be captured in a multi-agent model. This is the price one has to pay to maintain the transpar- ency that is an important aspect of the utility of dynamical systems models. Fig. 3 summarizes the effect of behavior on the model for two cases, a constant c = 0.14 inner limit cycle, and c set dynamically by Eq. 5 outer limit cycle. The inner limit cycle shows the nature of the overshoot and collapse when the population works too hard, but does not adjust its work level. Starting from the lower right-hand corner of the limit cycle, the population grows to a maximum over about 240 years as natural capi- tal maximum = 1 corresponding to unexploited state degrades from 0.9 90 of unexploited state to about 0.4 40 of unexploited state. The population then collapses over 60 years as natural capital further declines to less than 10 of the unexploited state. The resource base then recovers over several hundred years with a very low human population density. The outer cycle shows the same sequence when c is dynamically set. In this case, it takes 720 years for the population to reach its peak, but notice that the limit cycle is much flatter on the top. This corresponds to the population slowly increasing its work level to maintain the population as the resource base becomes ever more degraded. By rapidly increasing work effort between t = 720 and t = 735, the population can almost maintain itself. This does nothing more than set the popula- tion up for a more dramatic crash when the system finally does collapse. In a mere 6 years as compared to 60, the population collapses dra- matically. The fact that individuals attempt to meet food demand as opposed to working at a fixed level, makes the overshoot and collapse cycle more dramatic. This is an important point that I will develop in more detail in the extended Easter Island model. The analysis above does suggest that without the evolution of some type of institution, the Tsembaga could not achieve an equilibrium or small amplitude, short period limit cycle with their environment and may not have been there for Rappaport to study. The next question is under what circumstances the ritual cycle could prevent the degradation of the resource base. By adding the ritual cycle to the model, I isolated two key components: the parasitism of pigs, and the way the number of individuals to be killed during a ritual warfare bout is determined. The bulk of the responsibility of keeping pigs falls on Tsembaga women. They do most of the work in planting, harvesting and carrying the crops used to feed the pigs. In this sense, the pigs can be viewed as parasitizing Tsembaga women. They benefit from energy derived from the ecosys- tem, but do not contribute to obtaining that energy. It turns out that this relationship, in and of itself, is enough to help stabilize the ecosystem. Since the pigs absorb work that might otherwise be directed at feeding a larger population, they effectively increase the per-capita work level re- quired to maintain a particular population. As the pig population grows relative to the human popu- lation, the per-capita work level increases to a point beyond which it cannot be maintained by Tsembaga women. In this way, the pigs act as an ecosystem monitoring device. The ritual cycle involves periods of peaceful coexistence during which the human and pig pop- ulations grow, followed by the Kaiko, the festival at which most of the pigs are harvested, followed by ritual warfare between rival tribes. This ritual warfare is the key population regulation mecha- nism that prevents the Tsembaga from inevitably degrading their resource base. In order to model this institution we need to keep track of the pig population, the harvest rate, and the outbreak of war. I accomplish this by adding three state vari- ables, pt, ht, and wt which represent the pig population, pig harvest rate, and warfare inten- sity, respectively. The pig harvest and warfare intensity are zero most of the time and switch on and off based on the pig-to-person ratio. This ratio is the key ecosystem monitoring device. Thus, to Eqs. 1a and 1b we add dp dt = r p − hp 6a dh dt = f 1 pL, h 6b dw dt = f 2 pL, w 6c where r p is the growth rate of the pigs. The functions f 1 and f 2 generate ‘relaxation oscillators’ that turn the harvest and warfare on and off based on the ratio pL. A detailed description and analysis of these functions is beyond the scope of this paper, and I again refer readers to Anderies 1998a. The dynamics generated by Eqs. 6a, 6b and 6c are shown in Fig. 4. As shown in the graph on the right, between Kaikos, the harvest rate is nearly zero. During this time, the pig population grows exponentially as shown on the left. During the Kaiko, the harvest rate rises abruptly and the pig population falls correspond- ingly. The warfare intensity has the same form as that of the harvest shifted slightly to the right, i.e. just after the Kaiko, warfare begins. The last element of the model is to incorporate the impact of warfare on human population dy- namics by modifying Eq. 1 to read dL dt = GH,LL − wDL 7 where DL gives the mortality due to warfare as a function of population size. It turns out that the key requirement for the ritual cycle to be effective is that DL must increase nonlinearly with the population size. If DL is a linear function of L, the ritual cycle can’t stabilize the system. For example, the as- sumption of a constant proportion of males being killed during a warfare bout in other simulation models Shantzis and Behrens, 1973; Foin and Davis, 1984 will likely lead to the conclusion that the ritual cycle could not stabilize the Tsembaga model ecosystem regardless of the character of the rest of the model. The analysis of the more styl- ized model given by Eqs. 1a, 1b, 2, 3, 4, 5, 6 and 7, makes it immediately clear that the nonlinear relationship between deaths due to warfare and population size is absolutely essential. The ques- tion then is whether such a relationship does in Fig. 4. The dynamics of the ritual cycle. Between Kaikos, the harvest rate is very low. When the pig to person ratio exceeds the tolerable level, the harvest rate increases dramatically representing the pig slaughter associated with the Kaiko as shown in the graph on the right. Fig. 5. Limit cycle for the full model. fact exist in the Tsembaga system. Rappaport actually indicated that this was the case. As there are more pigs, people, and gardens, there are more ways for pigs to invade gardens and cause conflict. This increases the number of required blood revenge deaths during an active period of warfare. The number of ways a pig might invade an enemy’s garden rises much faster than linearly with increases in pig and garden numbers. Fig. 5 shows the limit cycles to which the model con- verges with DL = L 2 . Compare these limit cycles to those shown in Fig. 3. The human population density fluctuations are an order of magnitude smaller, falling from 0.49 to 0.41 during a bout of warfare versus from 0.8 to 0.1 during the collapse that occurs when the ritual cycle is not present. The difference in the fluctuations in biophysical capital between the two cases is even more dramatic. Without the ritual cycle, biophysical capital falls from 0.95 to 0.05 during the growth and collapse phase. The ritual cycle maintains biophysical capital in a very narrow range between 0.86 and about 0.88. So far I have illustrated how the application of a stylized dynamic model and bifurcation analysis can be used to help understand key drivers of the system. There are two main conclusions: “ the tendency of people to increase effort to attempt to meet food demands fundamentally destabilizes the model, “ the two key components that enable the Tsem- baga ritual cycle to re-stabilize the system is the ecosystem monitoring role of pigs, and the nonlinearity of DL. It is interesting to point out the relationship between the above results and, as Ostrom 1990 puts it, the three puzzles of common property resource governance: supply, commitment, and monitoring. These are the basic problems faced by any group of agents who use common property resources: how do the necessary institutions pub- lic goods get supplied, how is the commitment from participants maintained, and how is cheating prevented. The ritual cycle of the Tsembaga is so integral to their daily lives that supply and commitment are not issues at all. Monitoring is unnecessary because of pig husbandry. There is a very strong feedback between pressure on the forest and pigs. Increasing the number of pigs puts such a heavy burden on the Tsembaga women that there is no incentive to cheat and keep more pigs analogous to Hardin’s example of the herders. The resource management aspects of Tsembaga culture seem to be accidental side effects of the ritual cycle. The difficulty in crafting successful institutions for managing common property resources may reflect the relative importance of the characteristics of the underlying agents that participate in the com- mon property resource system. We now explore this issue further in the context of a simple model of Easter Island. 4 . 2 . Easter Island It is believed that Easter Island was settled by a small group of Polynesians around 400 AD at which time there was great palm forest on the island. The population grew rapidly with this abundant resource of palms which allowed them to eventually devote considerable time to carving and moving statues between about 1100 and 1500 AD. Pollen records indicate that the palm forest was almost entirely gone by 1400. By the time of the first European contact in the 18th century, the inhabitants of the island seemed incapable of carving and moving statues and indicated they had no knowledge of how to do so. The current explanation for this mystery is that the islanders degraded their environment to the point that it could no longer support them and the civilization that created the stone monuments died out, leav- ing only a small remnant to meet Dutch ships in 1722. Brander and Taylor 1998 note, however, that Polynesians almost always dramatically altered the environments of the islands they discovered, but did not collapse. In an attempt to explain why Easter Island culture collapsed while other Poly- nesian settlements did not, Brander and Taylor developed what they call a Ricardo – Malthus model of renewable resource use for the simple economics of Easter Island. The model is very similar in spirit to the Tsembaga model, but mod- els human behavior using the neo-classical eco- nomics approach of constrained optimization. The main finding in this work is that for a low intrinsic growth rate of the resource base, the model exhibits overshoot and collapse, while for larger values of the intrinsic growth rate parame- ter, the model exhibits monotonic approach to a steady state. This is consistent with the fact that the palm species on Easter Island is more slow growing than those on other islands that did not experience overshoot and collapse. This very sim- ple model helped isolate the main influence that might separate Easter Island from other Polyne- sian settlements. Aside from this stylized fact, the model might help explain other aspects of the Easter Island experience. One of these, as previously mentioned, is the question of why some institution for collec- tive action did not evolve to prevent collapse. The contribution of the model to this question is to highlight the importance of the relative time scales in operation in these resource systems. There are two important time scales in the Easter Island model, the archeological and the human lifetime. Although the forest disappeared very quickly on the archeological time scale, the forest stock would have decreased by no more than 5 over a typical lifetime. Such a change is imperceptible, making it almost impossible for islanders to rec- ognize that depletion was occurring. Brander and Taylor propose that this may have been a factor in the lack of institutional change. The slow time scale of ecological processes makes problems difficult to see and address when viewed on the fast timescale of a human lifetime. This interpretation, however, is not consistent with other archeological data from Easter Island that Brander and Taylor discuss. The appearance of a new tool in the archeological record around 1500 that is almost certainly a weapon, the move- ment of islanders into caves and fortified dwellings, and strong evidence of cannibalism suggest a more dramatic occurrence than the rela- tively slow and smooth decline predicted by the Brander and Taylor model. Further, the model predicts that a significant amount of manufactur- ing would continue throughout the evolution of the system which may be inconsistent with the abrupt cessation of statue building. These issues are related to the way in which human behavior is modeled. In this section we explore in detail how a relatively complex model of behavior more complex than in the Tsembaga model does not produce a very rich characterization of behavior in a dynamic context, and the implications this has for how the archeological data might be interpreted. In the Brander and Taylor version of the gen- eral model given by Eqs. 1a and 1b, they assume RS = rS1 − SK 8 with the intrinsic growth rate, r, carrying capacity, K, set at 0.04 and 12 000, respectively. Unlike the Tsembaga model, Brander and Taylor assume a linear relationship between fertility and resource intake, specifically, GH, L = b − d + f H L 9 where b and d are constant background birth and death rates, and fHL represents the variable growth component that depends linearly on per capita resource consumption. Representative agents maximize utility by con- suming two goods — bioresource goods agricul- tural output and fish, H, and manufactured goods tools, housing, and artistic output, M. Brander and Taylor assume a Cobb – Douglas utility func- tion, thus agents maximize uh, m = h b m 1 − b 10 subject to p h h + p m m 5 w 11 where h and m are per capita consumption rates of the bioresource and manufactured goods, p h and p m are their prices, w is the wage rate, and b defines the preferences for these goods. Agents make consumption decisions at the individual level, but the assumption that they are identical allows them to be aggregated. When I refer to individual be- havior, I am referring to the level at which deci- sions are made, not that each individual has different behavior. The resulting indirect per-capita demand func- tions are: h = b w p h and m = w1 − b 12 where w is the wage rate. Again, unlike the Tsem- baga model, the production structure is assumed to be linear: H = aSL H 13a M = L M 13b The bioresource harvest, H, is proportional to the product of the size of the resource stock and the quantity of labor devoted to obtaining it, L H, where a is analogous to the catchability coefficient often used in fishery models Clark, 1990. Manu- facturing output, M, depends on labor alone, and by choice of units, one unit of labor produces one unit of M. The linearity of the fertility and produc- tion functions greatly simplify the analysis of the model, but also limit the range of its possible behavior. Assuming that the only costs of production are due to labor, the per-unit supply prices are given by p h = wL H H 14a p m = wL M M 14b M is treated as a numeraire good whose price is normalized to 1. This with Eq. 13b implies that the wage rate is also 1. Then Eqs. 13a and 14a imply p h = 1 a S 15 which merely says that as the resource stock decreases, its supply price increases. Substituting the supply prices and wage rate into Eq. 12 yields the actual per-capita amounts of H and M pro- duced: h = abS 16a m = 1 − b 16b Combining this result with Eqs. 9 and 8 yields the full model studied by Brander and Taylor: dS dt = rS1 − SK − abSL 17a dL dt = b − d + fabSL 17b A glance at Eq. 17a reveals that they are equiva- lent to a Lotka – Volterra predator-prey system with a density-dependent prey growth rate. The model specified by Eq. 17a has one non- trivial equilibrium point S, L that satisfies S \ 0, L \ 0 and dSS, L dt = 18a dLS, L dt = 18b This equilibrium point is globally asymptoti- cally stable, the proof of which relies on a simple application of a theorem due to Kolmogorov relating to planar systems of this type see May, 1973; Edelstein-Keshet, 1988. Beginning from any interior initial condition, the system will con- verge to the steady state. Depending on parameter values, the steady state will either be a node or a spiral which will force the system to converge to the equilibrium either monotonically or through a series of damped oscillations. Of interest to Brander and Taylor is that for certain parameter values representative of the situation on Easter Island, the system will exhibit transitory oscilla- tory behavior which manifests itself in overshoot and collapse. Fig. 6 shows the human population and resource stock trajectories for an initial con- dition of 40 humans landing on Easter Island with the resource stock at carrying capacity. The units for the resource are a matter of scaling. Brander and Taylor 1998 choose a carrying capacity of 12 000 units for convenience. The remaining parameters were b = 0.4, f = 4 , r = 0.04, a = 0.00001, and b − d = − 0.1. As previously noted, the archaeological record indicates the first presence of humans at around 400 AD. The population increases which is ac- companied by a decrease in resource stock. The population and available labor peaks at around 1250 AD corresponding to the period of intense carving in the archaeological record. The popula- tion subsequently declines due to resource deple- tion. The model predicts a population of about 3800 in 1722, close to the estimated value of 3000. The model thus gives a reasonable qualitative picture of what may have happened to the culture on Easter Island. The culture became very pro- ductive and able to undertake the construction of major monuments, i.e. the labor force increased thus making L M large enough to complete such a large scale project. The population subsequently declined due to resource degradation which left the small population who knew nothing of the origin of the great monuments to meet the Dutch ships in the 18th century.

5. A closer look at Easter Island population decline