Perhaps the most revealing method of analysis of this dynamic system is through simulation. It is possible to estimate plausible values for the parameters in Eqs. (1) and (2) and simulate the evolution of Easter Island. The necessary parameters are a, b, n, r, and environmental carrying capacity K. It is also necessary to specify the size P(0) of the initial founding population.
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FIGURE 2 A flow diagram of stocks and flows. |
The environmental carrying capacity and the initial forest stock are taken to be the same. Of course, the size of this stock can be normalized to any value. For scaling purposes, it is convenient if the resource stock is normalized to be of approximately the same magnitude as the maximum population, so we assume the initial resource stock (and environmental carrying capacity) is 12,000 biomass units. The initial population probably arrived in one or a few large ocean-going canoes that had traveled a considerable distance in search of new land. Take the founding population to be 40, although it makes very little difference if the founding population were taken to be twice or three times as large or, for that matter, if it were smaller. The net fertility rate, n, is the proportional (or percentage) rate of growth in the absence of the resource stock. In the simulation, discrete periods of 10 years will be used. Using a value of —0.1 for n indicates that, without the biomass in place, the population would tend to decline at a rate of 10% per decade. Parameters a and b reflect particular underlying economic conditions related to harvesting technology and to the preferences of the Easter Islanders. Because Eqs. (1) and (2) are being used as the starting points for the formal analysis here, parameters a and b could be viewed as essentially arbitrary parameters that we choose appropriately to ‘‘fit’’ the model to the experience of Easter Island. However, Eqs. (1) and (2) can be derived from more fundamental economic structures that suggest plausible magnitudes for a and b. A plausible magnitude for a is 4 x 10—5 and a plausible magnitude for b is 1.6 x 10—4.
The key parameter of interest is the intrinsic growth rate, r. As it happens, Jubaea chilensis is a well-known species. It prefers temperate climate and grows slowly, normally requiring 40 to 60 years from first planting to the time it bears its first fruit. An intrinsic growth rate on the order of 4% per decade would be plausible. The Easter Island flora biomass consisted of more than just J. chilensis, but the forest was by far the most important part of the resource stock, so 4% (or 0.04) is used in the base case simulation for Easter Island.
To obtain simulations for Easter Island, Eqs. (1) and (2) are converted to discrete format. Thus, the two simulated equations are as follows:
St+1 = St + rSt{1 — St/K) — aStPt, (1A)
Pt+1 = nPt + bPtSt. (2A)
The units of time are decades and the model is run for about 1200 years. The results are shown in Fig. 3.
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As can be seen in Fig. 3, this relatively simply model tracks the known facts about Easter Island quite well. If it is assumed that first colonization occurred in 500 ad, then peak population occurs in the 14th century at just over 10,000. From there, the population decreases and is down to a little over 4000 by the time of first European contact in 1722—a little high relative to actual estimates, but quite close. The resource stock is down to less than half its initial level by 1400 ad. On the actual Easter Island, this was about the time the wine palm forest completely disappeared, so for this to be accurate it would need to be the case that the flora biomass in the absence of the palm forest was about half of the biomass with the forest. This is certainly high, because the original palm forest had far more than twice the biomass of whatever shrubs, grass, and other vegetation was left after the forest was gone. Still, the general qualitative pattern shown in Fig. 3 is a reasonable representation of the Easter Island experience.
A particularly interesting variation of this simulation arises from making one change in a model
parameter. Most of Polynesia did have palm forests. However, the palms growing on other islands grew much faster than the Chilean wine palm. The two most common large palms are the Cocos (coconut palm) and the Pritchardia (Fijian fan palm). These palms reach the fruit-growing stage after 7 to 10 years (much less than the 40- to 60-year period for the wine palm) and would have an intrinsic growth rate on the order of 35% per decade (rather than 4%). The resulting simulation is shown in Fig. 4, which illustrates a dynamic pattern strikingly different from that of Fig. 3. No boom-and-bust pattern is discernible. In fact, there is a very slight cycle that would not disappear entirely unless the intrinsic growth rate exceeded 71% per decade, but, even at 35%, the very gentle cycle would be much too damped to be evident to archaeologists and the approach to steady state is virtually monotonic. The forest stock declines smoothly to a new steady state that would correspond to having some standing forest in steady state. Low intrinsic growth rates, on the other hand, produce sharp cyclical fluctuations.
Therefore, in the predator-prey model described here, an island with a slow-growing resource base will exhibit overshooting and collapse. An otherwise identical island with a more rapidly growing resource base will exhibit a near-monotonic adjustment of population and resource stocks toward steady-state values. This fact alone can explain the sharp difference in experience between Easter Island and other Polynesian islands.
There are 12 so-called mystery islands in Polynesia that had been settled by Polynesians but were unoccupied at the time of first European contact. All these islands but one had relatively small carrying capacities. Applying our model, we see that if K (carrying capacity) is too small, then there is no interior steady state, implying that a colonizing population might arrive and expand but would eventually be driven to extinction.
