Figure 1.
Population Ecology in an Agrarian Society.
Components within the dashed rectangle make up the model itself; solid arrows depict the causal relationships that link them. Model parameters, those that must be set and can be manipulated for analytical purposes, are shown outside the dashed box. Dotted arrows show their point of operation in the logical structure of the model. Time is indexed implicitly in all model terms; x indexes age.
Table 1.
Model components and parameters.
Figure 2.
Growth in a Space-Limited Agrarian Population.
From top to bottom the graphs show concordant changes in population size (N), food availability (E) and population growth rate (r). At time zero a group of 20 people (N0 = 20) colonize a productive but space- and resource-limited agricultural landscape. High food availability (E>1) and an open frontier for settlement allow the population to grow at its maximum unconstrained rate for 346 years, when it becomes sufficiently large that the environment is filled and can no longer provide the amount of food energy optimal for baseline vital rates. As food availability falls below E = 1, convergence between per capita fertility and mortality rates causes a decline in growth rate (r) and the population asymptotically approaches an equilibrium. We describe this time course in terms of a copial phase, in which the population is demographically unconstrained and growing at this maximum rate; a short transitional phase, in which fertility declines and mortality increases; and, a Malthusian phase in which fertility matches mortality and growth ceases. Model parameters initially set to: TFRd (measured in daughters) = 2.44; average lifespan e0 = 45 years; agricultural area Am = 1000 ha; yield Y = 21,000 Kcal/ha/day; youths start work at age 10 and adults end working life at 65.
Figure 3.
Malthusian transition interval (MTI) and Quality of Life Demographic Variables.
(A) Food availability declines from E = 2.99, to E = 1 at year 347. It stabilizes at E = 0.668 approximately 51 years later. Coincident with the beginning of the transition phase: (B) total fertility rate (TFRd) declines sharply from 2.44 to replacement, 2.10; (C) average life span falls from e0 = 45 to = 30.0 years; and (D) the probability of a youth surviving to age five falls from 0.773 to 0.652. Although the production surplus declines continuously and gradually, Malthusian constraints impose themselves abruptly as variables representing demographic quality of life plummet.
Figure 4.
Effect of Settlement Population Size N0 on the Time Course of Vital Rates.
Increasing the size of the founding group by factors of 10, from 2 to 20 to 200, decreases the length of the copial phase by a constant 131 years each step, a log-linear relationship. During the copial phase per capita births and death rates, hence the reproductive rate, r, are unchanging. The transition phase in which birth and death rates converge (at b = d = 0.033) arrives sooner with larger settlement, but it otherwise is of the same form and duration for each scenario. At Malthusian equilibrium, the population numbers = 13,509 individuals, who have a life expectancy of
= 30.0 yrs; the food ratio is
= 0.668.
Figure 5.
Effect of Agricultural Area (Am) on the Time Course of Vital Rates.
Increasing the maximum area available for agricultural in-fill (Am) by factors of 10, from 100 to 1000 to 10,000 ha, increases the length of the copial phase by a constant 131 years each increment, a log-linear relationship. Habitat area has no effect on transition phase length; the passage time from E = 1 to the neighborhood of the Malthusian equilibrium is identical in all cases. At equilibrium, food availability ( = 0.668) and lifespan (
= 30) are unaffected by area. By contrast, total population size increases proportionally:
= 1351;
= 13,509;
= 135,090.
Figure 6.
Effect of Yield (Y) on the Time Course of Vital Rates.
The solid, low-yield, curves represent 7000 kcal/ha/day; the dashed, moderate yield curves a 3-fold increase to 21,000 kcal/ha/day; and, the dotted, high-yield curves another 3-fold increase to 63,000 kcal/ha/day. Because the low yield situation is below adequate kcal intake at settlement, there is no copial phase as fertility falls and mortality rises smoothly with further declines in food availability. The approach to equilibrium is slow, the transition phase extended. The moderate- and high-yield cases allow the population to grow at its maximum, constant rate until food availability drops below E = 1. They exhibit a lengthy copial phase. The moderate-yield population experiences hunger first, but in each case the approach to equilibrium is rapid once E has fallen below 1. At equilibrium, = 0.668 and
= 30.0 for each level of yield. Equilibrium population density, however, increases with yield;
= 2799;
= 13,509;
= 27,287.
Figure 7.
Effect of Background Mortality, Measured as Life Expectancy (e0), on the Time Course of Vital Rates.
The solid lines represent high background mortality corresponding to a short life expectancy of e0 = 30 years. The dashed curves depict a medium mortality, baseline life expectancy of e0 = 45 years; the dotted lines low background mortality, baseline expected lifespan of e0 = 60 years. In the high-mortality environment, the maximum growth rate is small and the copial phase lasts a lengthy 1204 years. Mortality starts high and it changes very little at equilibrium ( = 26.8) because equilibrium food availability is fairly high (
= 0.851). By contrast, the fast-growing, low-background-mortality population experiences a short copial phase of 237 years. Once it drops into hunger (E = 1) it approaches its Malthusian equilibrium very quickly, experiencing a large rise in mortality. At equilibrium, expected lifespan is longest in the most favorable environment (
= 34.0, having however dropped from e0 = 60). Hunger is acute in this case. In numbers: High mortality, Harsh Env.:
= 0.851;
= 26.8;
= 10,441. Moderate Mortality Env.:
= 0.668;
= 30.0;
= 13,509. Low Mortality, Propitious Env.:
= 0.562;
= 34.0;
= 16,075.
Figure 8.
Effect of Total Fertility Rate (TFRd) on the Time Course of Vital Rates.
Assessing TFRd at E = 1, the low-fertility population's TFRd is 2 daughters per mother (solid line), the moderate TFRd is 3 (dashed line) and the high fertility population has a TFRd of 4 (dotted line). Mortality tends to be more responsive than fertility in our parameterization of the model, so when fertility is high the gap between the vital rates is closed at equilibrium by a marked rise in morality. The result is higher fertility and mortality at equilibrium than in the other scenarios. Generally, lower maximum fertility also means lower per capita mortality at equilibrium, as shown by the per capita rate at which morality and fertility converge. The high-fertility population also grows quickly; it thus has a shorter copial phase, experiences hunger sooner, and has a steeper, more abrupt approach to equilibrium. It is the least well-fed at equilibrium and has a very short life expectancy. In numbers: High fertility population: = 0.579;
= 20.4;
= 15,956. Moderate fertility population:
= 0.624;
= 25.4;
= 14,601. Low fertility population:
= 0.731;
= 35.3;
= 12,247.
Figure 9.
Effect of Elasticity on the Time Course of Vital Rates.
(A) Elasticity of survival (mortality); (B) Elasticity of fertility. Elasticities play no role in the duration of the copial phase (much of which is elided in the graphs); they increase sensitivity of vital rates to hunger, shortening the duration of the transitional phase. Increasing the elasticity of mortality at E = 1 from half its empirical value (labeled “low” in the graphs) to its empirical value (normal) to double its empirical value (high) causes mortality to climb more quickly once E drops below 1, resulting in relatively high equilibration of per capita birth and death rates. Higher elasticity of fertility at E = 1 causes birth rates to drop more quickly with births and deaths. High elasticities are associated at equilibrium with more adequate Kcal intake, longer lifespan and lower population size. In numbers: Low survival elasticity: = 0.622;
= 31.8;
= 14,530. Normal survival elasticity:
= 0.668;
= 30.0;
= 13,509. High survival elasticity:
= 0.721;
= 28.5;
= 12,478. Low fertility elasticity:
= 0.652;
= 28.3;
= 13,894. Normal fertility elasticity:
= 0.668;
= 30.0;
= 13,509. High fertility elasticity:
= 0.694;
= 32.3;
= 12,959.