Lecture 15: Population Growth

Reading: Economy of Nature, pp. 326-331.

The integrated form of the exponential equation permits calculation of population doubling times.

= population size at time t
= population size at an earlier time
e = natural base = 2.71878 (a constant)
r = per capita rate of population change
t = time in the same units as r

When any population doubles in size, the ratio of / would be 2.0, so:

2.0 = and ln(2.0) = ln () which is ln(2.0) =

ln(2.0) is a constant = 0.69315

so the time it takes for an exponentially growing population to exactly double in size, regardless of the absolute size of the population will be:

in the time units of r

Since world human populations were growing at a per capita rate of 0.0156 in 1995, the doubling time for world human population, if it continues to grow exponentially at that rate would be:

Doubling times can be calculated for any population in the same manner given the assumptions of exponential growth and a constant per capita rate of change. However, these may not be good assumptions, the per capita rate of change may not be constant and may change as a function of population size (or density). This kind of response could result in population growth that becomes self-limiting.

Logistic Population Growth

Pearl and Reed (1920) observed that the per capita rate of change (r) in the human population of the United States was decreasing as population size (N) increased between the years 1790 to 1900 (Ricklefs, 1996, p 341, Fig. 15.8, 15.9).

Pearl and Reed hypothesized that the per capita rate of growth would change based on the following relationship:

r = observed per capita rate of growth
= intrinsic rate of increase (ideal per capita rate of change)
N = population size
K = carrying capacity of a given environment
Since a population growing exponentially would grow at the rate:

then a more realistic model of population growth, based on the observation that the per capita rate of change is not constant would be:

Recall that the right side of this equation is the observed per capita rate of growth:

In the logistic model, both the observed per capita rate of change (r), and the absolute rate of population growth (dN/dt) change with population size as a population grows (after Ricklefs, 1996, p 342, Fig. 15.10). When N = K, r = zero, and the absolute rate of population growth (dN/dt) is at a maximum when N = K/2.

The carrying capacity (K) can be viewed as the maximum sustainable population size for a given species in a given environment. This is the population size (or density) at which the environment is saturated. The portion of the logistic equation, (1-N)/K, is the proportion of the maximum sustainable population size that is yet unfilled. As unfilled "space" or opportunity for growth decreases, the absolute rate of growth (dN/dt) also decreases. When (1-N)/K = 0, then growth stops dN/dt = 0. This is a perfectly compensated growth limitation model of population growth.

Assumptions of the Logistic Model

Population growth may not be logistic in natural populations because these assumptions are not always met. A smooth approach to carrying capacity may be uncommon in natural populations. Yet, many natural populations are relatively stable (small density variation around a mean density) over time. In a closed system, a population equilibrium (circumstances of no population growth) will occur when ever birth rates equal death rates. This can happen if either birth rate, death rate, or both are density dependent.

Density dependence (that persists) will push a population to an equilibrium density (x) because differences between birth and death rates at non-equilibrium densities either (a) result in population growth when birth rate is greater than death rate, or (b) result in a population decrease when death rate is greater than birth rate. These processes also occur when only birth rate or only death rate is density dependent.

Some of the best evidence for density dependence in natural (open) populations comes from studies of agricultural pest species.

Cabbage Aphid (Brevicoryne brassicae):

density dependence in both birth and death rates
emigration by winged individuals produced in greater numbers as density increases and reduced fecundity with increased density

Sheep Blowfly (Lucilia cuprina):

birth rates are density dependent but death rates are density independent in laboratory populations

reduced fecundity with increased density, food competition

Larch Budmoth (Zeiraphera griseana):

death rates are density dependent but birth rates are density independent in natural populations

increased death rate with increased density due to increased rates of attack by wasp parasites and viruses

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