The Verhulst Model, named after Pierre-Francois Verhulst (1804-1849) of Belgium, improved upon the exponential growth model of Malthus by incorporating a limiting population value or "carrying capacity" that the environment can support. Above this value, lack of food or other resources causes the death rate to rise so that it equals the birth rate. It does not account for oscillations that may occur when food runs out suddenly, but is otherwise quite accurate, and has been shown to give a close match to real populations.
The Malthusian equation assumes that population growth is proportional
to the current population.
x'(t) = bx(t)
where b is the net growth rate per unit time and x is the population. This integrates to the familiar exponential
x(t) = x0 * exp(bt)
In the Verhulst model, competition occurs among individuals when they encounter other members of the population. This adds a quadratic term to account for the interactions between pairs of people, changing the differential equation to
x'(t) = bx(t) - dx2
where b and d are unknown constants. After integration (which is left as an exercise for the reader to carry out on some rainy day) we get
x = bx0 / ( dx0 + (b-dx0)exp(-bt))
In practice, b and d are related to the growth rate and the limiting population:
n = limiting population
g = growth rate
x = population
d = g / (n - x)
b = d * n
Luckily, n has never actually been measured in a human population, and its precise value actually doesn't greatly influence the calculations, especially if we are considering negative growth rates.
The growth rate g is the births - deaths per year per person. Note that if a female has two children, the birth rate would equal the death rate, and the population would remain constant. In reality, a fudge factor has to be added, because some children die before they grow up to have children. So in practice, a fertility rate of 2.1 is needed to keep the population constant.
The real death rate is not strictly an exponential constant since people don't die at random times. But the result should be close because we assume the death rate is constant over time.
For our example calculation, assume the limiting population n is 400,000,000. This is kind of a wild guess. If the population were to reach this value, the growth rate would drop to zero because people would eat each other's food. Note that this is not the observed limiting value (which arguably may have been already reached in Western Europe), but the physical limit on the population at which food becomes unavailable.
Some people have estimated the carrying capacity of the earth to be roughly 12 billion people. However, unlike fruit flies or squirrels, humans adapt to change by inventing new things and changing their behavior, so this number cannot be considered reliable.
The total fertility rate in the United States is 2.1, making the population
constant. So for our example, we will take Germany, where the population is
declining (fertility rate = 1.3; 1996 population = 82,000,000), and
assume an average lifespan of 75. In the calculation of the growth
rate g, we also assume that 50% of the population is female. This
makes the birth rate 1.3 / 2, or 0.65:
x0 = 82,000,000
g = 0.65/75 - 1/75 = -0.00466
d = g / (n - x0) = -1.4675e-11
b = d * n = -.00587
Substituting into the Verhulst equation
x = bx0 / ( dx0 + (b-dx0)exp(-bt))
x = x0 * exp(g*t)
x/x0 = exp(kt)
79.1/83 = 0.953 = exp(kt)
ln(0.953) = -0.0481 = kt
k = -0.0481 / 10 = -0.00481 = -0.48%
Spain has the lowest fertility rate in the world, of 1.15. If the average lifespan is 75 years, making g=-0.00566, Spain's population will be cut in half in 122 years.
In Russia, the fertility rate in 2000 was 1.21 and the average life
expectancy is around 58. Thus,
g=0.0104 - 1/58 = -0.0068
and the population will shrink by half in ln2 / 0.0068 = 101 years. If the fertility rate continues to decline, this point will occur even sooner. Zhirinovsky's proposal to ban abortion for 10 years and lower the legal marriage age from 18 to 15 might not be a bad idea. About 63% of pregnancies in Russia are terminated by abortion. Without abortion, Russia's fertility rate would be 3.24, well above the replacement rate.
Since the Verhulst Model does not take into account time-dependent changes in demographic age distribution, the true rate of population change may be slightly higher or lower than calculated by Verhulst Equation. For example, if the population is declining, the birth rate might decrease over time as the average age rises above childbearing age. This population momentum effect is not handled accurately by either model. Economic prosperity and education levels also can influence and be influenced by birth rates and death rates. If these factor are included, population models can get very complicated.