The evolution of population numbers over time has been studied widely both by anthropologists and population biologists. The basic observation is
The number of newborns and number of death depend on the number of people present. This means that the change in population is a function of the size of the population, which we can write in mathematical form as DN/ DT = C N where N is the number of people living at a given time and C is a constant of proportionality. The constant C is the difference between birth rate (b_o) and death rate (d_o), called r, the intrinsic rate of growth. The full expression then becomes dN/dT = r N (eq.1), which we can integrate to N = N_o e^rTand N_o is the people living at the time we started observing (T=0) and e is a constant (~2.718). The expression gives the number of people living at any given time for given b_o and d_ovalues. The nature of population growth is called exponential, because T is in the exponent of e. We can describe this type of growth by defining the "doubling time" (T_d), or the amount of time it takes to double a population. The statement 2N_o = N_o e ^{r Td} can be reduced to T_d = ln 2 /r = 0.69/r. The doubling time (years) becomes ~70/r (with r in %) and a growth rate of 2 % gives a T_d = 35 years; a growth rate of 5 % gives a doubling time of close to 15 years and so on. This type of growth is dramatically different from linear growth.
Exponential growth: we multiply the existing population N each year with a fixed number;
Linear growth: we add a fixed number to the population N each year.

Exponential growth pretty quickly runs out of hand and would rapidly create over-population in an environment. So it is generally assumed that the growth rate changes over time when the number of people changes. We can phrase the birth rate and death rate as initial constants b_o and d_o which are modified by a factor that depends on the population size N. In simple form, a linear dependency of the birth and death rates on N is assumed, and the factor r is then expressed as r = (b_o - k₁N) - ( d_o+k₂N) The change in a population can be expressed again as above in eq. 1 according to the equation
dN/dt = {(b_o - k₁N) - ( d_o+k₂N)} N. (eq.2) We can integrate that one as well which is more painful, but we can first look under what conditions dN/dt = 0. This happens when the growth rate is zero (population stable) and that occurs when (b_o - k₁N) = ( d_o+k₂N). We can solve this for N and rewrite to           N = (b_o-d_o) / (k₁ + k₂) and the stable population number that is reached when the growth rate=0 is known as the carrying capacity K. (Note that we use here a decreasing birthrate and increasing deathrate with N, while in our problem set we had both the birth rate and the death rate increasing at different rates with N).

The population growth curve for eq. 2 is called logistic growth and is the general model for populations that experience stresses both from the outside (environmental) and from within the population (competition). The fully integrated version of the logistic equation for population growth is
N = (K) / {1 + [(K-N_o)/N_o] e^-rt }

We distinguish two types of organisms:

R-strategists (e.g., weeds, bacteria) reproduce fast and often, have short life cycles and may overtake an environment rapidly.
K-strategists have few off spring, are well adapted to their environment, have longer life cycles and fewer off-spring (e.g., people?).

Graphs of the general shapes of the growth curves are given in the problem set.

For those of you who cannot stay away from math, check this page: logistic equation