In any population of living things, each year some die and others are born. If there is plenty of space and food available, then the number of each is roughly proportional to the size of the population.
For example, in a primitive society it might happen that every female between the ages of 20 and 40 has one baby each year and that one in three of these survive. If it happens that 40% of the population is in the 20-40 age group, with half of these being female, then the number of surviving offspring each year would be one third of 20% of the population.
Similarly, if the aged represent a constant fraction of the population, then the death rate will also be proportional to the total population.
If births exceed deaths, then the population grows each year by an amount proportional to its size. For example in the situation described above, if the death rate was 5% per year, then the net gain each year would be about 2% so over each year the population would grow to 1.02 times what it was at the start of the year.
In general, if the population
at some time
is P, then the net change (number of
births minus number of deaths) over one year is kP where k,
the constant
of proportionality, is independent of both population and time (in the
above
example k = 0.02), and the new
population at the end of the year will be
If the population t
years from now is
Each year the population gets
multiplied by
another factor of a, so after t years it will be given
by