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 , which we can write as .

 

If the population t years from now is , then it is now , and after one year it will be .   Now to predict the population at the end of the second year we need to add the second year’s growth which will be k times the population at the beginning of that year – i.e. at the end of year one. So . But , so

 

Each year the population gets multiplied by another factor of a, so after t years it will be given by .