a population is a group of organisms of the same species, occupying the same space at the same time, and actively interbreeding.

the biotic potential is the maximum rate at which a population can grow. No population anywhere can grow at its maximum rate for very long.

environmental resistance refers to all the factors, both biological and non-biological that prevent a population from growing at its maximum rate. Some such factors are:
disease, starvation, competition, fire, flood etc.
 

a population that is growing very rapidly is going through a phase of exponential growth (also called geometric growth). In exponential growth, the size of a population increases by a certain proportion after each cycle of reproduction.

e.g. 2, 6, 18, 54, 162 (increases by a factor of three - or triples in size after every cycle) etc. or 2, 4, 8, 16, 32, 64 etc. (increases by a factor of two - or doubles in size after every cycle)

a population that is growing, but not at its maximum rate, may be passing through a phase of linear (or arithmetic) growth. In this type of growth the size of the population increases by a fixed amount (not proportion) after each cycle:

e.g. 2, 4, 6, 8, 10, 12, 14, etc. (increases by two after each cycle)

or

2, 5, 8, 11, 14, 17 etc. (increases by three after each cycle)

Types of population growth

Populations in nature may grow in several different ways:

logistic growth:

In this type of growth, a population starts off with a small number of members. Since the numbers are small, the population increases in size very slowly. When the population increases above a certain level, it quite suddenly starts to grow very rapidly, possibly at its maximum rate of growth. This is exponential growth, and goes on for as long as there is enough room and enough food for all the members of this rapidly expanding population.

When the population gets quite large, space and food start running out. The individuals in the population become more crowded, food is scarce and disease spreads rapidly. The rate of population growth slows down and stabilises at a certain level called the carrying capacity.

The carrying capacity is the maximum population size that may be maintained indefinitely (for a long period of time) by the resources of a certain environment.

Irruptive growth (or Malthusian growth)

Some populations do not follow this pattern of growth and stabilisation. Some populations start off small, start to grow exponentially after a while and keep on growing. They may grow so rapidly and become so large that they may exceed the carrying capacity (the maximum amount that can be maintained by the environment). This is called overshoot. When a population overshoots, resources used by members of the population run out very quickly. This is therefore followed by a dieback, where high rates of mortality occur. This means that the population would fall back to its previous levels. This is called irruptive growth or Malthusian growth.

The chart below shows an example of irruptive growth: