Describe an exponential growth pattern include key factors such as growth factors?

One example of exponential growth and limiting factors is a basic population growth equation,

dN/dt=rN(1-N/K),

where N(t) is the population at time t, r is the populations growth rate at t=0, and K is the populations carrying capacity which is the limiting factor on the population's exponential growth. The population will increase exponentially until it starts to get close to K at which point the growth rate will slow down and the population will converge to K as t tends to infinity assuming no other factors influence the population.

This particular equation is known as a logistic model and in general doesn't represent exponential population growth very well in the real world due to numerous factors such as resources available, other species fighting for the same resources, natural factors such as disease or illness as well as others. This basic model just assumes that a population can grow to a capacity K without interruption and without external effects.