Growth Patterns

This post serves largely as background for a post I expect to write later today. This post is about different patterns of growth.

We say that something increases linearly when, over the same period of time, it increases by the same number of units. Here’s a graph of linear growth:

For a fixed run there is a fixed rise. Linear growth is commonplace, particularly in mechanical systems. Your printer, for example, may print 30 pages per minute (excluding things like warmup time, downtime, time to renew consumables, etc.). That’s how fast the gearing system advances the paper and how fast the printing mechanism prints.

We say that something increases exponentially when the rise (increase in units) is proportional to the run (elapsed time). Here’s a graph of exponential growth:

Exponential growth is also commonplace in all sorts of systems. Compound interest at a constant interest rate provides exponential growth of the capital. Ponzi schemes growth exponentially until they run out of new “investors”. The number of microorganisms in a culture medium will grow exponentially until an essential nutrient is exhausted.

Logistic growth is growth in an “S-shaped” curve. In logistical growth growth approximates exponential growth but over time saturation occurs, growth declines, and, eventually, stops. Here’s a graph of a logistic function:

Many real world systems exhibit logistic growth including the microorganisms mentioned above. Populations, tumor growth, the diffusion of innovation are all said to show logistic growth. A key concept in systems that exhibit logistic growth is that of carrying capacity. The system grows until it reaches the carrying capacity of the environment and then stops.