Exponential growth
Introduction
Exponential growth is a specific way that a quantity may increase over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time.
Mathematical Description
In an exponential growth process, the size of the growth is proportional to the current value. This can be described mathematically by the equation:
- N(t) = N0 * e^(rt)
where:
- N(t) is the quantity at time t,
- N0 = N(0) is the initial quantity, i.e., the quantity at time 0,
- r is the growth rate, and
- e is Euler's number, a mathematical constant approximately equal to 2.71828.
Growth Rate
The growth rate r in the exponential growth model is a rate, not a probability. It is often expressed on an annual basis. For example, if a population of mice doubles every year, it has an annual growth rate of 100%.
Doubling Time
The doubling time is time it takes for a quantity to double in size/value at a constant growth rate. We can find the doubling time for a population undergoing exponential growth by using the Rule of 70, which states that the doubling time (in years) is approximately equal to 70 divided by the annual growth rate (in percent).
Continuous Compounding
Exponential growth can be modelled in terms of continuous compounding. In the formula for continuous compounding, the number of compounding periods is taken to its logical extreme by letting it approach infinity. The formula for continuously compounded interest is A = P * e^(rt), where P is the principal amount, r is the annual interest rate, t is the time the money is invested for, and A is the amount of money that results from the investment.
Exponential Growth in Biology
In biology, when the reproduction rate of an organism is greater than its death rate, populations can grow exponentially. This is often the case with bacteria, where each individual divides to produce two offspring, leading to a doubling of the population size with each generation.
Exponential Growth in Physics
In physics, exponential growth is seen in processes such as chain reactions or nuclear fission. In a chain reaction, one event causes an average of more than one subsequent event, leading to an exponential increase in events.
Exponential Growth in Economics
In economics, exponential growth is seen in areas such as compound interest, economic growth, and inflation. In these cases, the growth rate is expressed as a percentage per time period.
Exponential Growth in Technology
In technology, exponential growth is often associated with Moore's Law, which predicts that the number of transistors on an integrated circuit (and thus the processing power of computers) will double approximately every two years.
Limitations of Exponential Growth
In the real world, exponential growth cannot continue indefinitely. Resources are finite, and at some point, growth will be limited by factors such as food, space, or light. This leads to a logistic growth model, where growth is exponential only up to a point, after which it slows and eventually stops as the carrying capacity of the environment is reached.