# Exponential Growth

Exponential growth refers to an increase in quantity in accordance with a law. It can also be defined as a growth model in which growth rate is directly proportional to the present amount.

This model has the following equation:

y=a (1+r) x

Where;

A= the initial amount

r= growth rate in percent

x= number of the time intervals have elapsed

Usually, a quantity growing at a fixed percent is said to have this kind of growth. For instance, when the birth rate in population is constant through time without disease or food limitation, then it has this growth. With this growth, only birth rate controls how slow or fast a population grows.

## When and why an exponential growth occurs

This growth occurs under certain ideal conditions. It mostly occurs when resources are unlimited. It occurs when individuals within a population reproduce constantly. This leads to continuous increase in the population until when it approaches the infinitely large size. In mathematical terms, this growth occurs when a mathematical function’s value has a growth rate that is proportional to the current value of the function. This results to its growth over time becoming an exponential function.

#### How this growth occurs

When a population grows exponentially, it means that every new generation of that population adds at a similar number of the new individuals like the previous generation. This growth also occurs as a new population of a generation increases at a similar rate as that of a previous generation. Additionally, this growth occurs when a new generation of a given population is of the same size as that of the generation that preceded it. A population is also said to have this growth when a new generation of that population increases by a differing amount.

#### What can slow down or even stop this growth

A population can stop growing exponentially or slow down the rate of this growth when:

• There is a limitation of resources
• There is an increased immigration rate
• Emigration rate decreases
• There is an increase in birth rate

#### Applications

In ecology of population, this growth is used in measuring the change in the total number of persons in a population over a given period of time. Population growth patterns can be shaped by different factors. Therefore, population biologists use this growth as a mathematical model or expression for describing the rate of population growth. It helps in making population projections, making it suitable for planning purposes. Additionally, this model acts as a foundation for other models like the model for logistic growth.

Example

The balance of a bank account, b for the account that starts with the s dollars amount with a yearly interest rate of r goes for a period of n years without being touched is calculated as follows b=s(1+r)n. You can be asked to find the balance of such a bank account to the nearest dollar when the account starts with a balance of \$100 with an annual interest rate of 4% staying untouched for a period of 12 years.

b=s (1+r) n

b=100(1+0.4)12

b=\$160.

#### Criticism of exponential growth model

Some critics argue that this is not a realistic model for determining population patterns where death rate and birth rates differ as a population size function. This is generally the case when there are limited resources. In most cases, populations exhibit death rates and birth rates that depend on density. Thus exponential population growth model cannot be put to broad use in population growth description. Nevertheless, the model has been used in predicting and describing population growth in cases where species are introduced to environments where resources are abundant without predators or competitors.