Exponential Growth

Exponential growth occurs when there is a constant birth rate an there is no limit by either disease or food. With this kind of growth, the birth rate is what controls how slow or fast a population grows. It also occurs when the value of mathematical function is proportional to the current value function.

Exponential decay takes place in the same manner when the growth rate is negative. The formula is used to calculate exponential growth is the variable X at the negative or positive growth rate R as time T goes on at discrete intervals.

Therefore x_t =x_o (1+r)^t where x_o is value of x at time 0.

Examples

Biology

• The microorganism’s number in a culture increase exponentially till an essential nutrient is exhausted. In a typical fashion, the first organism will get split into 2 daughter organisms which then can split to form 4, which also split to form 8 and the cycle goes on.
• A virus for instance small pox or SARS will spread exponentially at first if there is no artificial immunization available. Every infected person will also infect a couple more people.
• Genetic complexity of life on earth doubles every three hundred and seventy six million years. Extrapolating the exponential growth backwards indicates that life started 9.7 billion years ago and predates the earth by 5.2 billion years.

Physics

• Avalanche breakdown within dielectric material. Free electron becomes accelerated by an electrical field that is externally applied and which free it up so additional electrons collide with molecules or atoms of the dielectric material.
• Heat transfer experiments can yield results that are best fit in exponential decay curve

Economics

In economics, exponential growth is expressed in percentage terms that imply growth. For instance, the United States GDP per capita has grown at a rate of 2% since World War Two.

Finance

• Compound interest that is at a constant rate of interest provides exponential growth of capital
• Pyramid or Ponzi schemes allow exponential growth that result in high profits for initial investors and losses among a large number of investors.

It is important to note there are limitations to models. This means that exponential growth models that are of physical phenomena will only apply in limited regions. This is often as unbounded growth which is not physically realistic.

While initially the growth might be exponential, the model phenomena will enter into a region that has previously ignored negative feedback factors eventually that become significant. Alternatively, it can underlay assumptions of exponential growth model like instantaneous feedback, breakdown and continuity.

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