# Malthusian growth model

The Malthusian growth model, sometimes called the simple exponential growth model, is essentially exponential growth based on a constant rate of compound interest. The model is named after the Reverend Thomas Malthus, who authored An Essay on the Principle of Population, one of the earliest and most influential books on population.

## Formula

${\displaystyle {P(t)=P_{0}e^{rt}}\!}$

where P0 = Initial Population, r = growth rate, t = time.

## Exponential law

As noted by doctor Peter Turchin (Does population ecology have general laws?, 2001 and Complex Population Dynamics, 2003), this model is often referred to as The Exponential Law and is widely regarded in the field of population ecology as the first principle of population dynamics, with Malthus as the founder.

At best, it can be described as an approximate physical law as it is generally acknowledged that nothing can grow at a constant rate indefinitely (Cassell's Laws Of Nature, Professor James Trefil, 2002 - Refer 'exponential growth law'). Professor of Populations Joel E. Cohen has stated that the simplicity of the model makes it useful for very short-term predictions and of not much use for predictions beyond 10 or 20 years (How Many People Can The Earth Support, 1995). Philosopher Antony Flew - in his introduction to the Penguin Books publication of Malthus' essay (1st edition) - argued a "certain limited resemblance" between Malthus' law of population to laws of Newtonian mechanics. This view has been echoed by many other philosophers since. Also, "e: The Story Of A Number" by Eli Maor (1994) , "What Evolution Is" by Ernst Mayr, (2001) ,"The Complete Idiot's Guide To Calculus" by W. Michael Kelly (2002) and "The Galilean turn in population ecology" Mark Colyvan and Lev R. Ginzburg (2003).

## Malthusian law

The exponential law is also sometimes referred to as The Malthusian Law (refer "Laws Of Population Ecology" by Dr. Paul Haemig, 2005).

## Rule of 70

The Rule of 70 is a useful rule of thumb that roughly explains the time periods involved in exponential growth at a constant rate. For example, if growth is measured annually then a 1% growth rate results in a doubling every 70 years. At 2% doubling occurs every 35 years.

The number 70 comes from the observation that the natural log of 2 is approximately 0.7, by multiplying this by 100 we obtain 70. To find the doubling time we take the natural log of 2 by the growth rate. To find the time it takes to increase by a factor of 3 we would use the natural log of 3, approximately 1.1

## Logistic growth model

The Malthusian growth model is the direct ancestor of the logistic function. Pierre Francois Verhulst first published his logistic growth function in 1838 after he had read Malthus' An Essay on the Principle of Population. Benjamin Gompertz also published work developing the Malthusian growth model further.