A **Malthusian growth model**, sometimes called a **simple exponential** growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote *An Essay on the Principle of Population* (1798), one of the earliest and most influential books on population.^{[1]}

Malthusian models have the following form:

where

*P*_{0}=*P*(0) is the initial population size,*r*= the population growth rate, which Ronald Fisher called the*Malthusian parameter of population growth*in The Genetical Theory of Natural Selection,^{[2]}and Alfred J. Lotka called the*intrinsic rate of increase*,^{[3]}^{[4]}*t*= time.

The model can also be written in the form of a differential equation:

with initial condition:
P(0)= P_{0}

This model is often referred to as the *exponential law*.^{[5]} It is widely regarded in the field of population ecology as the first principle of population dynamics,^{[6]} with Malthus as the founder. The exponential law is therefore also sometimes referred to as the *Malthusian Law*.^{[7]} By now, it is a widely accepted view to analogize Malthusian growth in Ecology to Newton's First Law of uniform motion in physics.^{[8]}

Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources:

"Through the animal and vegetable kingdoms, nature has scattered the seeds of life abroad with the most profuse and liberal hand. ... The germs of existence contained in this spot of earth, with ample food, and ample room to expand in, would fill millions of worlds in the course of a few thousand years. Necessity, that imperious all pervading law of nature, restrains them within the prescribed bounds. The race of plants, and the race of animals shrink under this great restrictive law. And the race of man cannot, by any efforts of reason, escape from it. Among plants and animals its effects are waste of seed, sickness, and premature death. Among mankind, misery and vice. "

— Thomas Malthus, 1798.An Essay on the Principle of Population. Chapter I.

A model of population growth bounded by resource limitations was developed by Pierre Francois Verhulst in 1838, after he had read Malthus' essay. Verhulst named the model a logistic function.

## See also

- Logistic function for the mathematical model used in Population dynamics that adjusts growth rate based on how close it is to the maximum a system can support
- Albert Allen Bartlett – a leading proponent of the Malthusian Growth Model
- Exogenous growth model – related growth model from economics
- Growth theory – related ideas from economics
- Human overpopulation
- Irruptive growth – an extension of the Malthusian model accounting for population explosions and crashes
- Malthusian catastrophe
- Neo-malthusianism
- The Genetical Theory of Natural Selection

## References

**^**"Malthus, An Essay on the Principle of Population: Library of Economics"**^**Fisher, Ronald Aylmer, Sir, 1890-1962. (1999).*The genetical theory of natural selection*(A complete variorum ed.). Oxford: Oxford University Press. ISBN 0-19-850440-3. OCLC 45308589.`{{cite book}}`

: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)**^**Lotka, Alfred J. (Alfred James), 1880-1949. (2013-06-29).*Analytical theory of biological populations*. New York. ISBN 978-1-4757-9176-1. OCLC 861705456.`{{cite book}}`

: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)**^**Lotka, Alfred J. (1934).*Théorie analytique des associations biologiques*. Hermann. OCLC 614057604.**^**Turchin, P. "Complex population dynamics: a theoretical/empirical synthesis" Princeton online**^**Turchin, Peter (2001). "Does population ecology have general laws?".*Oikos*.**94**: 17–26. doi:10.1034/j.1600-0706.2001.11310.x.**^**Paul Haemig, "Laws of Population Ecology", 2005**^**Ginzburg, Lev R. (1986). "The theory of population dynamics: I. Back to first principles".*Journal of Theoretical Biology*.**122**(4): 385–399. Bibcode:1986JThBi.122..385G. doi:10.1016/s0022-5193(86)80180-1.

## External links

- Malthusian Growth Model from Steve McKelvey, Department of Mathematics, Saint Olaf College, Northfield, Minnesota
- Logistic Model from Steve McKelvey, Department of Mathematics, Saint Olaf College, Northfield, Minnesota
- Laws Of Population Ecology Dr. Paul D. Haemig
- On principles, laws and theory of population ecology Professor of Entomology, Alan Berryman, Washington State University
- Introduction to Social Macrodynamics Professor Andrey Korotayev
- Ecological Orbits Lev Ginzburg, Mark Colyvan