# Malthusian growth model

A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on a constant rate. 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:

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

where

• P0 = P(0) is the initial population size,
• r = the population growth rate, sometimes called Malthusian parameter,
• t = time.

This model is often referred to as the exponential law.[2] It is widely regarded in the field of population ecology as the first principle of population dynamics,[3] with Malthus as the founder. The exponential law is therefore also sometimes referred to as the Malthusian Law.[4]

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.