# Logistic Growth

Last updated: February 12, 2017

## What Does Logistic Growth Mean?

Logistic growth refers to any type of growth that exhibits a steady expansion followed by an abrupt slowdown.

Logistic growth is represented in graph form as an S-shaped curve.

## Safeopedia Explains Logistic Growth

Various phenomena can follow a logistic growth pattern. Population growth, industrial growth, the spread of resources, human or machine learning, and the way a rumor diffuses through a population can all be plotted on an S-shaped graph.

In population models, for instance, logistic growth occurs when an abundance of resources results in a population expansion that then slows down dramatically once resources become scarce.

### The S-Shaped Curve

The S-curve, named the Gompertz curve after the mathematician who discovered it, has four phases:

• Initiation/Birth (the initial slow increase)
• Acceleration/Growth (a rapid increase)
• Saturation (growth ceases or becomes negligible)

Source: The Foresight Guide

The growth point (GP), inflection point (IP), and saturation point (SP) are points on the curve where changes in growth conditions can be observed.

### The Logistic Growth Formula

The following formula is used for the logistic growth of a population:

dN/dt = rN (1 – N/K)

where

• dN is the change in population
• dt is the change in time
• N is the number of individuals at time t
• K is the carrying capacity of the population
• r is the intrinsic growth rate of the population

It is further assumed that

r = b – d

where

• b is the per capita birth rate
• d is the per capita death rate

From this formula, we can deduce that

• dN/dt indicates the growth of the population.
• The growth of the population depends on the population size (N) and the maximum carrying capacity (K)
• When the population is at maximum capacity or N=K, N/K = 1, 1 – N/K = 0, meaning there is zero growth

The model reveals that the population growth rate is determined by its biotic potential and population size as modified by natural resistance. As the size of the population gets closer to carrying capacity, the natural resistance increases. The logistic curve will continue up to the level of saturation (or carrying capacity), after which it starts declining. If a system is far below its carrying capacity, it will grow exponentially at first but this growth will gradually slow down as the system expands, finally stopping at the carrying capacity.