So, without further ado, let's talk about **logistic population growth! **

## Population growth

Populations consist of groups of individuals of a particular species living within a specified area. A **population's size** refers to the total number of individuals in that population in a specific area, while the **population's density** refers to the size of the population relative to the habitat it occupies (typically displayed as individual per unit of area, such as per km^{2}).

Population growth refers to an increase in the number of individuals within a population of a species over a period of time. Two types of population growth are recognized- exponential and logistic. **Exponential population growth** is very rare in nature, always temporary, and occurs when a given population's per capita growth rate remains constant, regardless of its size. Exponential growth is most often seen in experimental settings with bacteria, but it can be seen for brief periods in larger organisms (e.g., humans during the 20th and early 21st centuries). The reason it is always temporary is because populations are always being affected by external and internal factors that inevitably limit endless growth. We will be covering the more common population growth scenario, **logistic population growth**, throughout the remainder of the article.

## Logistic population growth definition

**Logistic population growth** is, by far, the most common kind of population growth and __occurs when the species population's per capita growth rate is decreased as its size increases.__ The population's growth rate slows as it approaches **carrying capacity**, which is influenced by density-dependent and independent limiting factors. **Density-dependent limiting factors** are often resource related. For example, a prey species that experiences a population explosion may also experience greater levels of predation, while a predator species that experiences a large increase in its population may experience starvation or increased competition between individuals. Density-dependent limiting factors may also include the increased spread of contagious disease, due to higher density populations with a greater number of individuals in close proximity to one another.

**Density-independent limiting factors** often involve catastrophic events, such as volcanic eruptions, forest fires or tsunamis. In humans, however, forest fires may actually be density-dependent, since more people equates to more potential chances for arson or accidental forest fire instigation. Both of these types of limiting factors result in a given populations maximum population size- its carrying capacity.

Populations also often vary significantly over periods of time. These variations are known as **population dynamics** and play an important role in the per capita growth rates of populations. The rates of birth, death, immigration, and emigration are collectively known as the **vital rates** of population dynamics. The bird and immigration rates alone are known as the population's **recruitment**.

**Carrying capacity**: A population's largest size, dictated by resource limitations and other limiting factors, is its carrying capacity. It is commonly referred to as "K".

**Density-dependent limiting factors**: These are factors that affect a given population's per capita growth rate to a greater degree as the population's density increases. Examples include resource limitations, increased spread of disease, and increased competition.

**Density-independent limiting factors**: There are factors that affect a given population's per capita growth rate regardless of the population's density. Examples include volcanic eruptions, wildfires, and tsunamis.

**Exponential population growth:** This occurs when the per capita rate of a population's growth remains the same independent of the population's size. Without any significant limiting factors, the population grows quickly and unchecked.

## Logistic population growth example

Coming up with examples for logistic population growth is very easy, as virtually all naturally occurring populations experience this type of growth, but we will provide an example so you can better understand the concept.

A fine example of logistic population growth that was observed by biologists in real time was the __recovery of the American alligator ( Alligator mississippiensis) in the southeastern United States during the latter half of the 20th Century.__ Many of you may be shocked to know that this currently abundant species (particularly in Florida and Louisiana) was

__once on the verge of extinction__. In 1967, alligators were classified as Endangered in the United States and were afforded protection. By 1987, however, their numbers had increased to such a degree that they were no longer considered threatened. Today, American alligators number in the millions, though they still face localized threats and are still recovering in some fringe portions of their range (e.g., southeastern Oklahoma).

As the alligator's population increased, prey abundance and habitat availability acted as density-dependent limiting factors impacting the species' carrying capacity. As areas of habitat reached capacity, alligators recolonized other areas of suitable habitat nearby. This process continued for decades and, over time, the species recolonized most of its known historic range. Further expansion and population growth is limited by both density-dependent (habitat and prey) and density-independent (cooler climate) factors.

For example, the alligator's northernmost natural range extends to Merchant's Millpond in northern North Carolina (near the border with Virginia) on the eastern coast, and to Holla Bend National Wildlife Refuge in central Arkansas and the Red Slough Wildlife Management Area in southeastern Oklahoma in the west (Fig. 1). __Colder climates and unsuitable habitat prevent further expansion__ northward into states like Virginia and Missouri, and __thus prevent further population growth__. In the southwest, however, other factors are involved. __Density-dependent factors such as competition with another species__ (the Morelet's crocodile, *Crocodylus moreletii*) __and limited habitat__ for both species prevent expansion of breeding populations into Mexico from southeastern Texas.

Figure 1: Current range of the American alligator. Source: Brandon Sideleau, own work

## Logistic model of population growth: which equation do we use?

Population growth can be modeled using both mathematical equations and plotting graphs. For logistic population growth we will look at the equation for the per capita growth rate and the type of curve produced when logistic growth is graphed.

The equation, or formula, for a population's per capita growth rate is __written as the difference in the population's size (N) divided by the time (t) difference:__ **dN/dt= rN**. For exponential population growth, this is all that is required, since the population is not being significantly affected by any limiting factors or carrying capacity.

However, __in logistic population growth, we must take into account the carrying capacity (K)__, in order to produce our logistic population rate of growth. This equation is written as **dN/dt=rN(1-N/K)**. See the table below to review what each variable represents.

Variable | Meaning |

K | Carrying capacity |

N | Population size |

r | Growth rate |

t | Time |

## Logistic population growth graph

When plotting a graph for logistic population growth, an **S-shaped curve** will be produced. This is because __the growth of the population gradually slows and levels off upon reaching carrying capacity.__ This is in contrast to exponential population growth, which produces a **J-shaped curve**, since the growth continues unchecked (Fig. 2). In the real world, all populations, even those that experience a brief period of exponential growth, will eventually produce an S-shaped growth curve.

## Logistic Population Growth - Key takeaways

- Logistic population growth is the most common kind of population growth.
- In logistic population growth, the population's growth rate slows as it approaches carrying capacity.
- A population's carrying capacity is influenced by density-dependent and independent limiting factors.
- The equation for logistic population growth is written as (K-N/K)N.
- Logistic population growth produces an "S-shaped" curve when plotted on a graph.

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