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# Exponential and Logistic Population Growth

Jun 14, 2023

Introduction:

### Resources

Any substance or object in the environment required by an organism for normal growth, maintenance, and reproduction can be defined as a resource. When a resource is consumed by one organism. As a result, it becomes unavailable to other organisms. Limited resources lead to increased competition, and some organisms’ populations can decrease.

### Resource Availability

Natural disasters, environmental changes, and even humans can affect the availability of resources.

Many plants and animals die when drought occurs. Animals and plants do not have a habitat when forests are cleared for human use.

A lack of resources can also make some individuals smaller or weaker.

### Predator-prey Relationship

Technically, some animals are a ‘resource’ for others. These populations go through cycles generally.

The predator-prey relationship comprises the interactions between two species and their consequent effects on each other.

One species is feeding the other species in a predator-prey relationship.

A predator is an animal that hunts, kills, and eats other animals for food. Prey is a term used to describe organisms that predators kill for food.

## Exponential Population Growth

A population can experience exponential growth when resources are unlimited. In this, its size increases at a greater and greater rate.

English clergyman Thomas Malthus published a book in 1798 stating that populations with unlimited natural resources grow rapidly.

However, when resources become depleted, population growth decreases.

This accelerating pattern of increasing population size is called exponential growth.

y(t) = aekt is the formula to calculate exponential growth, where a is the value at the start, t is time, k is the rate of growth or decay, and y(t) is the population’s value at time t.

### Example of Exponential Growth

Bacteria provide the best example of exponential growth.

They are prokaryotes and reproduce by prokaryotic fission, which takes about an hour for many bacterial species.

With an unlimited supply of nutrients, if 1000 bacteria are placed in a large flask, they will result in 2000 organisms after an hour.

In one day and 24 cycles of this type, the population would have increased from 1000 to more than 16 billion.

## Logistic Population Growth

Populations exhibit exponential growth when unlimited resources result in a J-shaped curve.

Populations exhibit logistic growth when resources are limited. In this type of population growth, population expansion decreases as resources become scarce.

When the environment’s carrying capacity is reached, it levels off, resulting in an S-shaped curve.

When infinite natural resources are available, exponential growth is possible, which is not the case in the real world.

In his description of the ‘struggle for existence, Charles Darwin recognized this and stated that individuals would compete (with members of their own or other species) for limited resources.

The ones who will survive will be considered successful and will pass on their own characteristics and traits (which we know now are transferred by genes) to the next generation at a greater rate. This process is known as natural selection.

### Carrying Capacity and the Logistic Model

Exponential growth cannot continue indefinitely in the real world with limited resources.

It may occur in environments with few individuals and plentiful resources, but when the number of individuals becomes large enough, resources will be depleted, slowing the growth rate.

The growth rate will plateau or level off ultimately.

The population size representing the maximum population size that a particular environment can support is called the carrying capacity, or K.

The formula which is used to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate.

The number of individuals who may be added to a population at a given stage is indicated by the expression ‘K-N.’

And when it is divided by ‘K,’ it becomes the fraction of the carrying capacity available for further growth.

If a graph is made of this equation, it will yield an S-shaped curve, and it is a more realistic model of population growth than exponential growth.

Three different sections are there to an S-shaped curve.

Carrying capacity is the number of organisms within a region that the environment can support sustainably.

Stable equilibrium is met when the population aligns with the carrying capacity line.

Slow growth occurs when natality is slightly above mortality, for fast growth natality is drastically greater than mortality.

The S-shaped logistic curve is formed when growth rate decreases as carrying capacity is approached by the population.

Because there are few individuals and ample resources available, that’s why the initial growth is exponential.

The growth rate decreases as resources begin to become limited.

Eventually, at the environment’s carrying capacity, growth levels off with little change in population size over time.

### Examples of Logistic Growth

A microscopic fungus called yeast used to make bread and alcoholic beverages, exhibits the classical S-shaped curve when grown in a test tube.

As the population depletes the nutrients required for growth, the population’s expansion slows.

However, in the actual world, where population size fluctuates, such as in wild populations of sheep and harbor seals, there are variations to this idealized curve.

### Role of Intraspecific Competition

Every individual within a population has equal access to resources and, thus, an equal chance for survival. The logistic model assumes this.

The amount of water, sunlight, nutrients, and the space to grow are important resources for plants, whereas, for animals, important resources include food, water, shelter, nesting space, and mates.

Some individuals are better adapted to their environment than others in the real world, resulting in intraspecific competition.

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