Various growth curves are used to model overall biological growth along with population dynamics. In this context, Logistics Growth curves are a family of mathematical models used to predict growth in many applications in different fields. These practices include modelling population growths, as well as studies that model yeast growth, regeneration of organs, and introduction of new products into the market. Therefore, logistics growth is used in many disciplines from economics to social sciences, from mathematics to biology and in many sector studies.

Logistics Growth curves were first made by in 1830. Pierre Verhulst was a Belgian mathematician and was working to develop methods for modelling population growth.

Verhulst based his studies on determining how the population of a particular country or region was growing, based on the work of Malthus. Malthus said that all life forms, including humans, have a propensity to exponential population growth when resources are abundant; however, that actual growth is limited by available resource.

Verhulst has obtained the Logistic Growth Model by adding a multiplicative factor to the Exponential Growth equation developed by Malthus. In the following years, many Prediction models have been prepared based on variations of the classical Verhulst Logistic Growth equation. Various growth curves have been developed to model both unpredictable intra-species population dynamics and more general biological growth.

Exponential Growth vs Logistic Growth

The exponential growth is not totally realistic because it doesn’t consider the environmental limits that have consequences on the population. According to this model, a population can grow with no limits.

The logistic growth is realistic than exponential growth because it considers those environmental limits that are density, food sickness, abundance, competition, resting place, parasites. The logistic growth taking into account data on all these factors then describe us that the population has a limit, thus those environmental factors.

For example, if you study a monkey population in a jungle, without hunters, of course, this population will reach a max number of individuals, because of the lack of food, the myxomatosis, parasites, inner and outer competition.

Exponential Growth:

It is the growth of a population by showing a geometric increase in an ideal environment where resources are not limited. It is any growth function which grows exponentially with time according to an equation which may be written in the form.

Population growth, which is relatively unlimited in populations with this type of growth, suddenly stops when a pressure factor comes into play. Said edition; It could be a shortage of food, frost, another seasonal factor, or a sudden cessation of the breeding season.

The J-type growth curve can be seen in some living populations such as algae in nature, some insect species of annual plants, and lemmings in the tundra.

Logistic Growth:

Overpopulation with population growth affects individuals’ ability to obtain sufficient resources for maintenance, growth, and reproduction. The population lives on the limited amount of resources available in the region. As a result, as the number of individuals increases, the amount of resources per individual decreases.

Four stages are observed in the S-Type growth curve.

1-Positive Increase Phase (Establishment Phase)

It indicates the time it takes for the population to settle and adapt to the new living environment. In this process, population growth is slow due to reasons such as finding food and shelter.

2-Logarithmic Increment Phase

The population has adapted to the environment. The number of individuals starts to increase at a geometric speed. At this stage, the growth rate of the population is maximum.

3-Negative Increment phase

With the limiting effect of natural resources, the rate of increase in the number of individuals decreases.

4-Equilibrium Phase

It is the equilibrium phase in which the increase of individuals in the population is zero. At this stage, the population has reached its carrying capacity.

The maximum population size that a given environment can support without habitat degradation in a given period of time is called the carrying capacity.

As the population size increases, environmental conditions resist, reducing the growth rate. This resistance is called environmental resistance. All elements, such as lack of water, diseases, competition, parasites, predator species, adverse climatic conditions constitute environmental resistance.

The destruction of habitats, which are the natural habitats of living things, causes the extinction of living species.

As a result of the destruction of their habitats, a large population that previously spread over a wide area can be divided into small populations. In these small populations that are separated, mating will occur between a limited number of individuals, with some alleles falling and others rising. In addition, a limited number of individuals will increase the likelihood of deleterious alleles coming together.

The extinction of some of the alleles and the assembly of deleterious genes reduce the inherited diversity within the species, reducing the species’ ability to adapt to changing environmental conditions. This is called genetic drift.

Settlement of invasive species with high ecological tolerance in different environments is also one of the important threats that can cause the extinction of some living species.

Limits On Growth

No community’s population grows unrestricted, so many populations have a so-called restrictive population determined by the carrying capacity of their environment.

The easiest way to model a restrictive population is to add a new term to our population model. This term is called an overcrowding term, and the coefficient of this term is called the coefficient of overcrowding. The simplest term overcrowding is proportional to the square of the current population.

Assuming A>0 in the equation below, the minus sign in the second term indicates that this term is reducing the population. This population model is called the logistic model.

dP / dt = k P – AP 2.

The population size, which represents the maximum population size that a given environment can support, is expressed as the “K”, the carrying capacity of the sea. The formula by which logistic growth is calculated adds the carrying capacity as a factor supporting the growth rate.

The expression “K – N” indicates how many people can be added to a population at a given stage. Dividing “K – N” by “K” shows the ratio of available carrying capacity for further growth. This factor constrains the exponential growth model to form the logistic growth equation:

Note that when N is very small, (KN) / K becomes close to K / K or 1; The right side of the equation reducing as: “r max N”, indicating that the population is increasing exponentially and is not affected by carrying capacity.

In the second scenario, when N is large, (KN) / K approaches zero, which means population growth will slow or even stop drastically.

Hence, population growth is greatly slowed by the K carrying capacity in large populations. This occurs when the number of individuals in the population exceeds the population’s carrying capacity, mainly because the (KN) / K value is negative.

All living organisms need certain resources to survive and reproduce, such as water, food, and environmental conditions suitable for their metabolism. These resources are not unlimited, and the population can only reproduce as long as the conditions in the region they live in are suitable. Thanks to this equation, no single species can multiply rapidly and spread throughout the world.

For example, imagine that there is a fish species that reproduce rapidly in the Mediterranean. After a while, as the food resources to be consumed by these fish in the Mediterranean ecosystem will be insufficient, their growing rate will decrease.

In a second possibility, imagine that this fish species continue to growing rapidly and some of them spread to other seas. The black sea in the north and the Atlantic Ocean in the west may be a migration area for this fish species. However, differences in sea salinity or seawater temperature  effects so this fish population can not survive in a northern sea or ocean. They may encounter species of fish that will hunt them. When we look at the world, it becomes clear that the balance in the ecosystem does not make it possible for a creature to overpopulate.

Population ecologists use a variety of mathematical methods (how populations change in size and composition over time) to model population dynamics. Some of these models represent growth without environmental constraints, while others include “ceilings” set by limited resources. Mathematical models of populations can be used to accurately describe changes occurring in a population and more importantly, to predict future changes.

Survival Capacity

Basically, all the resources needed for a species to survive can act as a boundary. Water and sunlight are decisive for plants. For a herd of bison, the meadows they graze on, the water and climate conditions are decisive.

The limited amounts of these resources lead to competition or intra-species competition between members of the same population. (Intra-specific competition) Inter-species competition for resources often results from the desire to own the territory, not the scarcity of resources. However, as the population size increases, competition intensifies and insufficient resources eventually balance the population of the population.

Logistics Growth Examples

Yeast, a microscopic fungus used to make bread and beer, creates a classic S-shaped curve when grown in a test tube. That is, it multiplies rapidly because the ambient conditions are suitable. However, when the mushroom population reaches a certain size, the food source is depleted and reproduction stops because the test tube is a closed environment. In the graph below we can see the stages of yeast growth.

The graph below, which illustrates population growth in harbor seals in Washington St. In the early part of the 20th century, seals accepted as harmful predators so they were actively hunted. In a couple of years seal population sharply decreasing. Since the seal hunting was stopped, seal populations have rebounded in a logistic pattern.

Even if the population of a community reaches the most ideal number and environmental conditions remain stable, increases or decreases will occur over time. However, when followed for a long time, it will be seen that the community has reached balance again.

Population Growth: Related words and terms

Population Ecology:  It is a sub-branch of ecology that examines the relationship of a population with its environment.

Population-specific characteristics: population density, birth rate, mortality rate, age distribution, growth rate.

Dynamic and Structural Features of the Population: Population Density, Population Distribution, Population Size, Age Distribution, Population Growth Patterns, Population Fluctuations

Population Dynamics: The branch of science that studies the change in population characteristics and the factors that cause this change is called Population Dynamics.

Carrying Capacity: It is the maximum population size an environment can carry as long as there is no intervention.

Environmental Resistance: All factors may limit the size of the population living in a habitat.

Population Fluctuations: These are sudden increases and decreases in population size.

Migrations: Inward migration: It is the arrival of homogeneous individuals from another area to the population. Emigration: The joining of individuals in a Population to a Population in another area.

Population Age Groups: These are grouping schemes made according to the developmental processes of individuals in a population.


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