Logistic Population Model - Explanation, Equation & Graph

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A brief history of the logistic model

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Raymond Pearl and Lowell Reed present the logical curve theory about population growth. The logistic function was published among the people by Pierre François Verhulst in a series of three papers between 1838 and 1847 and it is called the rational line. But this theory was not accepted in 1920 for some reasons. Due to this, the rationalist theory was later re-examined and this theory regained its popularity. Raymond Pearl developed the rationalist theory in his book "The Biology of Population". This theory is also called Raymond Pearl's rationalist theory. The logistics model of population growth has some permissive conditions.

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Terms of the logistic model

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No.	Terms
1	The population will increase up to a certain limit.
2	There is no possibility of population growth being infinite
3	The population does not even reach zero
4	The highest growth rate of the logistic curve will be limited to Point Inflection.

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A brief explanation of the logistic model

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The number of members or population in the living world will increase, this is one of the characteristics of the population. But when some organisms start living in a new environment, the process of population growth is slow. The rate of population growth tends to increase over time and the density of the population will stabilize as the rate of population growth reaches its maximum. The members of the population increase because the population increase through a mutual union.

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Again the level of the population goes down through mortality due to environmental barriers and resource depletion activities. The population thus stabilizes at a tolerable level which is limited to a level close to the capacity of the environment. The population in confined habitats continues to grow. As a result, fertility, longevity, developmental success and growth rate decline. According to logistical theory, the gradual decrease in population growth rate is considered as a linear function in density.

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Equation for Logistic Population Growth

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• If the amount of habitat is unlimited then the geometric growth of the population can be represented by the following formula: dN / dt = mN (i)

• The growth rate of the same population that grows in limited habitat will be: dN / dt = N (rm-CN) = rN (ii)

In the above equation, N is the density of the population and C is the fixed amount of decrease in the rate of increase of population as each animal is added in a single place. The growth ratio of the population tends to decrease as the linear function of N. The value of N gradually increases and when the value of N reaches the maximum, the value of CN reaches close to rm. Equation number two is a differential form of the logistic model. The symbol K can be used in this equation, which can indicate the maximum value of growth. As N moves towards the maximum value of growth, the rate of growth tends to move towards zero.

• When N = K then dN / dt = 0 and rm-CN = 0, Since rm-CK = (k = N). Therefore c = rm / k (iii)

• If we put the value of c in Equation 2
  dN / dt = N (rm - rm / k x N)
  => dN / dt = rmN (1- rm / k)
  = mN (K-N / k) (iv)

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^{Graph | Drawn & Clicked by Samsung S21 Ultra
Location}

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Here, N / K is considered an environmental barrier. This equation is also known as Logistic Equitation.

^{Reference - Population Ecology & Wikipedia}

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Thank you for reading

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