Logistic Growth Differential Equation, Model, Example - Calculus

The logistic growth differential equation is represented as dP/dt = rP(1 - P/K), where P is the population, t is time, r is the growth rate, and K is the carrying capacity, which represents the maximum population an environment can sustain. This equation models how a population's growth slows down as it approaches the carrying capacity, providing a more realistic model than simple exponential growth.

💡Components of the Equation
• dP/dt: The rate of change of the population with respect to time.
• P: The population size at any given time t.
• r: The intrinsic growth rate, which determines how quickly the population would grow without environmental limitations.
• K: The carrying capacity, the maximum population size the environment can sustainably support.

💡How the Equation Works
• When P is small: The term (1 - P/K) is close to 1, so dP/dt ≈ rP, resembling exponential growth.
• As P approaches K: The term (1 - P/K) becomes smaller, reducing the growth rate dP/dt.
• When P = K: The term (1 - P/K) becomes 0, and dP/dt = 0, indicating that the population has stabilized at the carrying capacity.
This model contrasts with exponential growth, where the growth rate continuously increases with the population size. The logistic equation accounts for limited resources, ensuring that the population growth eventually levels off.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link

💡Chapters:
00:00 Logistic growth model
01:15 Worked example

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