A New Approach to Population Growth Model Involving a Logistic Differential Equation of Fractional Order

Area of Science:

• Applied Mathematics
• Mathematical Biology
• Ecology

Background:

• Population dynamics are crucial for planning in ecology, economics, and public health.
• Malthus's exponential model highlighted population growth challenges, but was simplistic.
• Verhulst's logistic growth model, using a fractional differential equation (FDE), offers a more realistic approach.

Purpose of the Study:

• To solve the logistic differential equation using the Laplace decomposition method (LDM).
• To provide an efficient and accurate method for solving complex fractional partial differential equations (FPDEs).
• To graphically interpret the model's behavior and compare results with existing literature.

Main Methods:

• Application of the Laplace decomposition method (LDM).
• Solving the logistic differential equation, a type of FDE.
• Graphical analysis and comparison with exact solutions.

Main Results:

• The LDM successfully provided a solution for the logistic differential equation.
• Accurate and efficient computation of population dynamics.
• Graphical representations illustrated the model's behavior effectively.

Conclusions:

• The LDM is a powerful tool for solving complex differential equations, including those relevant to population dynamics.
• This method enhances the capabilities of applied mathematicians and scientists in modeling.
• The study validates the LDM's efficacy by comparing its results with established solutions.