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

Abstract

Population growth and its consequences remain one of the most pressing challenges of our time. The study of population dynamics, including factors like resource availability, disease, and environmental constraints, is fundamental for planning in various domains such as ecology, economics, and public health. One of the earliest models proposed to explain population growth was by Thomas Robert Malthus in the late 18th century. Malthus theorized that populations grow exponentially, while the food supply increases only in an arithmetic manner and that was explained by a mathematical model i.e. the population growth model. This imbalance, according to Malthus, could eventually lead to resource scarcity and population collapse. However, Malthus's model, though foundational, was simplistic in nature. Over time, a more refined and realistic model was developed by Pierre François Verhulst, a Belgian mathematician, which led to the formulation of the logistic growth model. This model involves a fractional differential equation (FDE) namely the logistic differential equation. Due to the significance of FDEs, several authors have proposed solutions for the model using different techniques. Our work finds this model's solution using the Laplace decomposition method (LDM) approach. The method represents a significant advancement in the tool case of applied mathematicians and scientists. Its ability to efficiently and accurately solve complex differential equations, especially FPDEs. The graphical interpretation of the behavior of the result is also mentioned and compare our results with exact solutions found in literature.