A Runge-Kutta numerical approach to modeling population dynamics with age-structured differential equations

Abstract

It is essential to have a solid grasp of population dynamics in order to comprehend demographic shifts, resource distribution, and policy planning. For the purpose of analyzing population growth, death rates, and fertility rates, this work constructs a mathematical model that makes use of age-structured Lotka-McKendrick partial differential equations (PDEs). In order to approximate numerical values, we use the Runge-Kutta (RK4) technique, which is of the fourth order. This is because analytical solutions are often intractable. The model is verified by utilizing demographic data from the United Nations (UN) Population Division, which is based on an actual population. RK4 is shown to be successful in forecasting population trends, variations in age distribution, and the influence of different birth and death rates, as shown by the results. Policymakers are provided with a powerful instrument for long-term demographic projection via the use of this technique.

This work gives a complete research on the modeling of population dynamics using age-structured differential equations. The primary emphasis of the study is on the application of the fourth-order Runge-Kutta (RK4) technique for numerical simulations. Models that are organized according to age provide a more accurate representation of the development of populations because they take into account birth and death rates that vary with age. For the purpose of representing the age dynamics, we develop the McKendrick–von Foerster partial differential equation, and then we discretize the model by using the technique of lines. The RK4 technique is then used to solve the system of ordinary differential equations that has been produced as a consequence. The potential of the model to capture major demographic trends is shown by numerical experiments, which include case studies on hypothetical and real-world population data. These experiments emphasize the benefits of RK4 in terms of accuracy and computing efficiency. A discussion on the consequences of age-structured modeling and ideas for future research paths, including stochastic extensions and geographical heterogeneity, are included in the conclusion of the work.

It is essential to have a solid grasp of population dynamics in order to comprehend demographic shifts, resource distribution, and policy planning. For the purpose of analyzing population growth, death rates, and fertility rates, this work constructs a mathematical model that makes use of age-structured Lotka-McKendrick partial differential equations (PDEs). In order to approximate numerical values, we use the Runge-Kutta (RK4) technique, which is of the fourth order. This is because analytical solutions are often intractable. The model is verified by utilizing demographic data from the United Nations (UN) Population Division, which is based on an actual population. RK4 is shown to be successful in forecasting population trends, variations in age distribution, and the influence of different birth and death rates, as shown by the results. Policymakers are provided with a powerful instrument for long-term demographic projection via the use of this technique.

This work gives a complete research on the modeling of population dynamics using age-structured differential equations. The primary emphasis of the study is on the application of the fourth-order Runge-Kutta (RK4) technique for numerical simulations. Models that are organized according to age provide a more accurate representation of the development of populations because they take into account birth and death rates that vary with age. For the purpose of representing the age dynamics, we develop the McKendrick–von Foerster partial differential equation, and then we discretize the model by using the technique of lines. The RK4 technique is then used to solve the system of ordinary differential equations that has been produced as a consequence. The potential of the model to capture major demographic trends is shown by numerical experiments, which include case studies on hypothetical and real-world population data. These experiments emphasize the benefits of RK4 in terms of accuracy and computing efficiency. A discussion on the consequences of age-structured modeling and ideas for future research paths, including stochastic extensions and geographical heterogeneity, are included in the conclusion of the work.