Numerical schemes for size-structured population equations

Area of Science:

• Mathematical biology
• Computational science
• Numerical analysis

Background:

• Population dynamics are often modeled using continuous variables, but real-world populations have discrete size structures.
• Existing numerical methods may not accurately capture the dynamics of size-dependent populations.
• Efficient and accurate computational tools are needed for analyzing these models.

Purpose of the Study:

• To develop and analyze novel numerical schemes for solving size-dependent population models.
• To establish theoretical convergence rates for the proposed discretization schemes.
• To validate the accuracy and performance of the schemes through numerical simulations.

Main Methods:

• Formulation of numerical schemes based on discretizing population size using a natural grid.
• Mathematical analysis of the schemes to derive optimal rates of convergence.
• Implementation of numerical experiments to test the schemes' accuracy and efficiency.

Main Results:

• The developed schemes effectively discretize size, introducing discrete dynamics into the population models.
• Optimal rates of convergence for the numerical schemes were theoretically derived.
• Numerical experiments confirmed the predicted accuracy of the schemes, aligning with theoretical results.

Conclusions:

• The proposed numerical schemes provide an accurate and efficient method for solving size-dependent population models.
• The discretization approach using a natural grid is effective in capturing the underlying population dynamics.
• The derived convergence rates offer a theoretical guarantee for the reliability of the numerical solutions.