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This research studies the application of the nth order Runge-kutta method as a numerical solution to ordinary differential equations. This method was chosen because it is able to provide high accuracy and flexibility in various PDB problems. We implement the nth-order Runge-Kutta algorithm in MATLAB and compare with other numerical methods, such as Euler's method. The results show that the nth order Runge-Kutta method is able to produce more accurate solutions, especially for nonlinear systems. This research makes a significant contribution to the development of numerical solutions for PDB and shows the potential of MATLAB as an effective tool for numerical simulation. Sensitivity analyzes of parameters and time steps were also performed to understand the impact of variations on stability and convergence.

This study explores the application of stochastic differential equations (SDEs) in modeling population dynamics. By introducing randomness into differential equations, we capture the inherent uncertainties in environmental and genetic factors affecting population growth. We derive analytical solutions for specific cases and provide numerical simulations to demonstrate how SDEs enhance predictions in ecological modeling. Our findings suggest that stochastic models provide a more robust framework for understanding population fluctuations in uncertain environments.