Mathematical modeling of biological systems using differential equations

Nonlinear interactions and temporal evolution in a biological system can be mathematically modeled by differential equations and can be analysed quantitatively.Combining biologically relevant growth, mortality, resource limitation, interaction, regulation, equilibrium, stability and numerical solution mechanisms, this study suggests an integrated framework.The novelty is to combine the three elements of nonlinear biological dynamics, Jacobian-based stability assessment, and comparative numerical evaluation, in a coherent modeling framework.The Euler, Heun, Midpoint, third-order Runge-Kutta (RK3) and fourth-order Runge-Kutta (RK4) methods were compared at a step size of 0.10.The results show that RK4 gives the highest accuracy of 98.8%, with the lowest RMSE of 0.012, and the lowest absolute error of 0.009 when compared to Euler with 91.6% accuracy and RMSE of 0.084.RK3 achieved 97.5% accuracy with 0.025 RMSE.These results confirm numerical reliability for studying nonlinear dynamics, equilibrium behavior, stability and responses of systems as a function of their parameters.