Asian Research Journal of Mathematics

This study investigates the mathematical achievement modeling in dynamical systems with its focus on parameters estimation and addressing uncertainty. The study considered a system of nonlinear first order ordinary differential equations. MATLAB ODE45 numerical scheme and Python were used for numerical solutions and simulations of the system.

The Runge-Kutta numerical method revealed that the steady-state solution is unstable with approximate values for the populations since the system is nonlinear and may have multiple solutions. The study showed different bifurcation for the variation of the parameters \(\gamma\), δ, ε, and K. The stability at critical point for the parameter variation is unstable saddle-point. It also revealed dynamic and nonlinear interactions among the state variables A(t), S(t), M(t) and D(t). The simulations from figure 1&2 showed locally asymptotically stable with its effective usage number (E_n < 1) ; while the four graphs depicts globally asymptotically unstable with (E_n > 1).  The system exhibited nonlinear dynamics and chaotic behaviour for specific parameter ranges. This chaotic behaviour was characterized by sensitivity to initial conditions, unpredictability complex and a periodic behaviour. The parameter estimation and uncertainty revealed that LSE and MLE can effectively estimate parameters, but MLE provides more accurate estimates and Gradient-based optimization converges faster. The numerical analysis also revealed that the estimates have reasonable uncertainty, Bootstrap resampling provides a robust uncertainty estimate and CIs and SEs provided a concise uncertainty summary.