Qualitative and numerical observations into the career trajectories of early career academics: A mathematical model approach

This study develops and analyzes a mathematical model that describes the interplay among different researcher groups within academia by focusing on how variables such as mentorship, access to funding, and academic peer influence affect the career trajectories of early-career, experienced, redundant, and successful researchers. Using stability theory and relevant theorems of differential equations, this study ensures the existence, uniqueness, positivity, and boundedness of the model's solutions. Additionally, the model’s equilibrium states, are analyzed to determine the local asymptotic stability. Numerical simulations using the Runge-Kutta fourth-order (RK4) method, implemented in Python, are performed to illustrate issues related to career growth within the academic community. The model simulations results indicate that the long-term sustainability of academic progression is governed by the dynamic balance among mentorship effectiveness, funding allocation efficiency, and institutional support intensity. Qualitatively, this corresponds to maintaining some model interaction parameters 𝛽1 , 𝛽3 , 𝛽4 (representing mentorship and funding influence rates) above their respective threshold values, while ensuring that the detrimental influence parameters 𝜂1 , 𝜂2 and rate at which experienced researchers leave the academic 𝛿𝑜 , remain sufficiently small to prevent destabilization of the equilibrium states. The simulations further reveal that an increase in the positive interaction coefficients enhances the asymptotic stability of the equilibrium, thereby promoting persistence of productive researcher populations over time. On the other hand, excessive negative interactions induce a shift toward instability and decline in academic performance. Hence, the model provides a quantitative framework through which policymakers and academic institutions can identify critical thresholds and parameter regimes that guarantee resilience, equitable opportunity, and long-term systemic stability within the academic ecosystem.