Migration Modified Stochastic Logistic Growth Model: Development, Characterisation Solutions and Applications to Nigeria Population

Abstract

This study explored and proposed the migration-modified stochastic logistic growth model (MM-SLGM), developed to capture the complex dynamics of population growth under the influence of both deterministic and stochastic forces. The model incorporates intrinsic growth rate, environmental carrying capacity, net migration effects and environmental noise, highly applicable to demographic systems in developing economies like Nigeria, where migration and uncertainty play significant roles. To approximate the solution of the underlying stochastic differential equation, we employed the Euler-Maruyama method, a well-established numerical technique that allows for time-step simulation of population trajectories. Complementing this is Kernel Density Estimation (KDE) applied to the simulated outcomes to infer the empirical probability distribution of the population over time, thus enabling robust uncertainty quantification. Both were applied to Nigeria’s annual population and net migration data from 1960 – 2023 obtained from data.worldbank.org and analysed with the aid of codes written in the R environment. Results from both methods were analysed and compared. The Euler-Maruyama simulation closely followed historical population trends, particularly in the earlier decades but began to underestimate observed values in more recent years, highlighting the limitations of fixed parameter models in dynamic migration contexts. KDE offered a non-parametric view of the distribution of population outcomes, revealing a broader probabilistic spread and emphasising the range of potential future scenarios. Together, these methods offer a complementary framework with Euler-Maruyama for pathwise analysis and forecasting and the KDE for distributional insight and risk assessment. This dual approach not only enhances model interpretability but also strengthens its application for population policy design in uncertain environments.