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In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three-variate low-order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID-19 pandemic in 2020. [ABSTRACT FROM AUTHOR]

Lotka-Volterra model continues to be of interest to numerous researchers, both in terms of its mathematical development and applications. This paper discusses an analytical solution of the Lotka-Volterra model near its equilibrium point. This solution, which is the main result of this paper, has not been previously explored by other researchers. The proposed solution is in the form of a series expansion with respect to parameter (1), representing a perturbed solution around the equilibrium point. The analytical solution is evaluated by comparing to both the corresponding exact orbit, which is directly derived from the model, and the corresponding numerical solution. Comparisons with the exact orbit reveal that the orbit formed by the analytical solution closely approximates the exact one. The orbits formed by the analytical solution are closer to the exact ones as approaches zero. Similarly, the analytical solution is also closer to the numerical solution as approaches zero. Additionally, as the solutions exhibit wave-like behavior, further observations are made regarding the 'amplitude' and 'wavelength' of the analytical solution. The wavelength and amplitude of the analytical solutions are closer to the ones of the numerical solutions as tends to zero. The amplitude and wavelength are crucial for optimizing harvesting strategies when applying the Lotka-Volterra model in integrated farming systems. [ABSTRACT FROM AUTHOR]