We present a fractional-order model of two-species facultative mutualism with harvesting. We investigate the stability of the equilibrium points of the model by using the linearization method for noncoexistence of equilibrium points and the Lyapunov direct method for the positive coexistence of an equilibrium point. In addition, we obtain sufficient conditions to ensure the local asymptotic stability and global uniform asymptotic stability for the model. Finally, we provide illustrated numerical examples to verify the stability results obtained in this study.

Ordinary kriging (OK) is a popular interpolation method for its ability to simultaneously minimize error variance and deliver statistically optimal and unbiased predictions. In this work, the adaptive moving window kriging with K-means clustering (AMWKK) technique is developed to improve the estimation obtained from the moving window kriging based on the K-means clustering proposed by Abedini et al. This technique specifically addresses the challenge of selecting appropriate windows for target points located near the borders, which can potentially be the source of errors. The AMWKK algorithm introduces a dynamic clustering approach within the moving window kriging, where each target site sequentially serves as a cluster centroid. The OK is then applied within the cluster encompassing the target point, ensuring localized and adaptive interpolation. The proposed method is compared with ordinary kriging and other moving window kriging variant approaches to estimate Thailand’s mean annual pressure and humidity in 2018. The results indicate superior estimation capabilities of the AMWKK approach in terms of distinct quantitative performance statistics. The advantage of using the AMWKK method for spatial interpolation can be attributed to the fact that it facilitates the automatic tuning of the window size at any estimation point. The algorithm is particularly effective when observations in the same cluster as target points are sparse.