The global climate has undergone great changes in recent decades, which has a significant impact on the vegetation system, especially in arid and semi-arid areas. Based on a dynamic model, this paper studied the response of vegetation pattern to climate change in Qinghai Lake, a typical semi-arid region. The conditions for Turing instability of the equilibrium were obtained by mathematical analysis. The numerical experiments showed the influence of different climitic factors (carbon dioxide concentrations [ $C{O}_2$], temperature and precipitation) on vegetation pattern. The results showed that the robustness of the vegetation system was enhanced as precipitation or [ $C{O}_2$] increased. Furthermore, we presented evolution of vegetation system under different climate scenarios to forecast the future growth of vegetation. We compared the various climate scenarios with representative concentration pathways (RCP2.6, RCP4.5, RCP8.5). The results revealed that RCP2.6 scenario was a desired climate scenario for Qinghai Lake. Our study also highlighted the measures to avoid desertification by the method of optimal control. We expect that this study will provide theoretical basis for vegetation protection.

In this paper, we investigate how to compute the minimum distance between a point and a parametric surface, and then to return the nearest point (foot point) on the surface as well as its corresponding parameter, which is also called the point projection problem of a parametric surface. The geometric strategy algorithm (hereafter GSA) presented consists of two parts as follows. The normal curvature to a given parametric surface is used to find the corresponding foot point firstly, and then the Taylor’s expansion of the parametric surface is employed to compute parameter increments and to get the iteration formula to calculate the orthogonal projection point of test point to the parametric surface. Our geometric strategy algorithm is essentially dependent on the geometric property of the normal curvature, and performs better than existing methods in two ways. Firstly, GSA converges faster than existing methods, such as the method to turn the problem into a root-finding of nonlinear system, subdividing methods, clipping methods, geometric methods (tangent vector and geometric curvature) and hybrid second-order method, etc. Specially, it converges faster than the classical Newton’s iterative method. Secondly, GSA is independent of the initial iterative value, which we prove in Theorem 1. Many numerical examples confirm GSA’s robustness and efficiency.