## 1 Introduction  
Species distribution models (SDMs) are widely used to predict current and future species distributions under various climatic models [1, 2, 3]. For instance, we can use the results of modeling to develop actionable recommendations for biological conservation and risk prevention and control [4, 5, 6]. Despite their important uses, there are still many technical issues involved with the use of SDMs [7, 8]. Addressing these issues will greatly improve the forecasting accuracy of SDMs, improve the feasibility of environmental management/policymaking, and create a bridge between modelers and decision-makers. A significant challenge is due to the fact that using SDMs to predict the current and future potential distributions of species gives different results at different resolution scales [9, 10, 11]. For instance, some climate refugia modeled at fine scales might be missed at coarser scales and reducing uncertainty in estimates of richness may improve the accuracy of modeling at fine scales [3, 11, 12]. However, a false sense of the accuracy of future climate scenarios might result in unreliable SDMs when using fine scales [13, 14], and some studies have used coarse resolutions to determine the robust potential distributions of species around the world [4; 5; 12; 15]. Moreover, climate factors are the best predictor of species richness at the finer scales of resolution because of the spatial scale effect [16]. Therefore, we aimed to put forward a solution to address the contradiction between the fine scale and the coarse scale and to balance results from different scales to develop robust distribution models [17; 18]. There is usually a linear relationship between fine scales and coarse scales in terms of the model outputs of SDMs [9; 17; 19]. Therefore, the results of SDMs can be balanced with different ecological interpretations based on different scales using multivariate statistics. For example, principal component analysis (PCA) offers the possibility to promptly generate results, integrating different data with linear regression [20; 21]. PCA is a statistical approach based on actual needs, from which several smaller aggregate variables are chosen, and reflects the information of the original variables as much as possible. PCA can be used to compact redundant data into fewer non-correlated and independent dimensions that are often more readily interpretable than the source data [20]. The linear relationship of SDM offers an opportunity to balance the various results across different spatial scales and then select an appropriate scale that is representative of all scales for further analysis, such as the prediction of the potential species distribution, risk assessments concerning invasive species, and the planning of priority protected areas [17; 18; 22; 23]. We therefore used a statistical approach to find solutions to determine the various potential distributions of species at different scales and used the appropriate scale to produce robust distribution models. Our method is meaningful for planning long-term management decisions to model the habitat quality of species across different spatial scales. Here, we developed a method to balance the various results based on different scales and used the appropriate scale to model the potential distributions of species. To show how our approach works, we considered the case of the invasion of invasive alien plant species (IAPS) into protected areas in mainland China. We used Maxent to model the current and future potential distributions of nine IAPS at different spatial scales in protected areas in the mainland of China (the scale range of grid resolutions was 2.5-30.0 arcminutes) and quantified the potential IAPS-invaded protected areas and the inequalities related to the invasion of protected areas by IAPS. We showed that our approach is better able to integrate different scales into species distribution models. Our approach can use different scales to project the potential distributions of species.