## 1 Introduction  
The IPCC AR6 (Sixth Assessment Report) highlights that the regions experiencing extreme weather events have expanded and become more pronounced as global warming intensifies. Notably, the annual maximum daily temperature and the annual wettest daily precipitation in mid-latitude and semi-arid regions worldwide have shown an upward trend compared to the past century [1]. In recent decades, the impact of climate change has become increasingly significant across most of China, marked by a rise in the frequency and intensity of extreme climate events, particularly extreme heat events, alongside a higher occurrence of extreme drought events [2]. Fragile ecosystems such as grassland ecosystems will be increasingly threatened [3]. Gross Primary Productivity (GPP), a fundamental ecological parameter that quantifies the total atmospheric carbon dioxide assimilated through vegetation photosynthesis, demonstrates significantly greater sensitivity to extreme temperature and precipitation variations in desert steppe ecosystems compared to other terrestrial biomes [4]. In the absence of research on the response mechanisms of desert steppe GPP to extreme climate events, a systematic investigation into their coupling and mutual feedback mechanisms holds significant importance for ecosystem conservation and sustainable development. The spatiotemporal dynamics of GPP are significantly influenced by climatic factors, and the anomalous GPP variations induced by climate change inevitably exert corresponding feedback effects on regional climate systems [5; 6]. Notably, these interactions exhibit distinct response patterns in terms of magnitude and direction across different seasons and along various climate gradients [7]. GPP exhibits heightened sensitivity to extreme temperatures in high-latitude and high-altitude regions. Extreme precipitation and temperature events are fundamental indicators for assessing extreme climate phenomena and exert significant influences on terrestrial ecosystem GPP [8; 9]. Extreme high temperatures adversely affect carbon sequestration by suppressing photosynthetic activity while simultaneously enhancing respiratory carbon release [10; 11; 12]. Conversely, extreme low precipitation events have been shown to reduce global GPP by 40-50% [13]. When normalized by global biome distribution, meadow vegetation demonstrates the most pronounced precipitation sensitivity among all vegetation biomes, particularly in the desert steppe [14; 15]. Moreover, the inherent uncertainties associated with projected extreme climate change scenarios are anticipated to exert more pronounced impacts on terrestrial ecosystem productivity. Methodologies for indicating the relationship between extreme weather events and the GPP can be categorized into two primary approaches: (1) prospective assessment, which involves identifying extreme climate events and subsequently analyzing their impacts [16], and (2) posterior assessment, which examines environmental conditions associated with extreme impacts and their co-occurrence patterns [17]. Recent findings by [18] demonstrate that over 60% of global terrestrial vegetation exhibits temporal lag effects in response to extreme climate events. Furthermore, when accounting for these temporal lag effects, climatic factors demonstrate enhanced explanatory power for global vegetation dynamics [19]. The direct impacts of extreme weather events, coupled with their legacy lag effects, can significantly exacerbate soil erosion, leading to reduced soil productivity and accelerated desertification in arid and semi-arid regions [20; 21]. However, a comprehensive understanding of the response mechanisms, encompassing both the direct and indirect effects of extreme weather on the carbon cycle and their associated lag effects, remains insufficiently explored [22]. Furthermore, the relationship between terrestrial ecosystem evolution and climate change exhibits variability across different spatial and temporal scales [16; 23].