## 4 Discussion  
A tree's taproot is typically the root system's central axis, anchoring the plant underground, preventing lodging, and supporting the entire plant system. In contrast, lateral roots play a crucial role in absorbing water and nutrients, as well as maintaining plant stability [52; 53; 54]. The growth of taproots and lateral roots is vital in enabling plants to adjust to varied environmental scenarios. Plants in arid or saline-alkali environments may enhance adaptability and stress resistance by adjusting the proportion of taproots and lateral roots [55]. Numerous studies have extensively investigated the QTLs for root growth and their environmental responses [56; 57]. However, traditional QTL mapping methods often focus on individual root traits or static growth development at specific time points [58]. Our emphasis lies in the holistic growth system of roots, investigating the dynamic relationships between different traits, such as the length of the taproot and lateral roots. Additionally, we aim to understand the genetic regulation of root growth under environmental stress and its systemic impact. This approach involves a systematic and structured analysis. In this study, we construct a system of differential equations, namely, the environment-induced differential interaction equation (EDIE), to investigate the growth of the taproot and lateral roots in _Populus euphratica_. We divided the overall growth into three components, independent growth, interactive growth between traits, and environmental impact, providing a comprehensive analysis of the growth dynamics. The construction of EDIE provides a systematic and highly accurate method for fitting root growth, offering an interpretable tool for exploring the biological significance of growth dynamics. Regardless of salt stress, the growth of both the taproots and lateral roots benefits from environmental influences. This conclusion is reliably evident, as the growth and development of plants rely on obtaining essential growth conditions such as nutrients and water from the environment. While exploring the interactions between traits, we observed that taproot growth inhibits the development of lateral roots. Under control conditions, taproots benefit from lateral root development, an interaction that seems to mirror parasitic relationships in nature. However, under salt-stress conditions, the situation is reversed, and lateral root growth begins to inhibit taproot development, a phenomenon similar to competitive relationships between species in nature. Furthermore, the inhibition effects between the taproots and lateral roots are enhanced. The changes in growth components under environmental stress suggest potential plant growth strategy adjustments. In QTL mapping, we embedded EDIE into the system mapping framework, successfully locating significant QTLs that regulate root system growth under both control and salt-stress conditions. These QTLs elucidate how _Populus euphratica_ regulates root system growth when confronted with different environmental stresses, contributing to identifying key genes or genetic factors that play pivotal roles in the plant's adaptability and stress resistance. Furthermore, these QTLs provide potential targets for molecular breeding and genetic improvement. By leveraging these QTLs, plant varieties can be selected and improved to exhibit superior performance under specific environmental conditions. The QTL mapping results indicate that multiple genes collectively influence the growth of the root system. Constructing an interaction network among these QTLs can aid in identifying and understanding the interactions between different genetic loci, revealing the synergistic effects between them. The flow of information within the network and the identification of hub nodes in the complex network structure unveil the intricate genetic mechanisms underlying quantitative traits. For example, Q7816, which regulates lateral root growth under control conditions, is promoted by Q7186 while being inhibited by Q825.