The citrate secreted by the rice (Oryza sativa L.) roots will promote the absorption of phosphate, and this process is described by the Kirk model. In our work, the Kirk model is divided into citrate sub-model and phosphate sub-model. In the citrate sub-model, we obtain the analytical solution of citrate with the Laplace transform, inverse Laplace transform and convolution theorem. The citrate solution is substituted into the phosphate sub-model, and the analytical solution of phosphate is obtained by the separation variable method. The existence of the solutions can be proved by the comparison test, the Weierstrass M-test and the Abel discriminating method.

中文翻译：

---

水稻（Oryza sativa L.）根的柠檬酸盐-磷酸盐耦合模型的解析解

水稻（Oryza sativa L.）根部分泌的柠檬酸盐会促进磷酸盐的吸收，这个过程用柯克模型来描述。在我们的工作中，柯克模型分为柠檬酸盐亚模型和磷酸盐亚模型。在柠檬酸盐子模型中，我们通过拉普拉斯变换、拉普拉斯逆变换和卷积定理得到柠檬酸盐的解析解。将柠檬酸盐溶液代入磷酸盐子模型，通过分离变量法得到磷酸盐的解析解。解的存在性可以通过比较检验、Weierstrass M 检验和 Abel 判别方法来证明。