大豆和玉米单、间作的氮养分吸收的微分方程建模

Differential Equation Modeling of Nitrogen Nutrient Uptake of Soybean and Maize in Their Mono- and Intercroppings

摘要: 大豆和玉米的氮素吸收量对于提高它们的产量有着重要的意义。在本文中，首先建立了大豆和玉米单作时的氮素吸收量的Logistic微分方程模型(称为单作模型)，单作模型的数值模拟分析得出大豆和玉米单作生长的几点结论。进一步，建立了大豆和玉米间作时氮素吸收量的二维微分方程模型(称为间作模型)，利用线性稳定性理论证明了唯一的正平衡点是稳定的结点。同时也在参数的一定条件下证明了间作玉米的氮吸收量，以及玉米和大豆的氮吸收总量都大于单作，因而使得间作最终呈现作物增产优势。间作模型的数值结果表明：1) 大豆的氮素吸收量的高峰时间出现在84天左右，玉米的氮素吸收量的高峰时间出现在71天左右。玉米与大豆对氮素吸收高峰时间是错开的，适合间作；2) 在选取合适的参数下，得到的唯一稳定正平衡点是(1.24, 4.29)。因此，在间作条件下，大豆的氮素吸收量为1.24 g，比单作大豆减少了26.19%；但是，间作玉米的氮素吸收量为4.29 g，比单作玉米增加了20.17%，特别地，间作大豆和玉米两者的氮吸收总量增加了5.33%，这就表明，间作产生了氮素吸量的优势。

Abstract: The amount of Nitrogen uptake of soybeans and maize is important for increasing their yields. In this paper, Logistic differential equation models (namely monocropping models) are established firstly for nitrogen uptake of soybean and maize as monocropping respectively. Several conclusions about the growth of soybean and maize monocroppers are given by numerical simulation analysis of the monocropping models. Furthermore, a two-dimensional differential equation model (namely intercropping model) is established for the nitrogen uptake of the soybean and maize as intercrop-ping. By using the linear stability theory, the unique positive equilibrium point is proved as stable node. At the same time, it is proved that the amount of Nitrogen uptake of intercropping maize, as well as the total amount of Nitrogen uptake of intercropping maize and soybean is larger than that of monocropping under certain condition of parameters, so the intercropping takes advantage in increasing productions finally. Numerical results of the intercropping model show that: 1) the peak nitrogen uptake of soybeans occurs around the 84th day, and the peak time for nitrogen uptake of maize occurs around 71st day. The peak time of nitrogen uptake of maize and soybean is staggered, so they are suitable for intercropping; 2) the unique stable positive equilibrium point is obtained (1.24, 4.29) under the selection of suitable parameters. Therefore, under intercropping conditions, the nitrogen uptake of soybeans is 1.24 g, which is less than that of monocropping soybeans by 26.19%; however, the nitrogen uptake of intercropping maize is 4.29 g, which is higher than that of monocropping maize by 20.17%, and the total nitrogen uptake of intercropping soybeans and corn increases by 5.33%. This shows that intercropping results in the advantages of nitrogen uptake.