自然条件下玉米与马铃薯单、间作生长的微分方程建模

Differential Equation Modeling of Monocropping and Intercropping Maize and Potato’s Growth under Natural Conditions

摘要: 在本文中，首先分别建立了自然条件下玉米和马铃薯单作生长时干物质积累的一维非自治微分方程模型。数值模拟结果表明：模拟值与试验数据非常吻合，因而所建的单作模型可分别用于描述玉米和马铃薯单作生长时的干物质积累过程。玉米的干物质积累主要集中在拔节–抽雄–灌浆期，而马铃薯的干物质积累主要集中在块茎形成–块茎膨大–淀粉积累期，两者的干物质积累过程都符合“S型”增长规律。进一步，考虑到玉米和马铃薯间作生长时有促进与抑制的作用，建立了两者间作生长时干物质积累的二维非自治微分方程模型，分析得出相应自治间作模型存在唯一的正平衡点，且为稳定结点，给出的相对增长率公式表明：间作玉米呈现增产优势，马铃薯呈现减产弱势，两者的总量呈现增长优势；非自治间作模型的数值模拟得出：若选取耦合项的参数为γ = 0.303，η = 0.0778，则两者作物的干物质积累量趋于稳定正平衡点N₄ (442.96, 266.55)，与单作相比，间作玉米的干物质积累量提高了26.76%，马铃薯减少了8.10%，而玉米和马铃薯两者总的干物质积累量却增加了11.09%，呈现出间作系统提高群体产量的优势。因此，在两者的种植方式上，可以优先选择间作栽培。

Abstract: In this paper, one-dimensional non-autonomous differential equation models of dry matter accu-mulation of mono-cropping maize and potato under natural conditions are established respectively. Numerical simulation results show that the simulated values are in good agreement with the ex-perimental data, so the monoculture models can be used to describe the dry matter accumulation process of maize and potato in monoculture growth respectively. The dry matter accumulation of maize concentrates mainly in jointing, tasting and filling stage, while the dry matter accumulation of potato concentrates mainly in tuber formation, tuber swelling and starch accumulation stage, both of which are in line with the “S-shaped” growth law. Furthermore, by considering the promo-tion and inhibition effect of intercropping maize and potato’s growth, a two- dimensional nonau-tonomous differential equation model of dry matter accumulation during their intercropping growth is established. It is concluded by analyzing the corresponding autonomous intercropping model that it has unique positive equilibrium point which is a stable node. The formulas of relative growth rate are given. They show that the intercropping maize increases production yield at a competitive advantage, the intercropping potato decreases production yield at a competitive dis-advantage, and their total production yield increases at advantage. Numerical simulation of the nonautonomous interplanting model shows that if the parameters of the coupling term of the model is chosen as γ = 0.303, η = 0.0778, then amount of dry matter accumulation of both crops tends to the stable positive equilibrium N₄ (442.96, 266.55). Compared with the monoculture maize and potato, the dry matter accumulation of intercropping maize increases by 26.76%, the potato de-creases by 8.10%, and the total dry matter accumulation corn of them increases by 11.09%. This is just the result of what the advantage of intercropping system increases production yield. Therefore, for the selection of planting methods, intercropping cultivation can be considered as priority.