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Lotka Volterra 方程

维基百科解释: Lotka Volterra

该方程组描述了 捕食系统模型 微分方程为:

dx/dt = Axy - Bx
dy/dt = Cy - Dxy
=>
dy/dx = (Cy - Dxy) / (Axy - Bx) (1.1)
{A, B, C, D} 均为模型系数

对微分方程(1.1)求解为: Lotka-Volterra-DSolveEq W 为Lambert W function

当系数为 {A -> 0.1, B -> 2, C -> 4, D -> 0.4, x -> {1,30}} y[x] Lotka-Volterra-XYEq

由于我们需要获得 x,y 关于自变量时间t,在模型系数{A,B,C,D}下的模型图,所以在Mathematica中对该方程组定义和设置模型系数 LotkaEq 求该方程的数值解 Lotka-Volterra-EqSolve 创建Modelica模型 在0时刻 x[0] = 10, y[0] = 5

model = CreateSystemMod["Hare", 
    Join[predatorPreyEq, {x[0] == 10, y[0] ==  5}], t,  
    Association[{"ParameterValues" -> params}]]

建模

Modelica 模型源码

model Hare
  parameter Real alpha = 0.1;
  parameter Real beta = 2;
  parameter Real gamma = 4;
  parameter Real sigma = 0.4;
  Real x;
  Real y;
initial equation
  x = 10;
  y = 5;
equation
  der(x) = (-1) * beta * x + alpha * x * y;
  der(y) = gamma * y + (-1) * sigma * x * y;
 »;
end Hare;

在SystemModeler中建模得到的x,y随着时间变化的关系图

Lotka-Volterra-ModelV1 Lotka-Volterra-ModelV1-Clean

特别的当 alpha = beta = gamma = sigma = 1

Lotka-Volterra-ModelV2 Lotka-Volterra-ModelV2-Clean.png

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