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1、4.在某种环境下猫头鹰的主要食物是田鼠，设田鼠的年平均增长率为，猫头鹰的存在引起的田鼠增长率的减少与猫头鹰的数量成正比，比例系数为；猫头鹰的年平均减少率为；田鼠的存在引起的猫头鹰减少率的增加与田鼠的数量成正比，比例系数为。建立差分方程模型描述田鼠和猫头鹰共处时的数量变化规律，对以下情况作图给出50年的变化过程。（1）设开始时有100只田鼠和50只猫头鹰。（2）同上，开始时有100只田鼠和200只猫头鹰。（3）适当改变参数（初始值同上）（4）求差分方程的平衡点，它们稳定吗？解：记第k代田鼠数量为xk，第k代猫头鹰数量为yk，
2、则可列出下列方程：xK+1=xK+(r1-yK*a1)*xKK+1=yK+(-r2+xK*a2)*yK运用matlab计算，程序如下：functionz=disiti(x0,y0,a1,a2,r1,r2)x=x0;y=y0;fork=1:49x(k+1)=x(k)+(r1-y(k)*a1)*x(k);y(k+1)=y(k)+(-r2+x(k)*a2)*y(k);endz=[x',y'];（1）z=disiti(100,50,0.001,0.002,0.2,0.3)plot(1:50,z(:,1));holdon;plot(
3、1:50,z(:,2),'r')得到如下结果：z=100.000050.0000115.000045.0000132.825041.8500153.831340.4125178.380840.7221206.793043.0336239.252547.9216275.637656.4758315.198370.6668355.963994.0149393.6908132.7422420.1696197.4383421.2459304.1220377.3850469.1057275.8285682.4408142.75768
4、54.181949.3682841.809217.6832672.38369.3300494.44836.5828355.34025.5602253.41645.2632180.20965.3674128.04365.753691.00516.380764.75087.243746.15188.358132.97499.754123.633611.474417.004613.574212.293516.12218.939219.20246.545722.91724.833327.38993.604932.76912.720
5、939.23382.082946.99881.621556.32241.287567.51431.046380.94650.873797.06510.7530116.40510.6733139.60770.6280167.44160.6150200.82690.6364240.86450.7011288.86850.8286346.40291.0587415.31671.4745497.76772.2570红线为猫头鹰数量曲线,蓝线为田鼠曲线(2)z=disiti(100,200,0.001,0.002,0.2,0.3)p
6、lot(1:50,z(:,1));holdon;plot(1:50,z(:,2),'r')z=100.0000200.0000100.0000180.0000102.0000162.0000105.8760146.4480111.5459133.5243118.9610123.2551128.0906115.6037138.9010110.5381151.3273108.0844165.2367108.3713180.3771111.6737196.3091118.4584212.3165129.4298227.29971
7、45.5610239.6737168.0647247.3278198.2066247.7713236.7886238.6561283.0909218.8260333.2864189.6595379.1639155.6793409.2388123.1052413.887296.7745391.624478.2302349.935666.5007299.706059.8702249.655556.8973204.652756.6326166.545258.5272135.445462.3054110.666367.871491
8、.256675.251976.267184.563164.865495.990556.3762109.777050.2865126.212146.2412145.618344.0412168.328743.6553194.646145.2556224.766549.2965258.639556.6680
