一、递推关系——酵母菌生长模型
代码:
import matplotlib.pyplot as plt
time = [i for i in range(0,19)]
number = [9.6,18.3,29,47.2,71.1,119.1,174.6,257.3,
350.7,441.0,513.3,559.7,594.8,629.4,640.8,
651.1,655.9,659.6,661.8]#19个数据 与i相对应
plt.title('Relationship between time and number')#创建标题
plt.xlabel('time')#X轴标签
plt.ylabel('number')#Y轴标签
plt.plot(time,number)#画图
plt.show()#显示
分析:
酵母菌数量增长有一个这样的规律:当某些资源只能支撑某个最大限度的种群 数量,而不能支持种群数量的无限增长,当接近这个最大值时,种群数量的增 长速度就会慢下来。
- 两个观测点的值差△p来表征增长速度
- △p与目前的种群数量有关,数量越大,增长速度越快
- △p还与剩余的未分配的资源量有关,资源越多,增长速度越快
- 然后以极限总群数量与现有种群数量的差值表征剩余资源量
 
认为delta_p为二次函数时
代码:
import numpy as np
import matplotlib.pylab as plt
p_n = [9.6,18.3,29,47.2, 71.1,119.1, 174.6,
257.3, 350.7, 441.0, 513.3, 559.7, 594.8, 629.4,
640.8, 651.1, 655.9, 659.6]
delta_p = [8.7, 10.7,18.2,23.9, 48,55.5,
82.7, 93.4, 90.3, 72.3, 46.4,35.1,
34.6, 11.4, 10.3,4.8,3.7,2.2]
plt.plot(p_n,delta_p)
poly = np.polyfit(p_n, delta_p, 2)
z = np.polyval(poly,p_n)
print(poly)
plt.plot(p_n, z)
plt.show()
[-8.01975671e-04 5.16054679e-01 6.41123361e+00]
b :把k(665-pn)看成一个整体
代码:
import numpy as np
import matplotlib.pylab as plt
p_n = [9.6,18.3,29,47.2, 71.1,119.1, 174.6,
257.3, 350.7, 441.0, 513.3, 559.7, 594.8, 629.4,
640.8, 651.1, 655.9, 659.6]
delta_p = [8.7, 10.7,18.2,23.9, 48,55.5,
82.7, 93.4, 90.3, 72.3, 46.4,35.1,
34.6, 11.4, 10.3,4.8,3.7,2.2]
p_n = np.array(p_n)
x= (665 - p_n) * p_n
plt.plot(x,delta_p)
ploy = np.polyfit(x,delta_p,1)
print(ploy)
z = np.polyval(ploy,x)
plt.plot(x,z)
plt.show()
[ 0.00081448 -0.30791574]
模型 :
为什么没有后面的b? 一开始酵母菌需要有一定的数量
预测曲线:
import matplotlib.pyplot as plt
p0 = 9.6
p_list = []
for i in range(20):
p_list.append(p0)
p0 = 0.00081448*(665-p0)*p0+p0
plt.plot(p_list)
plt.show()
预测与实际曲线
import matplotlib.pyplot as plt
number = [9.6,18.3,29,47.2,71.1,119.1,174.6,257.3,
350.7,441.0,513.3,559.7,594.8,629.4,640.8,
651.1,655.9,659.6,661.8]
time = [i for i in range(0,19)]
p0 = 9.6
p_list = []
for i in range(20):
p_list.append(p0)
p0 = 0.00081448*(665-p0)*p0+p0
plt.plot(p_list)
plt.scatter(time,number,s=100,alpha=1.0,marker='o')
plt.show()
二、显式差分——热传导方程
其中,k为热传导系数,第2式是方程的初值条件,第3、4式是边值条件,热传导方程如下:
绘制初值条件函数图像(第二个式子)
from matplotlib import pylab
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['Microsoft YaHei'] # 指定默认字体
mpl.rcParams['axes.unicode_minus'] = False # 解决保存图像是负号'-'显示为方块的问题
def initialCondition(x):
return 4.0 * (1.0 - x) * x
xArray = np.linspace(0, 1.0, 50)
yArray = np.array(list(map(initialCondition, xArray)))
pylab.figure(figsize=(12, 6))
pylab.xlabel('$x$', fontsize=15)
pylab.ylabel('$f(x)$', fontsize=15)
pylab.title(u'一维热传导方程初值条件')
pylab.plot(xArray, yArray)
plt.show()
三、马尔科夫链
构建差分方程组
代码:
import matplotlib.pyplot as plt
RLIST = [0.33333]
DLIST = [0.33333]
ILIST = [0.33333]
for i in range(40):
R = RLIST[i]*0.75+DLIST[i]*0.20+ILIST[i]*0.40
RLIST.append(R)
D = RLIST[i]*0.05+DLIST[i]*0.60+ILIST[i]*0.20
DLIST.append(D)
I = RLIST[i]*0.20+DLIST[i]*0.20+ILIST[i]*0.40
ILIST.append(I)
plt.plot(RLIST)
plt.plot(DLIST)
plt.plot(ILIST)
plt.xlabel('Time')
plt.ylabel('Voting percent')
plt.annotate('DemocraticParty',xy = (5,0.2))
plt.annotate('RepublicanParty',xy = (5,0.5))
plt.annotate('IndependentCandidate',xy = (5,0.25))
plt.show()
print(RLIST,DLIST,ILIST)