04-02 指数増殖，ロジスティック成長モデル

# 02-01. 指数増殖
import math

import matplotlib.pyplot as plt

a = 0.2
x0 = 10
dt = 0.1

t_list = []
x_list = []
for i in range(1001):
    t = dt * i
    x = x0 * math.exp(a * t)
    t_list.append(t)
    x_list.append(x)

plt.plot(t_list, x_list)

# 02-03. 指数増殖（数値解）
a = 0.2
x0 = 10

dt = 0.1

t = 0
x = x0
t_list = [t]
x_list = [x]
for i in range(1000):
    t = dt * (i + 1)
    x = x + a * x * dt

    t_list.append(t)
    x_list.append(x)

plt.plot(t_list, x_list)

# 02-04. 指数増殖（解析解と数値解の比較）
a = 0.2
x0 = 10
t_end = 100

dt_a = 0.1
i_end_a = int(t_end / dt_a)

dt_n = 0.1
i_end_n = int(t_end / dt_n)

# 解析解
x_list_a = []
t_list_a = []
for i in range(i_end_a):
    t = dt_a * i
    x = x0 * math.exp(a * t)
    t_list_a.append(t)
    x_list_a.append(x)

# 数値解
t = 0
x = x0
t_list_n = [t]
x_list_n = [x]
for i in range(i_end_n):
    t = dt_n * (i + 1)
    x = x + a * x * dt_n
    t_list_n.append(t)
    x_list_n.append(x)

# 可視化
plt.plot(t_list_a, x_list_a)
plt.plot(t_list_n, x_list_n)