Exponential growth function doesn't match example plot

Question

I've attempted to replicate the figure 4.4.6 in this 4.4: The Logistic Equation chapter

However, my plot doesn't match the above example, as the exponential function is pushed over to the right. I double-checked my work, but unsure where I went wrong. Any hints?

import numpy as np
import math
import matplotlib.pyplot as plt

x = np.linspace(-25,25,100)
# P(t)=\dfrac{1,072,764e^{0.2311t}}{0.19196+e^{0.2311t}}
a = [1072764*np.exp(0.2311*x) / (0.19196+np.exp(0.2311*x)) for x in x]
b = 900000*np.exp(0.2311*x) # P(t)=900000e^{(0.2311x)}
plt.axhline(y=1072764, color='r', linestyle='--', label = 'P=1,072,764')
plt.ticklabel_format(style='plain') 
plt.axis([-20, 20, 0, 1400000]) # let's zoom in to see this better

plt.plot(x, a, 'darkblue', label = '$P_0=900,000$') # plotting x, a separately 
plt.plot(x, b, 'turquoise', label = 'Exponential Growth') # plotting x, b separately 
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='lower right')
# plt.savefig('plots', dpi=300, bbox_inches='tight')
plt.show()

Answer 1

The figure that you are trying to replicate shows an exponential curve that matches value and slope at a point xref (here, -20). Such an exponential curve would have the form

where Lref and L' are value and slope at the reference point. These latter can be found exactly if you know the formula of your logistic curve L(x), but to a perfectly adequate approximation may be deduced from your already-computed numerical data.

Thus:

import numpy as np
import math
import matplotlib.pyplot as plt

x = np.linspace(-20,20,100)
a = [1072764*np.exp(0.2311*x) / (0.19196+np.exp(0.2311*x)) for x in x]

xref, Lref, Lprime = x[0], a[0], ( a[1] - a[0] ) / ( x[1] - x[0] )
b = Lref * np.exp( (Lprime/Lref) * ( x - xref ) )

plt.axhline(y=1072764, color='r', linestyle='--', label = 'P=1,072,764')
plt.ticklabel_format(style='plain') 
plt.axis([-20, 20, 0, 1400000]) # let's zoom in to see this better

plt.plot(x, a, 'darkblue', label = '$P_0=900,000$') # plotting x, a separately 
plt.plot(x, b, 'turquoise', label = 'Exponential Growth') # plotting x, b separately 
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='lower right')
# plt.savefig('plots', dpi=300, bbox_inches='tight')
plt.show()

produces output