Modeling Yeast Growth with a Discrete Logistic Equation

Yeast Growth

Problem

The table provides measurements of yeast culture biomass over time; the task is to build a mathematical model that simulates the growth process.

Analysis

Plotting the biomass against time reveals an S‑shaped curve that increases monotonically and appears to approach a stable value, suggesting a discrete Logistic model for the trend.

Model formulation

We adopt the discrete Logistic model where r is the intrinsic growth rate and N is the system's carrying capacity.

To solve the model, unknown parameters are estimated using the least‑squares method. By rewriting the equation and substituting the 19 observed values, we obtain an over‑determined linear system for the parameters.

Using Python, the parameters are fitted and the fitted curve is compared with the original data (see the figure below).

Verification and Evaluation

The model’s maximum relative error on known points is 31.47%, indicating that predictions for intermediate data points have a noticeable deviation and the model could be further improved.

The discrete Logistic model is not only suitable for yeast culture dynamics but can also be applied to other populations exhibiting S‑shaped growth.

Code

<code>import numpy as np
from matplotlib.pyplot import rc, plot, show, legend, figure
rc('font', size=16); rc('font', family='SimHei')

a = np.loadtxt("Pdata13_6.txt")
plot(np.arange(0, 19), a, '*')
show()

b = np.c_[a[:-1], -a[:-1]**2]
c = np.diff(a)
x = np.linalg.pinv(b).dot(c)

r = x[0]
N = x[0] / x[1]
print("r,s,N的拟合值分别为：", r, '\t', x[1], '\t', N)

Tx = np.zeros(19)
Tx[0] = 9.6
x0 = 9.6
for i in range(1, 19):
    xn = x0 + r * x0 * (1 - x0 / N)
    Tx[i] = xn
    x0 = xn

figure()
plot(np.arange(0, 19), a, '*')
plot(np.arange(19), Tx, 'o-')
legend(("原始数据点", "拟合值"), loc='best')
show()

delta = np.abs((Tx - a) / a)
print("所有已知点的预测值的相对误差", delta)
print("最大相对误差:", delta.max())
</code>

References

Python数学实验与建模 / 司守奎, 孙玺菁, 科学出版社