Unveiling the Discrete Stunted Growth Model: Predicting Population Limits

Discrete Stunted Growth Model

The discrete version of the stunted (logistic) growth model is expressed as a first‑order nonlinear difference equation:

x_{t+1}=x_t + r x_t \left(1-\frac{x_t}{N}\right)

where x_t denotes the population size at time step t , r is the intrinsic growth rate, and N represents the environmental carrying capacity.

When the intrinsic growth rate r is constant, the model reduces to a linear homogeneous difference equation whose solution is a geometric sequence, leading to unbounded exponential growth. However, limited resources impose a “stunting effect”, causing the growth rate to decline as the population approaches N , eventually reaching zero.

Assuming the growth rate decreases linearly with population size, the model becomes the discrete stunted growth model described above. The parameters r (intrinsic growth rate) and N (maximum capacity) determine the dynamics; when x_t = N , growth stops.

Alternative equivalent forms of the model can be written, for example:

x_{t+1}= (1+r) x_t - \frac{r}{N} x_t^2

To analyze equilibrium points, set x_{t+1}=x_t = x^* , yielding the algebraic equation x^* = (1+r) x^* - \frac{r}{N} (x^*)^2 . Solving gives two equilibria: x^*_1 = 0 and x^*_2 = N .

Theorem 1: The equilibrium x^*_1 = 0 is locally asymptotically stable if and only if -2 < r < 0, while the equilibrium x^*_2 = N is locally asymptotically stable if and only if -2 < r < 2.

The following Python script simulates the model for various values of k (analogous to r ) and plots the steady‑state population x^*(k) after discarding transients.

<code>import numpy as np
import matplotlib.pyplot as plt
plt.rc('text', usetex=True)
plt.rc('font', size=16)

logistic = lambda k, x: k * x * (1 - x)
kk = np.arange(0, 4.01, 0.01)
listk = []
listx = []
for k in kk:
    x = 0.5
    for i in range(1, 500):
        x = logistic(k, x)
        if i > 400:
            listk.append(k)
            listx.append(x)
plt.scatter(listk, listx, c='b', s=1)
plt.grid(True)
plt.xticks(np.arange(0, 4.01, 0.5))
plt.xlabel("$k$")
plt.ylabel("$x^*(k)$")
plt.show()
</code>

An illustrative plot of the steady‑state population versus the growth parameter is shown below.

Reference: Python Mathematics Experiment and Modeling, S. Shou‑kui & S. Xi‑jing, Science Press.