Solving the Logistic Equation with Python

The Logistic Equation serves as a ubiquitous model in several scientific disciplines to depict population growth within a resource-constrained environment. This article delves into how Python, aided by the SymPy, NumPy, and Matplotlib libraries, can efficiently tackle this equation both analytically and numerically.

The Logistic Equation

The Logistic Equation is mathematically expressed as:

$$\frac{dP}{dt}=rP(1-\frac{P}{K})$$

Where:

• P represents the population,
• t is the time,
• r is the growth rate,
• K is the carrying capacity of the environment.

Analytical Solution with SymPy

To ascertain the analytical solution, one can leverage the symbolic computation capabilities of the SymPy library.

Import Libraries and Define Variables

Firstly, SymPy is imported and the necessary variables are defined.

from sympy import symbols, Function, Eq, dsolve

t = symbols('t')
P = Function('P')(t)
r, K = symbols('r K')

logistic_eq = Eq(P.diff(t), r * P * (1 - P/K))

Integration and Solution

The general solution of the logistic equation can be found via integration:

$$\int \frac{dP}{P(1-P/K)}=\int rdt$$

Using SymPy, the analytical solution can be obtained as:

analytical_solution = dsolve(logistic_eq)

The analytical solution is:

$$P(t)=\frac{K\exp (C_{1}K+rt)}{\exp (C_{1}K+rt)-1}$$

Where $C_{\mathrm{1 }}$is an integration constant.

Numerical Solution with NumPy and Matplotlib

For a numerical solution, the Euler method can be employed to approximate the solution to the differential equation. NumPy is used for the computations and Matplotlib for visualization.

Code Implementation

import numpy as np
import matplotlib.pyplot as plt

r_value = 0.1
K_value = 1000
P0 = 100
T = 100
dt = 0.1
steps = int(T / dt)

P_numerical = np.zeros(steps)
P_numerical[0] = P0

for i in range(1, steps):
    dP = r_value * P_numerical[i-1] * (1 - P_numerical[i-1] / K_value) * dt
    P_numerical[i] = P_numerical[i-1] + dP

time = np.linspace(0, T, steps)
plt.plot(time, P_numerical, label='Numerical Solution')
plt.xlabel('Time')
plt.ylabel('Population')
plt.legend()
plt.grid(True)
plt.show()

Comparing Analytical and Numerical Solutions

A comparison between the two solutions confirms the accuracy of the numerical model. To do this, the numerical solution can be overlaid onto the analytically calculated values.

# Code to calculate and plot the analytical solution for a subset of points
# ...
plt.plot(time_reduced, P_analytical_reduced_real, label='Analytical Solution', linestyle='dashed')
plt.scatter(time, P_numerical, color='red', s=10, label='Numerical Solution')
plt.legend()
plt.grid(True)
plt.show()

Conclusion

We have observed that Python, with specialized libraries like SymPy, NumPy, and Matplotlib, can be a powerful tool for both analytical and numerical analysis of differential equations. Specifically, the logistic equation was solved using analytical and numerical methods, providing a complete understanding of population dynamics in a resource-limited setting.

This hybrid approach, blending analytical and numerical analysis, offers a robust strategy for understanding and solving complex problems across various applied science fields.