Mathematical Models of Asymptotes and Human Migration
Description
Description of Data and Graphs This code snippet creates three different models to illustrate various types of asymptotic behaviors: horizontal asymptote, vertical asymptote, and oblique asymptote. These models are represented graphically to understand their behavior over time, specifically in the context of "Quality of Life (Q(t))". Model 1: Horizontal Asymptote A horizontal asymptote depicts a scenario where a function approaches a constant value as time progresses indefinitely. The horizontal asymptote function used here shows how quality of life stabilizes at a certain level. Parameters: L: The limiting value that quality of life approaches. k: The rate constant that determines the steepness of the curve. t0: The time at which the midpoint of the growth occurs. Interpretation: As time increases, the quality of life approaches the value of L, representing a stabilization point. The horizontal line at y = L signifies the maximum attainable quality of life over time. Model 2: Vertical Asymptote A vertical asymptote indicates a point where a function becomes unbounded and shoots off to infinity, representing a critical threshold or a singularity. The vertical asymptote function shows how quality of life dramatically increases as it approaches a specific time point. Parameters: t0: The time at which the asymptote occurs. Interpretation: As time approaches t0, the quality of life increases dramatically, indicating a critical point where it becomes unmanageable. The vertical line at x = t0 signifies the threshold time beyond which the system becomes unstable or experiences a dramatic change. Model 3: Oblique Asymptote An oblique asymptote represents a function that approaches a linear behavior over time, with diminishing deviations. The oblique asymptote function shows how quality of life follows a linear growth pattern with minor fluctuations that decrease over time. Parameters: m: The slope of the linear part of the asymptote. b: The y-intercept of the linear part. c: The coefficient of the term that decreases over time. Interpretation: As time increases, the minor fluctuations diminish, and the quality of life approaches a linear function. The red line signifies the steady linear growth in quality of life with decreasing short-term fluctuations. Visualization and Interpretation The data is visualized using three subplots, each representing one of the asymptotic models: Horizontal Asymptote Plot: Displays how quality of life grows and eventually stabilizes at a maximum value L. The horizontal dashed line illustrates the asymptote. Vertical Asymptote Plot: Shows how quality of life increases rapidly as time approaches the critical threshold t0. The vertical dashed line emphasizes the point of critical change. Oblique Asymptote Plot: Demonstrates how quality of life follows a linear growth trend with diminishing short-term fluctuations. The dashed line represents the long-term linear behavior.
Files
Steps to reproduce
import numpy as np import matplotlib.pyplot as plt # Model 1: Horizontal Asymptote def horizontal_asymptote(t, L, k, t0): return L / (1 + np.exp(-k * (t - t0))) # Model 2: Vertical Asymptote def vertical_asymptote(t, t0): return 1 / (t - t0) # Model 3: Oblique Asymptote def oblique_asymptote(t, m, b, c): return m * t + b + c / t # Time range t = np.linspace(0.01, 10, 400) # Parameters L = 10 k = 1 t0 = 5 m = 1 b = 0 c = 5 # Compute values horizontal_values = horizontal_asymptote(t, L, k, t0) vertical_values = vertical_asymptote(t, t0) oblique_values = oblique_asymptote(t, m, b, c) # Plotting fig, axs = plt.subplots(1, 3, figsize=(18, 6)) # Plot Horizontal Asymptote axs[0].plot(t, horizontal_values, label='Horizontal Asymptote') axs[0].axhline(y=L, color='r', linestyle='--', label='Asymptote at y=L') axs[0].set_title('Horizontal Asymptote') axs[0].set_xlabel('Time (t)') axs[0].set_ylabel('Quality of Life (Q(t))') axs[0].legend() # Plot Vertical Asymptote axs[1].plot(t, vertical_values, label='Vertical Asymptote') axs[1].axvline(x=t0, color='r', linestyle='--', label='Asymptote at x=t0') axs[1].set_ylim(-10, 10) axs[1].set_title('Vertical Asymptote') axs[1].set_xlabel('Time (t)') axs[1].set_ylabel('Quality of Life (Q(t))') axs[1].legend() # Plot Oblique Asymptote axs[2].plot(t, oblique_values, label='Oblique Asymptote') axs[2].plot(t, m*t + b, color='r', linestyle='--', label='Asymptote at y=mt+b') axs[2].set_title('Oblique Asymptote') axs[2].set_xlabel('Time (t)') axs[2].set_ylabel('Quality of Life (Q(t))') axs[2].legend() plt.tight_layout() plt.show()