Mathematical Models of Asymptotes and Human Migration

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.