Predicting Growth Trajectories: The Power of Logistic Curves
In the world of population dynamics and forecasting, understanding the different growth patterns of various phenomena is essential. While in popular parlance the nonsensical “exponential growth” may dominate the headlines in early stages of an epidemic, it is the logistic curve that offers a more nuanced and realistic representation of growth dynamics, allowing us to predict the trajectory with relative ease.
In this blog post, I will delve (twice) into the predictability of growth curves by showing how to apply the correct growth model, particularly focusing on using inflection points to predict the shape of the curve when the exact values are still unknown. No math, but a fundamental understanding of calculus is needed to read along.
The Basics: Exponential vs. Logistic Growth
Exponential growth is characterized by a constant growth rate, resulting in an ever-increasing population without any constraints. The parable of the chessboard is a popular expression of such an unconstrained growth process.
A salient characteristic of the exponential function is that all of its derivatives are also exponential. Another characteristic is that exponential functions, including all their derivatives, don’t have inflection points or extreme values.
However, in the real world, populations tend to face limitations such as diminishing resources or other restraining factors that impede unrestricted growth. This is where the logistic growth model steps in.
Logistic Growth and Carrying Capacity
Unlike exponential growth, logistic growth accounts for the limitations and checks imposed by the environment. Initially, the growth rate is high, resembling exponential growth to the untrained eye, which accounts for much of the confusion and incoherence in the popular discourse about restricted growth processes.
Logistic growth is no more exponential than polynomial or periodic (sinusoidal) growth. This is a simple but widely repeated, nonsensical category mistake. Not all curves that bend upwards are exponential, indeed they rarely are. This is why picking the right growth model matters.
The logistic growth curve introduces the concept of a carrying capacity, representing the upper limit of the growth process, like a maximum sustainable population size or the share of the population potentially affected by an observed growth process.
Carrying capacity is typically unknown in advance and can vary between near-zero to the full population. Predicting the carrying capacity of a logistic wave is key to devising policy. Luckily it is not particularly hard with a fundamental understanding calculus.
Before the population approaches the carrying capacity, the growth rate gradually decreases until it reaches zero when the population stabilizes. This can be predicted early on by considering the inflection point of the logistic curve, and the extreme points of its derivatives.
Estimating the Carrying Capacity
Typically, most impeded growth processes slow down and stop long before they affect the whole population. This is true for processes in any domains such as technology adoption, invasive species, and epidemics. Some technologies capture a large share of the market, others remain niche products. But what if the actual size of that carrying capacity is initially unknown and needs to be estimated, or preferably forecast?
One fascinating aspect of logistic growth curves is their predictability, even in the absence of exact foreknowledge about the carrying capacity.
In standard logistic curves, the inflection point occurs exactly at half the carrying capacity. This means that if we can identify the inflection point, we can estimate the carrying capacity as twice the population size at that point. This might differ for generalized logistic trajectories or special types of logistic growth processes, such as Richards, Gompertz or Von Bertalanffy growth functions, but we can still apply the same principles.
Additionally, extreme values of the growth rate and its derivatives provide insights into critical periods of rapid growth or decline, helping us understand the approaching limits of growth.
Beyond the Inflection Point
While the inflection point is a significant milestone for estimating carrying capacity, it’s worth noting that additional data points and analysis can enhance accuracy and predictability of the process.
By considering the behavior of higher-order derivatives, such as the second derivative (acceleration) and third derivative (jerk), we can gain further insights into growth patterns and refine our predictions.
Indeed we can generalize this pattern as, “in logistic curves, the inflection point of each derivative corresponds to the extreme point of the subsequent derivative.” Or:
In a logistic growth process, the first maximum of the n+1ˢᵗ derivative occurs before the first maximum of the nᵗʰ derivative.
The corollary of this rule is that the better the underlying data, the earlier we can predict the shape of the curve and the eventual carrying capacity, since we can gain insights from higher-order derivatives.
Why It Matters
Logistic growth curves offer a realistic representation of population dynamics, accounting for carrying capacity and diminishing growth opportunities. Through the observation of inflection points and extreme values, logistic curves allow us to predict an unknown carrying capacity early on.
This predictive capability has far-reaching applications in understanding and forecasting population growth trajectories, helping us make informed decisions and prepare for the future. By embracing the power of logistic growth curves, we gain valuable insights into the complex dynamics of various phenomena and their potential for sustainable growth.
The Deep Dive: Understanding the Difference between Exponential and Logistic Growth
In many scientific fields, growth trajectories are studied to better understand the dynamics of various population phenomena. Two fundamental growth models are exponential growth and logistic growth. While both exhibit a growth trajectory, they differ significantly in terms of their underlying mechanisms and patterns.
Exponential Growth
Exponential growth refers to a theoretical rapid and unrestricted increase in quantity over time. It occurs when the rate of growth is proportional to the current value. This leads to a compounding effect, resulting in an ever-increasing curve.
Exponential growth can be observed in various natural and artificial systems when there is no underlying limiting factor, such as population growth in an unlimited world, or compound interest in finance.
If an exponential growth process exists in a limited environment, it will hit the capacity limit at the fastest growth point, rather than taper off.
Key Characteristics of Exponential Growth
- Constant growth rate. Exponential growth maintains a consistent growth rate throughout the process.
- Unbounded growth. The growth trajectory continues indefinitely without any constraints or limitations.
- J-shaped curve. When graphed, exponential growth forms a characteristic J-shaped curve, starting slowly and then accelerating rapidly.
Some Applications of Exponential Growth
- Population dynamics. Exponential growth is often used to model the theoretical growth of populations, assuming unlimited resources and ideal conditions.
- Financial investments. Compound interest is a classic example of exponential growth, where the initial investment grows exponentially over time.
Logistic Growth
Unlike exponential growth, logistic growth takes into account limitations that restrict growth as the quantity approaches a maximum value. It considers factors such as diminishing resources, increasing competition, and emerging environmental constraints.
Initially, logistic growth resembles exponential growth, but it gradually slows down and eventually reaches a stable equilibrium level, known as the carrying capacity. We can indeed predict this carrying capacity very early with the right knowledge.
Key Characteristics of Logistic Growth
- Slowing growth rate. As the quantity approaches the carrying capacity, the growth rate decreases due to constraints on resources and competition.
- Carrying capacity. Logistic growth introduces a maximum value that the quantity can attain, representing the system’s capacity or limit.
- S-shaped curve. The graph of logistic growth forms an S-shaped (or sigmoid) curve, characterized by an initial accelerating phase followed by a decelerating phase until reaching the conjectured carrying capacity.
Some Applications of Logistic Growth
- Population ecology. Logistic growth models are commonly used to study population dynamics, considering factors like limited food supply, space, and predation.
- Market saturation. When introducing a new product or service, it often experiences rapid growth (exponential phase), but eventually reaches market saturation (carrying capacity) as demand levels off.
- Adoption of innovations. The diffusion of innovations among a population often follows a logistic growth pattern, with an initial slow adoption phase, rapid growth, and finally, saturation.
Derivatives, Inflection Points, and Forecasting Growth Trajectories
An important aspect of understanding growth patterns lies in analyzing derivatives, inflection points, and extreme points. In exponential growth, the derivative remains constant, indicating a consistent rate of growth. However, since exponential growth is unbounded, there are no inflection or extreme points which would guide us toward a viable prediction model.
In a logistic growth process, where the population follows a sigmoid curve, we can extend the relationship between inflection points and extreme points from the first derivative (velocity) to higher-order derivatives such as the second derivative (acceleration), the third derivative (jerk), and beyond.
This principle allows us to make predictions about the behavior of the growth rate and its changes by examining the extreme points and inflection points of the corresponding derivatives.
By analyzing the inflection points of the first derivative, we can estimate the extreme points of the second derivative, which describe changes in the acceleration of growth. Extending this pattern, we can generalize it to higher-order derivatives, enabling predictions about the behavior of growth at different levels of abstraction.
It’s important to note that the accuracy of these predictions depends on the underlying assumption that the logistic growth model accurately represents the dynamics of the population being studied and that the observed data conforms reasonably well to the logistic curve. Additionally, other factors and variables influencing the growth process should be considered to provide a comprehensive understanding of the growth dynamics.
These insights are invaluable in various fields, including as they assist in understanding the limitations, potential plateaus, and optimal utilization of resources within a system.
Understanding the difference between growth models is essential for a wide range of fields, including urban planning, resource management, economic forecasting, population dynamics, epidemiology, and market research. By recognizing the distinctions between exponential and logistic growth, we can better comprehend and forecast the dynamics and trends that shape the world around us.
Getting this wrong might just lead to the costliest math mistake in human history.
This article uses snippets from a lengthy discussion with ChatGPT.
