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Michael Brenndoerfer

Integral Calculus and Differential Equations › The Logistic Equation: Bounded Growth

Michael BrenndoerferOctober 28, 20255 min read

Quantitative FinanceAll chapters

Part 1 of 15Mathematical Foundations

Chapter 7Integral Calculus and Differential Equations › The Logistic Equation: Bounded Growth

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A fixed carrying capacity can be a useful assumption when modeling adoption in a finite market. The logistic equation reduces proportional growth as adoption approaches that capacity; it does not identify an empirical competition or resource mechanism by itself:

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

where:

$P$ : the quantity being modeled (population, market share, number or fraction of adopters)
$t$ : time
$r$ : intrinsic growth rate (growth rate when $P$ is small)
$K$ : carrying capacity (maximum sustainable value)
$(1 - P/K)$ : saturation factor that approaches zero as $P$ approaches $K$

For $r>0$ and $0<P<K$ , the proportional rate $P'/P=r(1-P/K)$ falls as $P$ rises. Absolute growth $P'=rP(1-P/K)$ instead increases up to $K/2$ and decreases afterward. At $P=K$ , growth is zero.

Logistic Growth

When $P$ is small relative to $K$ , the model is close to exponential growth. Proportional growth decreases as $P$ increases, while absolute growth peaks at $K/2$ and tends to zero near capacity.

Assume $K>0$ , $r>0$ , and $0<P_0<K$ for an increasing path. A future accelerating-then-decelerating S-shaped segment additionally requires $P_0<K/2$ . If $P_0\ge K/2$ , the future starts at or beyond the inflection. $P_0=0$ is an equilibrium, while $P_0>K$ decreases toward $K$ from above.

Under $0<P_0<K$ , the logistic equation has the analytical solution:

P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}

where:

$P(t)$ : quantity at time $t$
$P_0$ : initial quantity at $t = 0$
$K$ : carrying capacity
$r$ : intrinsic growth rate
$(K - P_0)/P_0$ : ratio determining how far the initial value is from capacity

The full logistic solution is sigmoid on a linear vertical scale. The example starts at $P_0=0.01K$ , so its future includes both sides of the inflection. Its comparison figure uses a logarithmic vertical axis to retain the unmodified exponential curve; that geometry is not the ordinary-scale S shape. As $t\to\infty$ , the exponential term tends to zero and $P(t)$ approaches $K$ .

An S-shaped curve can model technology adoption, market penetration, or the spread of a financial product when a fixed carrying capacity is a reasonable assumption. In that model, early adoption is close to exponential, the pool of potential new adopters shrinks, and adoption approaches the specified capacity. The curve is a modeling choice, not a historical claim that every product follows a logistic path.

In[30]:

Code

import numpy as np


def logistic_growth(t, P0, r, K):
    """Analytical solution to the logistic equation."""
    if not np.isfinite([K, r, P0]).all() or K <= 0 or r <= 0 or not 0 < P0 < K:
        raise ValueError("Require K > 0, r > 0, and 0 < P0 < K")
    return K / (1 + ((K - P0) / P0) * np.exp(-r * t))


# Model market adoption of a new financial product
P0 = 1000  # Initial adopters
K = 100000  # Market capacity (100,000 potential customers)
r = 0.5  # Growth rate

t = np.linspace(0, 20, 200)
P = logistic_growth(t, P0, r, K)

# Compare with exponential growth
P_exponential = P0 * np.exp(r * t)

Out[31]:

Visualization

Line chart on a logarithmic vertical axis comparing a logistic curve that levels near 100,000 adopters with an exponential curve that continues above it. — Logistic versus exponential growth on a logarithmic vertical scale. Logistic growth approaches market capacity, whereas the unmodified exponential curve continues increasing.

The displayed model starts with nearly exponential proportional growth. Absolute adoption rises to its maximum at $P=K/2$ and then falls. For $0<P_0<K/2$ , that future inflection occurs at:

Out[32]:

Console

Market adoption analysis:
  Initial adopters: 1,000
  Market capacity: 100,000
  Inflection point: 50,000.0 adopters at t = 9.2 years
  Time to reach 50% capacity: 9.2 years
  Time to reach 90% capacity: 13.6 years

Out[33]:

Visualization

Plot showing the logistic growth rate dP/dt as a function of P, with maximum at K/2. — Growth rate in the logistic model peaks at the inflection point (P = K/2) and declines toward zero as the market saturates.