Growth Models

Growth models are mathematical descriptions of how a quantity changes over time, ranging from constant-rate (linear) to constant-proportion (exponential) to self-limiting (logistic) behaviour. Key result: when a quantity's rate of change is proportional to its current size, it grows or decays exponentially; when growth is throttled by limited resources, it follows a logistic S-curve. Applies to populations, money, radioactivity, epidemics, and chemical reactions. Developed from Malthus (1798) and Verhulst (1838) onward.

A growth model is an equation — usually a difference equation or differential equation — that predicts the future value of a changing quantity from its present value and a rule for how fast it changes. These models matter because nearly every measurable thing that grows or shrinks (bacteria, savings accounts, viruses, isotopes) obeys one of a small number of canonical patterns, so mastering a handful of equations lets you forecast an enormous variety of real systems. The most important distinction is between linear growth (add the same amount each period) and exponential growth (multiply by the same factor each period). The surprising fact: exponential growth, no matter how slowly it starts, will eventually overtake any polynomial growth — a doubling penny beats a million-dollar head start within weeks.

The Core Insight

The behaviour of a growing quantity is dictated by what its rate of change depends on:

If the rate is constant → linear growth (straight line).
If the rate is proportional to the current amount → exponential growth (the rich get richer, faster and faster).
If the rate is proportional to the current amount but capped by a ceiling → logistic growth (fast at first, then leveling off into an S-shape).

One question — "What is the growth rate proportional to?" — selects the whole model.

Formal Statement

Linear Growth

A quantity $N$ grows linearly if it increases by a fixed amount $m$ per unit time:

N (t) = N_{0} + m t

⟳integrate the constant-rate equation

\frac{d N}{d t} = m

with

N (0) = N_{0}

Here $N_{0}$ is the initial value (at $t = 0$ ), $m$ is the growth rate (amount per unit time, can be negative for decay), and $t$ is time. The defining feature is that the per-period change $Δ N = m$ is constant.

Exponential Growth

A quantity grows exponentially if its rate of change is proportional to its current size:

\frac{d N}{d t} = k N

⟳this is the statement "growth speed ∝ amount"; the constant of proportionality

k

is the per-capita rate.

Solving this gives the closed form:

N (t) = N_{0} e^{k t}

⟳separate variables —

\int \frac{d N}{N} = \int k d t

gives

ln N = k t + C

; exponentiate and set

N (0) = N_{0}

Here $k$ is the continuous growth constant (units: 1/time). If $k > 0$ the quantity grows; if $k < 0$ it decays. The discrete (per-period) version is:

N (t) = N_{0} (1 + r)^{t}

⟳each period multiplies by the growth factor

(1 + r)

; applying it

t

times gives the power.

where $r$ is the fractional growth per period and $b = (1 + r)$ is the base or growth factor.

Doubling Time

A quantity growing exponentially at continuous rate $k$ doubles in time

T_{2} = \frac{ln 2}{k} \approx \frac{0.693}{k}

⟳set

N (T_{2}) = 2 N_{0}

in equation [3], so

e^{k T_{2}} = 2

, then take logs.

For a percentage rate $p$ (where $r = p /100$ ), the practical Rule of 70 gives $T_{2} \approx 70/ p$ .

Logistic Growth

When growth is limited by a maximum sustainable size $K$ (the carrying capacity), the model becomes:

\frac{d N}{d t} = k N (1 - \frac{N}{K})

⟳take exponential growth

k N

and multiply by the "braking factor"

(1 - N / K)

, which is near 1 when

N

is small but drops to 0 as

N \to K

The closed-form solution is:

N (t) = \frac{K}{1 + A e ^{- k t}}, A = \frac{K - N _{0}}{N _{0}}

⟳separate variables in [6] using partial fractions

\frac{1}{N ( 1 - N / K )} = \frac{1}{N} + \frac{1/ K}{1 - N / K}

, integrate, and solve for

N

Symbols: $K$ is carrying capacity, $A$ encodes the initial condition, and the curve is steepest (fastest growth) when $N = K /2$ .

Why It Holds / Derivation

Solving the Exponential Equation

We derive equation [3] from the defining relation [2].

Start with $\frac{d N}{d t} = k N$ .
Separate variables (collect all $N$ on one side, all $t$ on the other): $\frac{d N}{N} = k d t$ .
Integrate both sides: $\int \frac{1}{N} d N = \int k d t \Rightarrow ln ∣ N ∣ = k t + C$ .
Exponentiate to undo the logarithm: $N = e^{k t + C} = e^{C} e^{k t}$ .
Apply the initial condition $N (0) = N_{0}$ : at $t = 0$ , $N_{0} = e^{C} \cdot 1$ , so $e^{C} = N_{0}$ .
Conclude: $N (t) = N_{0} e^{k t}$ . $■$

Why Logistic Growth Levels Off

Examine the braking factor $(1 - N / K)$ in equation [6].

When $N$ is small ( $N ≪ K$ ), $N / K \approx 0$ , so $(1 - N / K) \approx 1$ and the equation reduces to $\frac{d N}{d t} \approx k N$ — pure exponential growth.
When $N$ approaches $K$ , the factor $(1 - N / K) \to 0$ , so $\frac{d N}{d t} \to 0$ — growth stalls.
The maximum of $\frac{d N}{d t}$ occurs where its derivative vanishes; treating $\frac{d N}{d t} = k N - \frac{k}{K} N^{2}$ as a downward parabola in $N$ , the vertex is at $N = K /2$ .
Therefore the population grows fastest at half the carrying capacity, producing the characteristic S-curve. $■$

Pierre François Verhulst | 1804–1849

A Belgian mathematician, Verhulst introduced the logistic equation in 1838 in response to Thomas Malthus's claim that human populations grow without bound. He recognized that real populations face resource limits and modified the exponential model with a self-damping term, coining the word logistique. His work was largely forgotten until rediscovered in the 1920s by Raymond Pearl and Lowell Reed, who applied it to U.S. population data. The logistic curve now underpins ecology, epidemiology, and even machine learning (the logistic/sigmoid function).

Building Intuition

Think of interest in a bank. Linear growth is like keeping cash under a mattress where a friend slips in the same \$10 every week — steady, predictable, a straight line. Exponential growth is compound interest: the bank pays interest on your interest, so the bigger your balance, the faster it grows. That's why the curve curls upward ever more steeply — the amount you add depends on how much you already have.

Logistic growth is what happens when the bank has a vault that can only hold so much. At first you grow like compound interest, but as you near the vault's limit, each addition gets smaller until you flatline at the ceiling. The key mental switch is always: "How does the speed of growth depend on the current size?" Constant speed → line. Speed ∝ size → explosion. Speed ∝ size but choked near a cap → S-curve.

Albert A. Bartlett | 1923–2013

A physics professor famous for his lecture Arithmetic, Population, and Energy, delivered over 1,700 times. His signature line — "The greatest shortcoming of the human race is our inability to understand the exponential function" — captured how steady percentage growth (in population, energy use, debt) leads to deceptively sudden crises. He popularized the doubling-time rule as a public-literacy tool.

Comparison of linear, exponential, and logistic growth curves over time

Worked Examples

Bacterial Colony — Exponential Growth

Given: A bacterial culture starts with

N_{0} = 500

cells and doubles every 3 hours. Find the population after 12 hours and the continuous growth constant

k

. Step 1 — Use the doubling form. Doubling every 3 h means after

t

hours the number of doublings is

t /3

N (t) = 500 \cdot 2^{t /3} .

Step 2 — Evaluate at $t = 12$ . Number of doublings

= 12/3 = 4

N (12) = 500 \cdot 2^{4} = 500 \cdot 16 = 8000 cells .

Step 3 — Find $k$ from equation [5],

T_{2} = ln 2/ k

with

T_{2} = 3

k = \frac{ln 2}{3} = \frac{0.693}{3} \approx 0.231 per hour .

Answer:

N (12) = 8000 cells, k \approx 0.231 h^{- 1}

Fish Pond — Logistic Growth

Given: A pond has carrying capacity

K = 1000

fish, starts with

N_{0} = 100

fish, and has growth constant

k = 0.5 yr^{- 1}

. Find the population after 4 years. Step 1 — Compute $A$ from equation [7]:

A = \frac{K - N _{0}}{N _{0}} = \frac{1000 - 100}{100} = 9.

Step 2 — Plug into the logistic solution at

t = 4

N (4) = \frac{1000}{1 + 9 e ^{- 0.5 \cdot 4}} = \frac{1000}{1 + 9 e ^{- 2}} .

Step 3 — Evaluate. Since

e^{- 2} \approx 0.1353

N (4) = \frac{1000}{1 + 9 ( 0.1353 )} = \frac{1000}{1 + 1.218} = \frac{1000}{2.218} \approx 451 fish .

Answer:

N (4) \approx 451 fish

(The population has passed the inflection point

K /2 = 500

? Not quite — it's still accelerating toward it.)

Common Misconceptions

"Exponential just means very fast growth." No — exponential refers to the mechanism (constant percentage rate / proportional rate), not the speed. A quantity growing at 0.1% per year is still exponential, just slow. Conversely, a fast-but-decelerating quantity is not exponential. Always ask whether the percentage change per period is constant, not whether the absolute numbers look big. Confusing the discrete base $b$ with the continuous rate $k$ . In

N = N_{0} b^{t}

the base is

b = (1 + r) = e^{k}

, so

k = ln b

. A 5% annual rate gives

b = 1.05

but

k = ln 1.05 \approx 0.0488

, not 0.05. Mixing these up produces small-but-compounding errors. Linear is not "the small-growth limit" of exponential. One might assume that for tiny

k

, exponential growth becomes linear. While

e^{k t} \approx 1 + k t

for small

k t

(early times), this approximation fails for large $t$ no matter how small

k

is. Take

k = 0.001

: by

t = 10000

e^{k t} = e^{10} \approx 22026

, wildly larger than the linear prediction

1 + 10 = 11

. Exponential always wins eventually — the curves genuinely differ in kind, not just degree.

Connections

Growth models sit at the crossroads of several areas. The exponential equation $\frac{d N}{d t} = k N$ is the simplest Differential Equations and the prototype for all separable ODEs. The closed-form solution relies on properties of the Exponential Function and its inverse, the Logarithm. Discrete growth $(1 + r)^{t}$ is governed by Geometric Sequences and Series, while linear growth is just an Arithmetic Sequence in continuous disguise. Radioactive decay uses the same equation with $k < 0$ , linking to Half-Life and Radioactive Decay. The logistic function reappears in statistics and AI as the Sigmoid Function used in Logistic Regression.

Real-World Applications

Compound interest & finance — Savings, loans, and credit-card debt all follow $A = P (1 + r / n)^{n t}$ , a discrete exponential model; understanding it is essential for personal finance.
Epidemiology (early COVID-19, flu) — Early disease spread is exponential; the full epidemic curve is logistic as susceptibles run out — the basis of the SIR model.
Radioactive dating — Carbon-14 decay ( $k < 0$ ) lets archaeologists date organic remains via Half-Life and Radioactive Decay.
Population ecology — The Verhulst logistic model predicts how fish stocks, deer herds, or yeast cultures saturate at carrying capacity, guiding sustainable harvesting.
Moore's Law / technology — Transistor counts doubled roughly every two years, a famous exponential trend used in computing forecasts.
Pharmacokinetics — Drug concentration in the bloodstream decays exponentially, setting dosing intervals.
Viral marketing & social media — Adoption of new products often traces a logistic curve: slow start, rapid uptake, saturation.

Historical Development

1683 — Jacob Bernoulli, studying compound interest, discovers the constant $e$ as the limit of $(1 + 1/ n)^{n}$ .
1798 — Thomas Malthus publishes An Essay on the Principle of Population, arguing populations grow geometrically (exponentially) while food grows arithmetically (linearly).
1825 — Benjamin Gompertz proposes the Gompertz curve for mortality, an asymmetric cousin of the logistic.
1838 — Pierre François Verhulst introduces the logistic equation to correct Malthus's unbounded model.
1845–1847 — Verhulst publishes further papers naming the curve logistique and fitting it to Belgian and French population data.
1920 — Raymond Pearl and Lowell Reed rediscover the logistic model and fit it to U.S. census data, sparking modern mathematical demography.
1926 — Alfred Lotka and Vito Volterra extend growth models to interacting species (predator–prey equations).
1927 — Kermack and McKendrick formulate the SIR epidemic model, embedding logistic-type dynamics into disease spread.
1976 — Robert May shows the discrete logistic map $x_{n + 1} = r x_{n} (1 - x_{n})$ produces chaos, revolutionizing dynamical systems theory.

Subtleties & Limits

Real systems rarely obey a single clean model. Time-varying rates (seasonal birth rates, changing interest), stochastic noise (random deaths in small populations), and delays (gestation periods causing oscillations rather than smooth approach to $K$ ) all break the basic equations. The discrete logistic map famously shows that even a "simple" growth rule can produce chaotic, unpredictable behaviour for large growth parameters — an active frontier in nonlinear dynamics. Modelling multiple interacting populations, spatial spread (reaction–diffusion equations), and evolving carrying capacities (climate-driven $K$ ) remain open challenges where no single formula suffices and numerical simulation is required.