Growth Models

⚡ Key Takeaways

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.

💡 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.

DefinitionLinear Growth

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

1

\quad N(t) = N_0 + m\,t

Derivation hint: integrate the constant-rate equation $\dfrac{dN}{dt}=m$ 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 $\Delta N = m$ is constant.

DefinitionExponential Growth

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

2

\quad \frac{dN}{dt} = k\,N

Derivation hint: this is the statement "growth speed ∝ amount"; the constant of proportionality $k$ is the per-capita rate. Solving this gives the closed form:

3

\quad N(t) = N_0\,e^{k t}

Derivation hint: separate variables — $\int \frac{dN}{N} = \int k\,dt$ gives $\ln N = kt + C$ ; 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:

4

\quad N(t) = N_0\,(1+r)^t

Derivation hint: 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.

TheoremDoubling Time

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

5

\quad T_2 = \frac{\ln 2}{k} \approx \frac{0.693}{k}

Derivation hint: set $N(T_2)=2N_0$ in equation [3], so $e^{kT_2}=2$ , then take logs. For a percentage rate $p$ (where $r=p/100$ ), the practical Rule of 70 gives $T_2 \approx 70/p$ .

DefinitionLogistic Growth

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

6

\quad \frac{dN}{dt} = k\,N\!\left(1 - \frac{N}{K}\right)

Derivation hint: take exponential growth $kN$ 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:

7

\quad N(t) = \frac{K}{1 + A\,e^{-k t}}, \qquad A = \frac{K-N_0}{N_0}

Derivation hint: 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 $N = K/2$ .

ProofSolving the Exponential Equation

We derive equation [3] from the defining relation [2]. 1. Start with $\dfrac{dN}{dt} = kN$ . 2. Separate variables (collect all $N$ on one side, all $t$ on the other): $\dfrac{dN}{N} = k\,dt$ . 3. Integrate both sides: $\displaystyle\int \frac{1}{N}\,dN = \int k\,dt \;\Rightarrow\; \ln|N| = kt + C$ . 4. Exponentiate to undo the logarithm: $N = e^{kt+C} = e^C e^{kt}$ . 5. Apply the initial condition $N(0)=N_0$ : at $t=0$ , $N_0 = e^C\cdot 1$ , so $e^C = N_0$ . 6. Conclude: $N(t) = N_0 e^{kt}$ . $\blacksquare$

∎

ProofWhy Logistic Growth Levels Off

Examine the braking factor $(1-N/K)$ in equation [6]. 1. When $N$ is small ( $N\ll K$ ), $N/K\approx 0$ , so $(1-N/K)\approx 1$ and the equation reduces to $\frac{dN}{dt}\approx kN$ — pure exponential growth. 2. When $N$ approaches $K$ , the factor $(1-N/K)\to 0$ , so $\frac{dN}{dt}\to 0$ — growth stalls. 3. The maximum of $\frac{dN}{dt}$ occurs where its derivative vanishes; treating $\frac{dN}{dt}=kN-\frac{k}{K}N^2$ as a downward parabola in $N$ , the vertex is at $N=K/2$ . 4. Therefore the population grows fastest at half the carrying capacity, producing the characteristic S-curve. $\blacksquare$

∎

mathematician · 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).

🧠 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.

AA

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.

Figure

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

ExampleBacterial 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 \text{ cells}.

Step 3 — Find $k$ from equation [5], $T_2=\ln2/k$ with $T_2=3$ :

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

Answer:

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

ExampleFish Pond — Logistic Growth

Given: A pond has carrying capacity $K = 1000$ fish, starts with $N_0 = 100$ fish, and has growth constant $k = 0.5\ \text{yr}^{-1}$ . 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 \text{ fish}.

Answer:

\boxed{N(4) \approx 451 \text{ fish}}

(The population has passed the inflection point $K/2=500$ ? Not quite — it's still accelerating toward it.)

⚠️ Common mistake

"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.

⚠️ Common mistake

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.

Counterexample

Linear is not "the small-growth limit" of exponential. One might assume that for tiny $k$ , exponential growth becomes linear. While $e^{kt}\approx 1+kt$ for small $kt$ (early times), this approximation fails for large $t$ no matter how small $k$ is. Take $k=0.001$ : by $t=10000$ , $e^{kt}=e^{10}\approx 22026$ , wildly larger than the linear prediction $1+10=11$ . Exponential always wins eventually — the curves genuinely differ in kind, not just degree.

connection

Growth models sit at the crossroads of several areas. The exponential equation $\frac{dN}{dt}=kN$ 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.

🔬 Where this appears

⏳ History

- 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}=rx_n(1-x_n)

produces chaos, revolutionizing dynamical systems theory.

open-problem

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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also