Logistic & Other Models (AI HL)
A logistic model describes growth that starts almost exponential but then slows as it approaches a fixed ceiling - a population reaching an island's food supply, a rumour saturating a school, a product's sales plateauing. This topic covers reading the carrying capacity and initial value from a logistic function, finding where growth is fastest, and working with related "other" bounded models that approach a limit without the full S-shape.
What the syllabus says
This topic is an AHL extension to the modelling content, examined only at Applications & Interpretation HL.
| Code | Syllabus content |
|---|---|
| AHL2.9 | Logistic models: \(f(x) = \dfrac{L}{1+Ce^{-kx}}\), with \(L, C, k > 0\). The horizontal asymptote at \(f(x)=L\) is often referred to as the carrying capacity - used where growth faces a natural restriction, such as a population on an island, bacteria in a petri dish, or the height of a growing seedling. |
| AHL2.9 | In examinations, students may be expected to interpret and use other models that are introduced directly in the question, alongside the exponential, natural logarithmic, sinusoidal and piecewise models also covered at AHL. |
Enrichment link: the logistic equation solves the differential equation \(\dfrac{dP}{dt}=kP\left(1-\dfrac{P}{L}\right)\) with \(P=P_0\) at \(t=0\), giving \(C = \dfrac{L}{P_0}-1\).
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a logistic model?
A logistic model is a function of the form \(f(x) = \dfrac{L}{1+Ce^{-kx}}\) that grows quickly at first, like an exponential, but then slows and levels off towards a fixed ceiling \(L\) as \(x\) increases. It's used whenever growth faces a natural limit.
e.g. For \(P(t) = \dfrac{100}{1+9e^{-0.5t}}\), \(P(0) = \dfrac{100}{1+9} = 10\).
What is the carrying capacity?
The carrying capacity is the horizontal asymptote \(L\) that a logistic curve approaches as \(x\to\infty\), but never actually reaches. It represents the maximum sustainable level - the largest population an island can support, or the total market a product could saturate.
e.g. For \(P(t) = \dfrac{500}{1+4e^{-0.3t}}\), the carrying capacity is \(500\).
What is the initial value of a logistic model?
The initial value \(P(0)\) is found by substituting \(x=0\), which makes the exponential term equal 1: \(P(0) = \dfrac{L}{1+C}\). It's the starting population or quantity before any growth has occurred.
e.g. For \(P(t) = \dfrac{500}{1+4e^{-0.3t}}\), \(P(0) = \dfrac{500}{1+4} = 100\).
What is a saturation (bounded growth) model?
A saturation model, one of the "other" models on this topic, rises from a starting value and approaches a ceiling from below without the initial slow start of a full logistic curve - for example \(S(t) = L(1-e^{-kt})\), which is really the exponential model \(f(x)=ke^{rx}+c\) written the other way up.
e.g. For \(S(t) = 800(1-e^{-0.3t})\), \(S(0) = 0\) and \(S \to 800\) as \(t\to\infty\).
What is the inflection point of a logistic curve?
The inflection point is where the logistic curve switches from speeding up to slowing down - the point of fastest growth. It always occurs at exactly half the carrying capacity, \(P = L/2\).
e.g. For \(N(t) = \dfrac{400}{1+7e^{-0.25t}}\), the inflection is at \(N=200\), when \(t \approx 7.78\).
Key formulas
One core model, one related "other" model, and two properties that let you read off key features without solving anything.
Formula reference
Both the logistic model and the general exponential model used for "other" bounded models are listed in the AI formula booklet under functions and modelling.
| Formula | Used for | Booklet? |
|---|---|---|
| \(f(x) = \dfrac{L}{1+Ce^{-kx}}\) | Logistic model | ✓ Yes |
| \(f(x)=ke^{rx}+c\) | Exponential form used for "other" bounded models | ✓ Yes |
| \(\displaystyle\lim_{x\to\infty} f(x) = L\) | Definition of carrying capacity | Not in booklet - definition |
| Fastest growth at \(P = L/2\) | Locating the inflection point | Not in booklet - derived property |
| \(\dfrac{dP}{dt}=kP\left(1-\dfrac{P}{L}\right)\) | Logistic differential equation (enrichment) | Not in booklet - HL enrichment |
Logistic vs unrestricted exponential growth
Both models start similarly, but they diverge completely once growth is well underway - this is the whole point of using a logistic model over a plain exponential one.
| Feature | Exponential growth | Logistic growth |
|---|---|---|
| Long-term behaviour | Grows without bound | Approaches carrying capacity \(L\) |
| Shape | Ever-steepening curve | S-shaped (sigmoid) |
| Early behaviour | Exponential | Near-exponential (small \(P\) relative to \(L\)) |
| Realistic for | Short time frames only | Populations, epidemics, sales with a ceiling |
Reading a logistic model
Every question on this topic starts with correctly identifying \(L\), \(C\) and \(k\) from the given function - get this right and the rest is substitution or solving.
Carrying capacity
\[L = \lim_{x\to\infty} f(x)\]
Read \(L\) directly off the top of the fraction - it's the value the model approaches but never reaches.
Not in the formula booklet - definitionInitial value
\[P(0) = \dfrac{L}{1+C}\]
Substitute \(x=0\) so \(e^{-k(0)}=1\); this also lets you solve for \(C\) if \(L\) and \(P(0)\) are known.
Not in the formula booklet - substitutionGrowth constant \(k\)
\[e^{-kt} = \text{isolated value} \Rightarrow t = \dfrac{\ln(\dots)}{-k}\]
\(k\) controls how quickly the curve rises - isolate the exponential term, then take logs to solve for time or for \(k\) itself.
Not in the formula booklet - solving techniqueOther bounded models
Not every "levelling off" context is a full logistic S-curve - some start at their steepest and slow down from the very first instant.
Saturation growth
\[S(t) = L(1-e^{-kt})\]
Rises from \(S(0)=0\) towards the ceiling \(L\) - no slow start, because it's steepest right at \(t=0\).
✓ Written from the booklet's exponential modelBounded recovery
\[F(t) = L - Ae^{-kt}\]
Same shape as saturation growth, shifted to start from any initial value rather than zero - useful for a stock recovering towards a maximum.
✓ Written from the booklet's exponential modelKey logistic properties
These two properties let you answer "when is growth fastest" questions without any calculus.
Point of fastest growth
\[P = \dfrac{L}{2} \text{ at the inflection point}\]
Growth accelerates up to half the carrying capacity, then decelerates as the curve levels off.
Not in the formula booklet - derived propertyLong-term behaviour
\[x\to\infty \Rightarrow f(x)\to L\]
As \(x\) grows, \(Ce^{-kx}\to0\), so the fraction tends to \(L/1=L\) - this is exactly what "carrying capacity" means.
Not in the formula booklet - limit behaviourWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
For \(P(t) = \dfrac{60}{1 + 5 e^{-0.7t}}\):
(a) State the carrying capacity.
(b) State the initial value \(P(0)\).
(c) State the long-term behaviour.
Worked solution
(a) Carrying capacity \(= 60.\) A1
(b) \(P(0) = \dfrac{60}{1+5}\) M1
\(= 10.\) A1
(c) As \(t\to\infty,\ P \to 60.\) A1
A population is \(P(t) = \dfrac{2000}{1 + 19 e^{-0.4t}}\), \(t\) in years.
(a)(i) State the carrying capacity.
(a)(ii) State the initial population.
(b) Find when the population reaches 1000.
Worked solution
(a)(i) Carrying capacity \(= 2000.\) A1
(a)(ii) \(P(0) = \dfrac{2000}{20} = 100.\) A1
(b) \(1 + 19e^{-0.4t} = 2 \Rightarrow e^{-0.4t} = \tfrac{1}{19}.\) \(t = \dfrac{\ln 19}{0.4}\) M1
\(\approx 7.36\) yr. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Reading \(C\) as the carrying capacity. In \(f(x)=L/(1+Ce^{-kx})\), \(L\) is the carrying capacity - \(C\) only controls the initial value and how quickly the curve moves away from it.
- Assuming fastest growth happens at \(t=0\). A full logistic curve is slowest at the very start and fastest at the inflection point, when \(P=L/2\) - not immediately.
- Confusing a saturation model with a logistic model. \(S(t)=L(1-e^{-kt})\) is steepest at \(t=0\) and has no S-shape - don't apply the "fastest at half capacity" rule to it.
- Forgetting the carrying capacity is never reached exactly. \(P\) gets arbitrarily close to \(L\) as \(t\to\infty\) but the model never actually equals \(L\) for any finite \(t\).
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Some questions give a table of values and ask you to recognise and fit a logistic or other bounded model - your GDC's regression menu handles this directly, in the same place as quadratic, cubic and exponential regression.
- Enter the data in two lists (x and y).
- STAT → EDIT to enter L1/L2, then STAT → CALC → QuadReg / CubicReg / ExpReg / PwrReg / SinReg (Logistic sits in the same menu on models that support it).TI-84
- In a Lists & Spreadsheet page enter the data, then menu → Statistics → Stat Calculations → choose the regression type, including Logistic Regression.Nspire
- Main menu → Statistics, enter the data in lists, then CALC → REG and pick X² / X³ / Exp / Power / Sin, or Logistic regression where available.Casio
Tip: Turn DiagnosticOn (TI-84: 2nd → 0 → DiagnosticOn) to see R². Choose the model with the best R² that also makes sense for the context.
After fitting several candidate models, you need to decide which fits the data best - R² is the key tool.
- Fit each candidate model in turn and note the R² value each time.
- Turn DiagnosticOn first (2nd → 0, scroll to DiagnosticOn, ENTER) - then R² appears after every regression.TI-84
- R² is shown automatically after each regression calculation in the Statistics menu.Nspire
- R² (displayed as r²) appears in the regression output; run CALC → REG for each model type and compare.Casio
- The model with R² closest to 1 explains the most variation in y - but also consider whether the model makes sense for the context.
Tip: R² alone doesn't tell you whether the model is appropriate - always look at the scatter plot too. A high R² on a model that shouldn't apply is meaningless.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Logistic & other models questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What makes a model "logistic" rather than exponential?
An exponential model grows without limit. A logistic model \(f(x) = L/(1+Ce^{-kx})\) starts by growing almost exponentially, then slows as it approaches a fixed ceiling \(L\), the carrying capacity - because something in the real situation (space, food, market size) restricts further growth.
What is the carrying capacity and how do I find it?
The carrying capacity is the horizontal asymptote \(L\) that the logistic function approaches as \(x\to\infty\). For \(f(x)=L/(1+Ce^{-kx})\) you can read it straight off - it's the constant on top of the fraction.
Where does a logistic curve grow fastest?
At the inflection point, when the value equals exactly half the carrying capacity, \(L/2\). Before that point growth is speeding up; after it, growth is slowing down as the curve levels off towards \(L\).
Is the logistic model in the formula booklet?
Yes. \(f(x) = L/(1+Ce^{-kx})\) is listed in the AI formula booklet under models, along with the other AHL modelling functions. You don't need to memorise it.
Sub-topics
Logistic & Other Models broken down into its individual skills, each with its own focused page.
Related topics
More Functions & Modelling topics from the same AI HL syllabus unit, in case you want to keep going.