Exponential Growth Models: How Mathematics Predicts Population Growth (Class 11 Maths)

Exponential Growth Models: How Mathematics Predicts Population Growth

How does a government agency estimate how many people will live in a country ten years from now? The CBSE Class 11 NCERT Appendix on Mathematical Modelling answers this with a population growth model built from birth rates, death rates, and one of the most important functions in mathematics — the exponential function. This post derives the model step by step and closes out the appendix by summarising the full modelling cycle.

Step 1: Formulating the Problem

Suppose a population control unit wants to find out how many people will be in a certain country after 10 years. The population clearly changes with time: it increases through births and decreases through deaths.

Let t denote time in years, where t = 0 is the present time, t = 1 is the next year, and so on. Let P(t) denote the population at time t. If B(t) is the number of births between year t and t + 1, and D(t) is the number of deaths in that same interval, then the population one year later is:

P(t + 1) = P(t) + B(t) − D(t)

Defining Birth Rate and Death Rate

Two useful ratios are defined from this relation:

• B(t) / P(t) is called the birth rate for the interval t to t + 1

• D(t) / P(t) is called the death rate for the same interval

Two Simplifying Assumptions

To turn this into a workable model, two assumptions are made:

1. The birth rate is the same for every interval, and so is the death rate. This means there exist constants b (birth rate) and d (death rate) such that b = B(t)/P(t) and d = D(t)/P(t) for all t ≥ 0.

2. There is no migration into or out of the population — the only sources of population change are births and deaths.

Step 2: Deriving the Growth Equation

Using these two assumptions, the population relation becomes:

P(t + 1) = P(t) + B(t) − D(t) = P(t) + bP(t) − dP(t) = (1 + b − d) P(t)

Setting t = 0 gives P(1) = (1 + b − d) P(0). Setting t = 1 gives P(2) = (1 + b − d) P(1) = (1 + b − d)² P(0), since P(1) can be substituted from the previous step. Continuing this pattern for every value of t produces the general formula:

P(t) = (1 + b − d)ᵗ P(0)

The constant (1 + b − d) is commonly abbreviated as r and called the growth rate, or the Malthusian parameter, named after Robert Malthus, who first popularised this style of population model. In terms of r, the formula becomes:

P(t) = P(0) rᵗ, for t = 0, 1, 2, …

This is an example of an exponential function — any function of the form c·rᵗ, where c and r are constants, is exponential. This equation is the complete mathematical formulation of the population problem.

Step 3: Solving with Real Numbers

Suppose the current population is 250,000,000, with a birth rate b = 0.02 and a death rate d = 0.01. This gives a growth rate r = 1 + 0.02 − 0.01 = 1.01. To find the population after 10 years, calculate P(10):

P(10) = (1.01)¹⁰ × 250,000,000 = 1.104622125 × 250,000,000 = 276,155,531.25

Step 4: Interpretation and Validation

This result is immediately flagged as slightly unrealistic — no country can have 0.25 of a person. This is a normal and expected feature of mathematical models: the underlying assumptions (constant birth and death rates, zero migration) are simplifications of a far more complex reality, so the model’s output needs sensible rounding and interpretation. After rounding, the population is approximated as 276,155,531. The small mismatch between the model’s precision and reality is not a flaw to panic over — it is simply the cost of simplifying the real world enough to make it mathematically tractable.

The Complete Mathematical Modelling Cycle

Having worked through the tower-height, Konigsberg bridge, pendulum, food-mix, and population examples, the NCERT appendix closes by summarising the entire process as a repeating cycle:

Assumptions/Axioms → Formulation → Solution → Interpretation → Validation → Satisfied?

If the answer to “Satisfied?” is yes, the process stops — the model is accepted as a good enough representation of reality. If the answer is no, the cycle loops back to the assumptions stage: perhaps an essential parameter was left out, an assumption was too simplistic, or the formulation needs adjustment. A new formulation, new mathematical manipulation, and a fresh evaluation follow, and the loop repeats until the model’s predictions are close enough to observed facts.

This is the central lesson of the entire appendix: the real world is far too complex to describe completely. Modelling works by deliberately picking out just one or two dominant factors, building a simplified model around them, and refining that model repeatedly as better data or sharper reasoning becomes available. Every equation covered in this appendix — from h = d tan θ to P(t) = P(0)rᵗ — is a snapshot of this ongoing refinement process, not a final, perfect truth.

Keep Learning

• Linear Programming as a Mathematical Model: The Farm Food-Mix Problem

• The Four Steps of Mathematical Modelling Explained with the Simple Pendulum

• What Is Mathematical Modelling? Meaning, Need & Real-Life Examples

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Test Your Knowledge

Q1.In the population model, the birth rate for the interval t to t + 1 is:

P(t) / B(t)B(t) − D(t)B(t) / P(t)B(t) / D(t)

Q2.What does the model assume about migration?

There is no migration in or outMigration exactly offsets the deathsMigration grows at the rate r each yearMigration is counted together with births

Q3.The growth rate r (the Malthusian parameter) equals:

b − d1 − b + db ÷ d1 + b − d

Q4.What kind of function is P(t) = P(0)rᵗ?

LinearExponentialQuadraticLogarithmic

Q5.In the modelling cycle, what happens when the answer to "Satisfied?" is no?

The model is discarded for goodThe solution is simply rounded offThe cycle returns to the assumptions stageThe observed data is replaced by averages