Is it correct to say: Leaving out immigration and emigration in the growth equation?

N(t+1) = N(t) + [(B + I) − (D + E)]. Immigrants add to the population and emigrants leave it.

Is it correct to say: Writing the logistic factor as (N − K)/K?

It is (K − N)/K. While N is below K this factor is positive and shrinks towards zero as N approaches K.

Is it correct to say: Entering r as a percentage, e.g. r = 2 instead of 0.02?

r is a per capita rate. Convert any percentage to a decimal and keep r and t in the same time units.

Is it correct to say: Marking parasitism as (+, 0) or commensalism as (+, −)?

Parasitism is (+, −): the host is harmed. Commensalism is (+, 0): one benefits and the other is neither harmed nor benefited.

Is it correct to say: Thinking predators only harm the prey population?

Predators keep prey populations under control and help maintain species diversity. When the starfish Pisaster was removed from an intertidal area, more than 10 invertebrate species became extinct within a year due to competition.

Is it correct to say: Saying total number is always the best measure of population size?

For a few large organisms (one huge banyan tree versus 200 Parthenium plants) per cent cover or biomass is more meaningful.

📄 Formula sheet (PDF)

Organisms and Populations

Population attributes, birth and death rates, age pyramids, the population growth equation, exponential and logistic growth, life history variation and population interactions — NCERT Class 12 Biology Ch 11

📐 14 formulas✏️ 4 examples🎯 6 practice⚖️ Unit X Ecology and Environment: 10 marks (with Ch 12 and 13)🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Birth and death rates are per capita rates. Divide by the starting population and write the unit in words (offspring per lotus per year, deaths per fruit fly per week).
→In the growth equation N(t+1) = N(t) + [(B + I) − (D + E)], immigration adds and emigration subtracts. List all four numbers before substituting.
→Sketch both growth curves on one set of axes: the J-shaped exponential curve and the S-shaped logistic curve that levels off at K. Label lag, acceleration, deceleration and asymptote.
→Learn the interaction table with signs (+, −, 0) and one NCERT example for each. Parasitism and predation are both (+, −) — the difference is in how the host or prey is used.
→For 'why is logistic growth more realistic?', the key phrase is that resources are finite and become limiting sooner or later.

📐 Formulas(14)

Population Density

N = \frac{number of individuals}{unit area or volume}

Symbol	Meaning
$N$	Population density (individuals per unit area or volume)

Birth Rate (Natality Rate)

b = \frac{number of births in a period}{population at the start}

Symbol	Meaning
$b$	Per capita birth rate (births per individual per unit time)

Death Rate (Mortality Rate)

d = \frac{number of deaths in a period}{population at the start}

Symbol	Meaning
$d$	Per capita death rate (deaths per individual per unit time)

Age Pyramids

Expanding (growing); Stable; Declining

Population Growth Equation★ Board fav

N_{t + 1} = N_{t} + [(B + I) - (D + E)]

Symbol	Meaning
$N_{t}$	Population density at time t
$N_{t + 1}$	Population density one time step later
$B$	Number of births (natality)
$I$	Number of immigrants (same species coming in)
$D$	Number of deaths (mortality)
$E$	Number of emigrants (leaving the habitat)

Exponential Growth (Differential Form)★ Board fav

\frac{d N}{d t} = (b - d) \times N = r N

Symbol	Meaning
$N$	Population size (density)
$r$	Intrinsic rate of natural increase, r = b − d (per unit time)
$b, d$	Per capita birth and death rates

Exponential Growth (Integral Form)

N_{t} = N_{0} e^{r t}

Symbol	Meaning
$N_{t}$	Population density after time t
$N_{0}$	Population density at time zero
$e$	Base of natural logarithms (2.71828)
$t$	Time elapsed

NCERT Values of r

r_{Norway rat} = 0.015; r_{flour beetle} = 0.12; r_{India, 1981} = 0.0205

Logistic Growth (Verhulst–Pearl)★ Board fav

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

Symbol	Meaning
$N$	Population density at time t
$r$	Intrinsic rate of natural increase
$K$	Carrying capacity — the maximum population the habitat can support

Limits of the Logistic Equation

N ≪ K \Rightarrow \frac{d N}{d t} \approx r N; N = K \Rightarrow \frac{d N}{d t} = 0

Growth Curve Shapes

Exponential \Rightarrow J-shaped; Logistic \Rightarrow S-shaped (lag, acceleration, deceleration, asymptote)

Life History Variation

Breed once: Pacific salmon, bamboo; Breed many times: most birds and mammals

Population Interactions★ Board fav

Interaction Mutualism Competition Predation Parasitism Commensalism Amensalism Species A + - + + + - Species B + - - - 00

Gause's Competitive Exclusion Principle

Two closely related species competing for the same resources cannot co-exist indefinitely

✏️ Solved Examples

1Solved Exampleeasy2 steps

A pond had 50 lotus plants at the start of a year, and 15 new plants were added by reproduction during the year. In a laboratory, 12 of 80 fruit flies died in one week. Calculate the birth rate of lotus and the death rate of fruit flies.

Birth rate is births divided by the starting population.

b = \frac{15}{50} = 0.3

2Solved Exampleboard3 steps

A deer population has density 500 at the start of a year. During the year there are 60 births, 40 deaths, 15 immigrants and 10 emigrants. Find the density at the end of the year and state whether it is growing. (2 marks)

Write the growth equation.

N_{t + 1} = N_{t} + [(B + I) - (D + E)]

3Solved Exampleboard3 steps

A bacterial culture starts with 1000 cells and grows exponentially with r = 0.2 per hour under unlimited resources. Estimate the number of cells after 10 hours.

Use the integral form of exponential growth.

N_{t} = N_{0} e^{r t}

4Solved ExampleHOTS4 steps

A population has r = 0.4 per year and carrying capacity K = 1000. Compare its growth rate dN/dt under the exponential and logistic models when N = 250. What is the logistic growth rate at N = 1000?

Exponential model.

\frac{d N}{d t} = r N = 0.4 \times 250 = 100 per year

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Leaving out immigration and emigration in the growth equation
✓N(t+1) = N(t) + [(B + I) − (D + E)]. Immigrants add to the population and emigrants leave it.
2
Writing the logistic factor as (N − K)/K
✓It is (K − N)/K. While N is below K this factor is positive and shrinks towards zero as N approaches K.
3
Entering r as a percentage, e.g. r = 2 instead of 0.02
✓r is a per capita rate. Convert any percentage to a decimal and keep r and t in the same time units.
4
Marking parasitism as (+, 0) or commensalism as (+, −)
✓Parasitism is (+, −): the host is harmed. Commensalism is (+, 0): one benefits and the other is neither harmed nor benefited.
5
Thinking predators only harm the prey population
✓Predators keep prey populations under control and help maintain species diversity. When the starfish Pisaster was removed from an intertidal area, more than 10 invertebrate species became extinct within a year due to competition.
6
Saying total number is always the best measure of population size
✓For a few large organisms (one huge banyan tree versus 200 Parthenium plants) per cent cover or biomass is more meaningful.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
6 new individuals are added through births to a population of 30 in a year. What is the birth rate?
Q2
N(t) = 1200, B = 90, D = 70, I = 20, E = 40. Find N(t+1) and comment.
Q3
A population of 200 grows exponentially with r = 0.1 per day. Estimate its size after 10 days (e ≈ 2.718).
Q4
Under logistic growth with r = 0.5 per year and K = 400, what is dN/dt when N = 400?
Q5
Name the interaction in which one species benefits and the other is unaffected, and give one example.
Q6
State Gause's competitive exclusion principle.

📝 Notes

Organisms and Populations

This chapter treats a population as a unit with its own attributes — birth rate, death rate, sex ratio and age distribution — that individuals do not have. It then asks how populations grow and how species interact.

From attributes to the growth equation

Birth and death rates are always per capita: divide by the starting number. The four processes that change density combine in one bookkeeping equation:

$N_{t+1} = N_t + [(B + I) - (D + E)]$

In board numericals, list B, I, D and E first, then substitute. Equal inflow and outflow give a stable population.

Two growth models

Exponential — $dN/dt = rN$ , or $N_t = N_0 e^{rt}$ . It assumes unlimited resources and gives a J-shaped curve. Here $r = b - d$ is the intrinsic rate of natural increase.
Logistic — $dN/dt = rN(K - N)/K$ . Resources are limited, so growth slows near the carrying capacity K and the curve is S-shaped, with lag, acceleration, deceleration and asymptote phases.

A useful check: for small N the logistic factor is close to 1 and the two models agree; at N = K the logistic growth rate is exactly zero.

Interactions — learn the signs

Write the six interactions with their (+, −, 0) signs and one example each. Predation and parasitism share the same signs. Competition can occur between totally unrelated species (flamingoes and resident fishes competing for zooplankton) and even when resources are not limiting (interference competition). Gause's principle and resource partitioning (MacArthur's warblers) often appear together in a single answer.

Life history

Organisms that breed once (Pacific salmon, bamboo) and those that breed many times (most birds and mammals) show how populations evolve to maximise reproductive fitness in their habitat.