Population — Concepts, Formulas & Examples

Population is a group of individuals of the same species in a given area. Population ecology studies how numbers change with time and environment. CBSE Class 12 and NEET both test growth models and interactions.

Core Concepts

Population attributes

Size, density, age structure, sex ratio, natality (birth rate), mortality (death rate), immigration, emigration. Growth = (births + immigration) - (deaths + emigration).

Population density can be measured as:

Number per unit area: 50 trees per hectare, 200 deer per km $^2$
Biomass per unit area: When counting individuals is impractical (grass, plankton)
Cover percentage: For plants — what fraction of ground is covered

Age structure shows the proportion of individuals in three categories:

Pre-reproductive (young)
Reproductive (adults)
Post-reproductive (old)

A population with many young individuals has a growing age pyramid (broad base). A population with roughly equal proportions at all ages has a stable pyramid. A population with few young has a declining pyramid (narrow base). India has a growing pyramid; Japan has a declining one.

Sex ratio: The ratio of males to females. In India, the child sex ratio has been a concern — some states report fewer than 900 females per 1000 males. The expected natural ratio at birth is approximately 105 males to 100 females.

Exponential growth

When resources are unlimited. Rate is $dN/dt = rN$ , where r is the intrinsic rate of increase. Gives a J-shaped curve. Rarely sustained for long in nature.

\frac{dN}{dt} = rN

Integrated form: $N_t = N_0 e^{rt}$

Where $N_t$ = population at time $t$ , $N_0$ = initial population, $r$ = intrinsic rate of natural increase, $e$ = base of natural logarithm.

The intrinsic rate of increase ( $r$ ) depends on the species:

Bacteria: $r$ can exceed 60 per hour
Insects: $r \approx$ 1-5 per year
Birds: $r \approx$ 0.1-1 per year
Large mammals: $r \approx$ 0.02-0.1 per year

When does exponential growth happen in nature? When a species colonises a new, resource-rich habitat with no competitors. The cane toad introduction in Australia (1935) is a textbook case — from 102 individuals to over 200 million today. Bacterial growth in a fresh nutrient medium also follows exponential growth initially.

Logistic growth

Resources are limited. Rate is $dN/dt = rN(K-N)/K$ , where K is carrying capacity. Gives an S-shaped curve that levels off at K. More realistic than exponential.

\frac{dN}{dt} = rN\left(\frac{K-N}{K}\right)

Where $K$ = carrying capacity, $(K-N)/K$ = environmental resistance factor. When $N$ is small, $(K-N)/K \approx 1$ and growth is nearly exponential. As $N$ approaches $K$ , $(K-N)/K$ approaches 0 and growth slows to zero.

Phases of the S-curve:

Lag phase: Small population, slow growth (few individuals reproducing)
Acceleration phase: Population grows rapidly (exponential-like)
Deceleration phase: Growth slows as resources become limiting
Stationary phase: Population stabilises around $K$

Growth rate is maximum at $N = K/2$ : Taking the derivative and setting it to zero shows that the logistic growth rate peaks when the population is exactly half the carrying capacity. This fact appears frequently in NEET.

Population interactions

Mutualism (both benefit, +/+), commensalism (+/0), parasitism (+/-), predation (+/-), competition (-/-), amensalism (0/-). NCERT lists all six; NEET loves asking which is which.

Interaction	Species A	Species B	Example
Mutualism	+	+	Bee and flower, lichen (fungus + alga)
Commensalism	+	0	Remora on shark, orchid on tree
Parasitism	+	-	Plasmodium in human, Cuscuta on plant
Predation	+	-	Tiger and deer, snake and frog
Competition	-	-	Flamingo and fish competing for plankton
Amensalism	0	-	Penicillium mould inhibiting bacteria

Predation is more than just killing: Predators control prey populations (preventing overgrazing), keep prey populations healthy (removing weak/sick individuals), and drive evolution (faster prey survive, sharper predators succeed). The removal of predators can cause ecosystem collapse — the classic example is the reintroduction of wolves in Yellowstone, which restored river channels by controlling elk grazing.

Parasitism: Unlike predators, parasites usually do not kill their host (a dead host = no home). Ectoparasites live on the surface (lice, ticks). Endoparasites live inside (tapeworm, liver fluke, Plasmodium). Brood parasitism is the cuckoo strategy — lay eggs in another bird’s nest and let the host raise your chicks.

Co-evolution: Predator-prey and parasite-host interactions drive co-evolution. As prey evolves better defences, predators evolve better attack strategies. This is the Red Queen hypothesis — you must keep running (evolving) just to stay in the same place.

Carrying capacity

The maximum population a habitat can support indefinitely. Determined by food, water, space and predation. Populations overshoot and crash, or stabilise around K.

Factors that determine K:

Available food and water
Space and shelter
Predation pressure
Disease prevalence
Climate and seasonal variation

Overshoot and crash: Sometimes populations exceed K temporarily (when resources are temporarily abundant or predators are absent). The excess population degrades the resources, carrying capacity drops, and the population crashes below the original K. Reindeer on St. Matthew Island is the textbook example — introduced population grew to 6000, then crashed to 42 when they overgrazed the lichen.

Life history strategies

r-selected species (r-strategists): High reproductive rate, many offspring, little parental care, short lifespan, rapid maturation. Thrive in unstable environments. Examples: bacteria, insects, mice, weeds.

K-selected species (K-strategists): Low reproductive rate, few offspring, extensive parental care, long lifespan, slow maturation. Thrive in stable environments near carrying capacity. Examples: elephants, whales, humans, oak trees.

Feature	r-selected	K-selected
Offspring	Many, small	Few, large
Parental care	Minimal	Extensive
Maturation	Fast	Slow
Lifespan	Short	Long
Population size	Fluctuates	Stable near K
Survivorship curve	Type III	Type I

Key Formulas

dN/dt = rN(K-N)/K

The $(K-N)/K$ term makes growth slow as $N$ approaches $K$ , giving the S-curve.

\text{Growth rate} = (b + i) - (d + e)

Where $b$ = birth rate, $d$ = death rate, $i$ = immigration rate, $e$ = emigration rate. If growth rate > 0, population increases.

Worked Examples

Introduced in 1935 with no natural predators. Population exploded from about 100 individuals to billions today. The classic example of unchecked exponential growth.

Predator and prey populations oscillate — predators increase, prey decrease, predators then decrease, prey recover. The cycles are out of phase and can persist for many years.

Given: $N_0 = 100$ , $r = 0.5$ per generation, $t = 5$ generations.

N_t = N_0 e^{rt} = 100 \times e^{0.5 \times 5} = 100 \times e^{2.5} = 100 \times 12.18 = 1218

The population grows from 100 to about 1218 in 5 generations �� a 12-fold increase.

The logistic growth rate is $rN(K-N)/K$ . This is a quadratic in $N$ with maximum at $N = K/2$ .

If $K = 1000$ and $r = 0.1$ : Maximum growth rate = $0.1 \times 500 \times (1000-500)/1000 = 0.1 \times 500 \times 0.5 = 25$ individuals per unit time.

This is a standard NEET numerical.

Solved Problems (Exam Style)

Problem 1 (NEET pattern): In logistic growth, the growth rate is maximum when: (a) $N = K$ (b) $N = K/2$ (c) $N = K/4$ (d) $N$ is very small

The logistic equation $dN/dt = rN(K-N)/K$ is maximised when $N = K/2$ . At this point, the population is large enough to reproduce rapidly but resources are not yet severely limiting. Answer: (b)

At $N = K$ , the growth rate is zero (population is at carrying capacity).

Problem 2 (NEET pattern): Which interaction is shown by the remora fish attached to a shark?

The remora benefits (gets food scraps, free transport, protection). The shark is neither harmed nor helped (the remora is too small to matter). One benefits, other unaffected = commensalism (+/0).

Common Mistakes

Confusing exponential and logistic. Exponential is unlimited; logistic levels off at K.

Mixing up commensalism and mutualism. Commensalism is one benefits, other unaffected; mutualism both benefit.

Writing that parasitism always kills the host. Most parasites keep the host alive — dead hosts stop giving food.

Thinking growth rate is maximum when population reaches K. Growth rate at $N = K$ is zero. Maximum growth rate occurs at $N = K/2$ .

Confusing predation and parasitism. Predators usually kill the prey immediately. Parasites live on or in the host for an extended period without immediately killing it.

Exam Weightage and Revision

This topic is a repeat performer in board papers and entrance exams. NEET typically asks one to two questions on the core mechanisms, CBSE boards give three to six marks, and state PMT papers often include a diagram-based long answer. The PYQs cluster around a small set of facts — lock those and you clear the topic.

NEET 2023 asked about the logistic growth equation and maximum growth rate. NEET 2022 tested population interactions (commensalism example). CBSE boards ask about growth curves and their comparison. Population ecology gives 2-3 guaranteed questions in NEET.

When a question gives a scenario, identify the core mechanism first, then match it to the concepts above. Most wrong answers come from reading the scenario too quickly.

Sketch both growth curves on one axis. The J and S shapes together are worth two marks on every NEET paper.

Practice Questions

Q1. What is the significance of the term $(K-N)/K$ in the logistic equation?

$(K-N)/K$ represents the environmental resistance — the fraction of resources still available. When $N$ is small, this fraction is close to 1 (plenty of resources, growth is near-exponential). As $N$ approaches $K$ , this fraction approaches 0 (resources are exhausted, growth slows to zero). This single term converts exponential growth into logistic growth.

Q2. Give one example each of r-selected and K-selected species.

r-selected: Mosquito — short lifespan, hundreds of eggs per batch, no parental care, rapid maturation, population fluctuates dramatically with seasons. K-selected: Elephant — long lifespan (60-70 years), one calf every 4-5 years, extensive parental care, slow maturation (reaches reproductive age at ~15 years), population stable near carrying capacity.

Q3. What is brood parasitism?

Brood parasitism is when one species lays its eggs in another species’ nest, tricking the host into raising the parasite’s offspring. The cuckoo (Cuculus canorus) is the classic example — it lays eggs that mimic the host bird’s eggs in colour and pattern. The host incubates and feeds the cuckoo chick, often at the expense of its own offspring.

Q4. A population of 500 rabbits has $r = 0.02$ per day and $K = 2000$ . Calculate the growth rate.

\frac{dN}{dt} = rN\frac{K-N}{K} = 0.02 \times 500 \times \frac{2000-500}{2000} = 10 \times \frac{1500}{2000} = 10 \times 0.75 = 7.5 \text{ rabbits/day}

Q5. What is the Red Queen hypothesis?

Named after the Red Queen in Lewis Carroll’s Through the Looking-Glass, who says “you must run as fast as you can just to stay in the same place.” In ecology, it means species must constantly evolve (adapt) just to maintain their fitness relative to co-evolving competitors, predators, parasites, or prey. A prey species that stops evolving will be caught by an evolving predator.

FAQs

What is the difference between density-dependent and density-independent factors? Density-dependent factors become more severe as population density increases — competition for food, disease transmission, predation, waste accumulation. They are biological in nature and create the S-curve. Density-independent factors affect populations regardless of size — floods, fires, droughts, temperature extremes. They are often abiotic and can cause sudden population crashes unrelated to the carrying capacity.

Why do some populations oscillate instead of stabilising at K? Time lags in the response to resource availability can cause overshooting. The population grows past K, degrades resources, then crashes below K. If the lag is long, oscillations can be dramatic. Predator-prey cycles (like lynx and snowshoe hare) show this pattern, with peaks roughly every 10 years.

What does $r = 0$ mean for a population? If $r = 0$ , birth rate equals death rate (zero population growth, ZPG). The population size stays constant. This can happen at any population size, not just at K. Many developed countries are approaching ZPG.

How is population ecology relevant to conservation? Understanding K, r, and growth curves helps conservationists set sustainable harvest limits, identify species at risk (those with small K or low r), design reserves of adequate size, and predict the impact of habitat loss on population viability.

Population ecology is where math meets biology. The simple equations predict real outcomes — lion populations in Gir, locust swarms in Rajasthan, rats in Mumbai.