Population Dynamics (Lotka-Volterra) Notes & Questions | Class 12 Biology | Drishti OS

What you'll learn

The difference between exponential and logistic population growth
What carrying capacity means and how it limits growth
The Lotka-Volterra predator-prey equations and what they predict
The r-strategist vs K-strategist life history spectrum
Biotic and abiotic factors that regulate population size

Key concepts

Population Growth Models

Exponential Growth (J-curve)

Occurs when resources are unlimited. Population size grows at a constant rate per individual.

dN/dt = rN

Symbol	Meaning
N	Current population size
r	Intrinsic rate of natural increase (birth rate − death rate)
dN/dt	Rate of change in population size

Populations rarely sustain exponential growth for long — resource limits eventually apply.

Logistic Growth (S-curve)

Growth slows as the population approaches the environment's carrying capacity (K).

dN/dt = rN × (K − N) / K

When N is small → growth is nearly exponential
When N approaches K → growth rate approaches zero
When N = K → population is stable (dN/dt = 0)

Phase	Description
Lag phase	Slow initial growth, small population
Exponential phase	Rapid growth, resources plentiful
Deceleration phase	Growth slows as N approaches K
Plateau	Population stabilises near K

Carrying Capacity (K)

The maximum population size an environment can sustainably support, given available food, space, water, and other resources.

Factors that lower K: habitat destruction, pollution, disease, competition. Factors that raise K: improved food supply, reduced predation, habitat restoration.

Lotka-Volterra Predator-Prey Model

Describes the cyclic interaction between a prey population (N) and a predator population (P).

Prey equation:

dN/dt = αN − βNP

Predator equation:

dP/dt = δNP − γP

Symbol	Meaning
α	Prey birth rate (in absence of predators)
β	Rate at which predators consume prey
δ	Predator birth rate per prey eaten
γ	Predator death rate (in absence of prey)

Key predictions:

Prey and predator populations oscillate in cycles.
Predator numbers peak slightly after prey numbers.
Neither population goes extinct in ideal conditions.

Classic example: Canadian lynx (predator) and snowshoe hare (prey) show 10-year population cycles that match Lotka-Volterra predictions.

r-Strategists vs K-Strategists

Feature	r-Strategists	K-Strategists
Environment	Unpredictable	Stable, resource-limited
Offspring	Many, small	Few, large
Parental care	Little	Extensive
Lifespan	Short	Long
Population growth	Rapid, near exponential	Slow, near K
Examples	Insects, annual plants, mice	Elephants, whales, humans

Population Regulation Factors

Type	Density-Dependent	Density-Independent
Definition	Effect increases as population grows	Effect same regardless of density
Examples	Predation, disease, competition, starvation	Floods, fires, earthquakes, harsh winters
Effect	Stabilises population near K	Can cause sudden crashes

Quick check

Write the logistic growth equation and explain what happens to the growth rate (dN/dt) when the population size N equals K.
In a Lotka-Volterra model, prey numbers peak first, followed by a peak in predator numbers. Explain why this time lag occurs.
Compare the reproductive strategies of a mosquito and an elephant using r/K selection theory. What trade-offs are involved?
A deer population is affected by a harsh winter that kills 30% of individuals regardless of herd size. Is this a density-dependent or density-independent factor? Justify your answer.
List three ways in which carrying capacity (K) can be altered in a natural ecosystem, and state whether each raises or lowers K.

Population Dynamics (Lotka-Volterra)