Population Growth

The increase in the number of individuals of a species in a given area over a period of time is called population growth.

The rate at which new individuals are added to a population through reproduction is called natality.

The rate at which individuals die in a population over a given period is called mortality.

The rate of change in population size per individual per unit time is called the population growth rate.

The maximum growth rate of a population under ideal conditions with maximum reproduction and minimum mortality is called the intrinsic rate of growth.

The reproductive capacity of a population, measured as the number of births during a given period, is called fecundity.

The addition of individuals to a population from neighbouring populations is called immigration.

The movement of individuals out of a population to other areas is called emigration.

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

\[\frac{\mathrm{dN}}{\mathrm{dt}}=\mathrm{rN}\]

Integral form: \[\mathbf{N_{t}}=\mathbf{N_{0}}\mathbf{e^{rt}}\]

Verhulst-Pearl Logistic Growth:

\[\frac{\mathrm{d~N}}{\mathrm{d~t}}=\mathbf{rN}\left[\frac{\mathrm{K}-\mathrm{N}}{\mathrm{K}}\right]\]

• Population growth is the increase in the number of individuals of a species in a particular area over time.
• The size of a population is not static; it changes continuously depending on food availability, predation pressure, weather, and other environmental conditions.
• Changes in population density help us understand whether a population is increasing, stable, or declining.

Population density changes because of four basic processes:

• Natality (B): Number of births in a population during a given time.
• Mortality (D): Number of deaths in a population during a given time.
• Immigration (I): Entry of individuals of the same species from another area into the habitat.
• Emigration (E): Movement of individuals out of the habitat to another place.

Formula for population density

• Here, N_t is the population density at time tt, and N_t+1​ is the density at time t + 1.
• Population increases when births + immigration are greater than deaths + emigration.
• Under normal conditions, births and deaths are usually more important than immigration and emigration.

• Exponential growth occurs when resources such as food and space are unlimited.
• Under these ideal conditions, a species can realise its full reproductive potential, and the population increases very rapidly.
• The population grows exponentially or geometrically.

Equation:

\[\frac{dN}{dt}=rN\]

Where:

• N = population size
• r = intrinsic rate of natural increase
• `"dN"/"dt"`​ = rate of change of population size

Integral form:

Where:

• N_t​ = population density after time tt

• N₀​ = initial population density

• e = base of natural logarithms

Key features of exponential growth

• Occurs under ideal environmental conditions.
• Resources are unlimited.
• The population increases very fast.
• The graph formed is a J-shaped curve.
• It can continue only for a limited period in nature because resources are not unlimited.

Exponential growth curve of population

• In nature, resources are limited, so no population can grow exponentially forever.
• As population density increases, competition for food, space, and other resources also increases.
• Every habitat can support only a certain maximum number of individuals, called the carrying capacity (K).

Equation

Where:

• N = population density
• r = intrinsic rate of natural increase
• K = carrying capacity

Features of logistic growth:

• Occurs when resources are limited.
• Competition takes place among individuals.
• Population growth slows down as the population approaches the carrying capacity.
• The graph is an S-shaped (sigmoid) curve.
• This growth is also called Verhulst–Pearl Logistic Growth.
• It is considered the more realistic growth model for natural populations.

Phases of logistic growth

• Lag phase: Initial slow growth.
• Acceleration phase: Rapid increase in population.
• Deceleration phase: Growth rate slows down.
• Asymptote / stationary phase: Population stabilises at carrying capacity.

Logistic growth curve of population

• Population growth is studied mainly by two models: exponential growth and logistic growth.
• Population density (N) changes due to four factors: Natality (B), Mortality (D), Immigration (I), and Emigration (E).
• Exponential growth → Occurs when resources are unlimited; population increases rapidly and forms a J-shaped curve.
• Logistic growth → Occurs when resources are limited; growth slows down and forms an S-shaped (sigmoid) curve.
• Phases of logistic growth → Lag phase → Log (exponential) phase → Diminishing growth phase → Stationary phase.
• Carrying capacity (K) → Maximum population size that the environment can support; population stabilises at this level.