MP Board · Class 12 · Biology · Organisms and PopulationsExplain the different population growth models (Exponential and Logistic growth) with their mathematical equations, graphical representations, and biological significance.

Introduction to Population Growth\nThe size of a population for any species is not a static parameter. It keeps changing depending on various factors including food availability, pressure from predators, and weather. The two main population growth models are:

1. Exponential Growth Model

• Condition: Occurs when resources (food and space) are unlimited in a habitat.
• Equation: $\frac{dN}{dt} = rN$
  • Where $N =$ Population density, $t =$ Time, $r =$ Intrinsic rate of natural increase, and $e =$ Base of natural logarithms.
• Integral form: $N_t = N_0 e^{rt}$
• Graph: When population density ($N$) is plotted against time ($t$), the exponential growth curve results in a J-shaped curve.
• Significance: Realistically, resources are finite, so exponential growth can only occur for short periods under exceptional conditions (e.g., algal blooms).

2. Logistic Growth Model

• Condition: In nature, a given habitat has enough resources to support a maximum possible number, beyond which no further growth is possible. This is called carrying capacity ($K$).
• Equation: $\frac{dN}{dt} = rN \left( \frac{K - N}{K} \right)$
  • Where $K =$ Carrying capacity.
• Graph: A population growing in a habitat with limited resources shows an initial lag phase, followed by phases of acceleration and deceleration, and finally an asymptote, resulting in a Sigmoid curve (S-shaped curve).
• Significance: Verhulst-Pearl Logistic Growth is more realistic because no population of any species in nature has at its disposal unlimited resources to permit exponential growth.

Comparison

• Exponential growth is idealistic and unbounded (J-shaped).
• Logistic growth is realistic, bound by carrying capacity ($K$), and forms an S-shaped curve.