49. Among the following which curve represents logisticgrowth?

(1) A (2) B
(3) C (4) D

What Is Logistic Growth?

Logistic growth describes how a population increases rapidly at first, then slows as it approaches a maximum limit imposed by limited resources in the environment. This maximum is called the carrying capacity. Unlike exponential growth, which produces a J-shaped curve and continues indefinitely (in theory), logistic growth results in an S-shaped (sigmoid) curve.

Identifying the Logistic Growth Curve

Given the options—A, B, C, and D—the curve that represents logistic growth is the one shaped like the letter S. It is also known as the sigmoid curve.

• Option A, B, C, or D:

  • J-shaped curve: Represents exponential growth (not logistic).

  • S-shaped curve: Represents logistic growth.

  • Other shapes (L, M, etc.): Do not represent classic logistic growth.

So, the correct answer is the option that matches the S-shaped curve—typically, this is the letter associated with the S-curve in your question or diagram.

Why Is the S-Shaped Curve Important?

The S-shaped curve is crucial because it reflects real-world limitations. As a population grows, resources such as food, space, and mates become scarcer. This leads to increased competition within the species (intraspecific competition), slowing the rate of population growth until it stabilizes at the carrying capacity.

Phases of the Logistic Growth Curve

The logistic growth curve can be divided into four main phases:

1. Initiation/Birth:

   • The population starts small.

   • Growth is slow due to limited numbers and resources.

2. Acceleration/Growth:

   • As resources become available, the population grows rapidly.

   • The growth rate increases, and the curve steepens.

3. Deceleration/Maturing:

   • The population approaches the carrying capacity.

   • Resources become limited, and competition increases.

   • The growth rate slows down.

4. Saturation:

   • The population stabilizes at the carrying capacity.

   • Births and deaths are balanced, and the population remains relatively constant.

Mathematical Representation

The logistic growth model is described mathematically by the equation:

dNdt=rN(1−NK)dtdN=rN(1−KN)

Where:

• $N$ = population size at time $t$

• $r$ = intrinsic growth rate

• $K$ = carrying capacity

This equation shows that as $N$ approaches $K$, the growth rate slows and eventually stops.

Real-World Examples

• Yeast in a Test Tube:

  • Yeast cells multiply rapidly at first, but as nutrients are depleted, growth slows and stabilizes, forming an S-shaped curve.

• Wild Animal Populations:

  • Species like sheep and harbor seals exhibit logistic growth, with population sizes fluctuating around the carrying capacity.

• Technology Adoption:

  • The adoption of new technologies often follows an S-curve, with rapid initial adoption, followed by a slowdown as the market becomes saturated.

Comparison: Exponential vs. Logistic Growth

Key Terms

• Carrying Capacity ($K$):

  • The maximum population size an environment can support.

• Intraspecific Competition:

  • Competition between individuals of the same species for limited resources.

• Inflection Point:

  • The point on the S-curve where growth rate is fastest, typically at half the carrying capacity.

Why Choose the S-Shaped Curve?

When faced with a question like, “Which curve represents logistic growth?” always look for the S-shaped (sigmoid) curve. This curve is the hallmark of logistic growth, reflecting the balance between rapid initial expansion and the eventual limitation imposed by the environment.

Summary

• Logistic growth is represented by an S-shaped (sigmoid) curve.

• Exponential growth is represented by a J-shaped curve.

• The S-curve reflects real-world population dynamics, where growth slows and stabilizes as resources become limited.

• The correct answer among A, B, C, and D is the S-shaped curve.

Additional Insights

The S-curve is not just limited to biology. It is widely used in economics, technology adoption, and even social sciences to model how innovations spread and how populations stabilize. Understanding this curve helps in predicting future trends and managing resources effectively.

Conclusion

When identifying which curve represents logistic growth, remember: the S-shaped curve is your answer. This curve illustrates how populations grow rapidly at first, then slow down and stabilize as they reach the carrying capacity of their environment. Whether you are studying biology, economics, or technology, the logistic growth model is a fundamental concept that explains how growth is limited by real-world constraints.