63. Among the following which graph best represent relationship between intrinsic rate of growth ‘r’ and generation time ‘t

Introduction

In population ecology, two fundamental parameters—intrinsic rate of growth (r) and generation time (t)—play a crucial role in shaping how populations expand or contract. Understanding the relationship between these variables is essential for predicting population trends, managing wildlife, and studying evolutionary strategies.

What Are Intrinsic Rate of Growth and Generation Time?

• Intrinsic Rate of Growth (r):
  This is the maximum potential growth rate of a population under ideal conditions, reflecting how quickly a population can increase in size per individual per unit time.

• Generation Time (t):
  This is the average time between the birth of individuals and the birth of their offspring, essentially measuring the pace of generational turnover in a population.

The Mathematical Relationship

Ecological theory and life-table analysis reveal a negative relationship between r and t: as generation time increases, the intrinsic rate of growth decreases. This relationship is often expressed mathematically as:

r=ln⁡R0Gr=GlnR0

Where:

• $r$ = intrinsic rate of growth

• R0R0 = net reproductive rate

• $G$ = generation time

This means that for a given reproductive output, populations with shorter generation times can achieve much higher growth rates than those with longer generation times.

What Does the Graph Look Like?

When plotting intrinsic rate of growth (r) on the Y-axis against generation time (t) on the X-axis:

• The graph shows a negative, hyperbolic (rectangular hyperbola) curve.

• As generation time increases (moves right along the X-axis), the value of r decreases rapidly at first and then levels off toward zero.

• This curve visually demonstrates that species with short generation times (like bacteria or insects) have high intrinsic growth rates, while those with long generation times (like elephants or whales) have low intrinsic growth rates.

Biological Interpretation

• Fast Life Histories:
  Species with short generation times and high r (e.g., mice, fruit flies) can rapidly increase their populations, making them resilient to disturbances but also prone to population crashes.

• Slow Life Histories:
  Species with long generation times and low r (e.g., elephants, whales) have stable but slow-growing populations, making them more vulnerable to overexploitation or environmental change.

Why This Relationship Matters

• Conservation:
  Understanding this inverse relationship helps in assessing extinction risk and recovery potential for endangered species.

• Pest Management:
  Species with high r and short t can rebound quickly, requiring different management strategies than slow-growing species.

• Evolutionary Ecology:
  The r-t relationship underpins the classic r/K selection theory, explaining why some species favor rapid reproduction while others invest in longevity and survival.

Conclusion

The graph that best represents the relationship between intrinsic rate of growth (r) and generation time (t) is a negative, hyperbolic curve. As generation time increases, the intrinsic rate of growth decreases. This inverse relationship is a cornerstone of population ecology, shaping the life-history strategies of all living organisms.