When Do Mice on an Island Reach Their Highest Population Growth Rate?

Introducing a small group of animals to a new environment is a classic scenario in ecology. When a few mice are released on an uninhabited island, their population dynamics can teach us a lot about how species colonize new habitats. This article explores the growth patterns of such a mouse population, focusing on when their growth rate is highest and what factors influence this critical point.

The Scenario: Mice on an Island

Imagine that 10 mice are introduced to an uninhabited island. With plenty of food, shelter, and no predators, their population grows rapidly at first. After three years, the population reaches 520 mice and then stabilizes. This pattern—rapid initial growth followed by stabilization—is typical of many real-world populations.

Understanding Population Growth Models

There are two main models for population growth: exponential and logistic.

1. Exponential Growth

When resources are unlimited, populations grow exponentially. The growth rate is constant, and the population size doubles at regular intervals. The equation is:

N(t)=N0ertN(t)=N0ert

where:

• $N(t)$ = population at time $t$

• N0N0 = initial population

• $r$ = intrinsic growth rate

• $t$ = time

2. Logistic Growth

As resources become limited, growth slows and the population stabilizes at the carrying capacity (the maximum population the environment can support). The logistic growth equation is:

dNdt=rN(1−NK)dtdN=rN(1−KN)

where:

• $K$ = carrying capacity

• $N$ = current population size

In this scenario, the population grows exponentially at first (when $N$ is much less than $K$) and then slows as it approaches the carrying capacity.

When Is the Growth Rate Highest?

In exponential growth, the per capita growth rate is constant, but the absolute growth rate (number of new mice per year) increases as the population grows. However, this only continues as long as resources are unlimited.

In logistic growth, the absolute growth rate is highest at the inflection point—when the population is half the carrying capacity ($N=K/2$). After this point, the growth rate slows as the population approaches the carrying capacity.

Applying the Models to the Mouse Population

• Initial phase: 10 mice introduced. With abundant resources, the population grows exponentially.

• After 3 years: Population reaches 520 and stabilizes. This suggests that 520 is the carrying capacity ($K$).

• Growth rate: The absolute growth rate is highest when the population is half the carrying capacity ($N=K/2=260$).

Evaluating the Options

Let’s analyze each option:

1. When the mice population was first introduced.

   • Analysis: At introduction, the population is small. The per capita growth rate is high, but the absolute growth rate (number of new mice per year) is low because there are few mice to reproduce.

   • Conclusion: Not the point of highest absolute growth rate.

2. When the population size is 260.

   • Analysis: 260 is half of 520, the carrying capacity. This is the inflection point of logistic growth, where the absolute growth rate is highest.

   • Conclusion: Correct.

3. Their population growth rate, remains constant throughout.

   • Analysis: This would be true only for pure exponential growth with unlimited resources. In reality, resources become limited, and the growth rate slows as the population approaches the carrying capacity.

   • Conclusion: Incorrect.

4. When the population size reaches 520.

   • Analysis: At the carrying capacity, the growth rate is zero because the population cannot grow further.

   • Conclusion: Incorrect.

Why Is the Inflection Point Important?

The inflection point ($N=K/2$) is crucial for several reasons:

• Maximum sustainable yield: In resource management, harvesting at this point allows for the highest sustainable yield.

• Population management: Understanding this point helps in predicting population peaks and planning interventions.

• Ecological modeling: It is a key feature in models of population dynamics.

Real-World Implications

• Conservation: Knowing when a population grows fastest helps in planning reintroductions and managing invasive species.

• Agriculture: Understanding growth rates helps in managing pest populations.

• Wildlife management: It informs decisions about hunting and harvesting quotas.

Common Misconceptions

• Growth rate is highest at introduction: While the per capita rate may be high, the absolute rate is low because the population is small.

• Growth rate remains constant: This is only true for exponential growth with unlimited resources.

• Growth rate is highest at carrying capacity: At carrying capacity, the growth rate is zero.

Summary Table

Conclusion

For a mouse population introduced to an uninhabited island, the absolute growth rate is highest when the population is half the carrying capacity—in this case, when there are 260 mice. This is the inflection point of the logistic growth curve, where the population grows at its fastest rate before slowing as it approaches the carrying capacity.

Correct answer:
(2) When the population size is 260.