Problem 7 - Campbell Biology

Question

According to the logistic growth equation $$ \frac{d N}{d t}=r N \frac{(K-N)}{K} $$ (A) the number of individuals added per unit time is greatest when \(N\) is close to zero. (B) the per capita population growth rate increases as \(N\) approaches \(K\) (C) population growth is zero when N equals \(K\). (D) the population grows exponentially when \(K\) is small.

Step-by-Step Solution

Answer

Option (C) is correct: population growth is zero when \(N\) equals \(K\).

Step 1: Understand the Logistic Growth Equation

The logistic growth equation is \ \[ \frac{d N}{d t}=r N \frac{(K-N)}{K} \]. Here, \(N\) represents the population size, \(r\) is the intrinsic growth rate, and \(K\) is the carrying capacity of the environment.

Step 2: Identify Population Growth When N Equals K

When \(N = K\), the term \((K - N)\) becomes zero. Therefore, the equation becomes: \ \[ \frac{d N}{d t}=r N \frac{(K-K)}{K}=r N \cdot 0 = 0 \] \ So, the population growth is zero when \(N = K\).

Step 3: Evaluate Other Given Options

(A) Incorrect: The number of individuals added per unit time is greatest not when \(N\) is close to zero but when \(N = \frac{K}{2}\). \ (B) Incorrect: The per capita population growth rate doesn’t increase as \(N\) approaches \(K\); it decreases because the term \( \frac{(K-N)}{K} \) approaches zero. \ (D) Incorrect: Exponential growth doesn’t depend on \(K\) being small; the logistic equation inherently takes into account the carrying capacity, \(K\).

Step 4: Conclusion

Based on the logistic growth equation, only option (C) is correct: the population growth is zero when \(N\) equals \(K\).

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population over time. In the context of the logistic growth equation, this growth is regulated by both the intrinsic growth rate and carrying capacity. The intrinsic growth rate, denoted as \( r \), is the rate at which the population can grow when there are no limiting factors.
During the initial phase when the population size \( N \) is small, the growth is approximately exponential because resources are abundant and competition is minimal. This can be seen in the term \[ \frac{d N}{d t}=r N \frac{(K-N)}{K} \], where \( N \) is much smaller than \( K \), making the fraction \( \frac{(K-N)}{K} \) close to 1.
However, as the population grows and \( N \) approaches \( K \), the growth rate starts to slow down. This is because resources become limited and competition increases. Ultimately, the growth rate reaches zero when the population size \( N \) equals the carrying capacity \( K \).

Carrying Capacity

Carrying capacity \( (K) \) is the maximum number of individuals in a population that an environment can support sustainably. This concept is key to understanding the logistic growth model. The environment has finite resources, and it can only support a certain number of individuals before those resources become too limited.
In the logistic growth equation \[ \frac{d N}{d t}=r N \frac{(K-N)}{K} \], the carrying capacity \( K \) acts as a regulatory factor that limits the population growth when \( N \) is large. As the population size \( N \) nears the carrying capacity \( K \), the growth rate \( \frac{d N}{d t} \) reduces because the term \( \frac{(K-N)}{K} \) gets smaller.
When the population size \( N \) equals \( K \), the equation simplifies to \( \frac{d N}{d t}=0 \), meaning the population growth stops. This is the balance point where the number of births equals the number of deaths.

Intrinsic Growth Rate

The intrinsic growth rate, often denoted as \( r \), is a measure of the inherent capacity of a population to grow under ideal conditions. It's a constant that represents the rate of increase of the population per individual per unit time, without the influence of environmental limitations.
In the logistic growth equation \[ \frac{d N}{d t}=r N \frac{(K-N)}{K} \], the intrinsic growth rate \( r \) determines the speed at which the population grows when resources are abundant.
In the context of population dynamics, if the value of \( r \) is high, the population has a greater potential for rapid growth, assuming other conditions remain ideal. Conversely, a lower \( r \) means slower potential growth. However, as the population size \( N \) approaches the carrying capacity \( K \), the impact of \( r \) diminishes as the term \( \frac{(K-N)}{K} \) approaches zero.

Population Dynamics

Population dynamics refer to the variations in population size and composition over time and the processes that cause these changes. The logistic growth equation is a key model in understanding these dynamics. This model accounts for the natural limits set by the environment, which is critical for predicting future population trends.
Several factors influence population dynamics, including birth rates, death rates, immigration, and emigration. The logistic growth equation \[ \frac{d N}{d t}=r N \frac{(K-N)}{K} \] illustrates how these factors interact. When \( N \) is small, the growth rate is nearly exponential, showing rapid population increase. However, as \( N \) grows, environmental resistance (limited resources, competition) slows the growth rate.
Ultimately, when the population size \( N \) equals the carrying capacity \( K \), population growth halts, showing a stable population size. This stability suggests a balance between birth and death rates, affected by the carrying capacity and the intrinsic growth rate. Understanding these dynamics is crucial for managing wildlife populations, conducting conservation efforts, and studying human population trends.