Problem 7 - Campbell Biology

Question

Level \(2 :\) Application/Analysis 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

The correct answer is (C). Population growth is zero when \(N = K\).

Step 1: Understanding the Logistic Growth Equation

The logistic growth equation is given by: \[\frac{dN}{dt} = rN \frac{(K-N)}{K}\] Here, \(N\) is the population size, \(r\) is the intrinsic rate of increase, and \(K\) is the carrying capacity.

Step 2: Identifying the Zero Population Growth Point

Set the right-hand side of the equation to zero to find when population growth is zero:\[rN \frac{(K-N)}{K} = 0\]This equals zero when either \(N = 0\) or \(N = K\). Hence, population growth is zero when \(N = K\). This eliminates choice (A), (B), and (D).

Step 3: Verify Population Growth at Carrying Capacity

When \(N\) equals \(K\), then \[ \frac{(K-N)}{K} = \frac{(K-K)}{K} = 0 \]Hence, \(\frac{dN}{dt} = 0\).Therefore, option (C) is correct.

Key Concepts

Population Growth

When discussing population growth, we refer to how the size of a population changes over time. In the context of the logistic growth equation, population growth is represented by \(\frac{dN}{dt}\).

• \(N\) is the population size.
• \(r\) is the intrinsic rate of increase, which we'll delve into later.
• \(K\) is the carrying capacity, another crucial factor we'll discuss.

The logistic growth equation shows that early on, when the population size \(N\) is much smaller than the carrying capacity \(K\), the population grows rapidly. This is because there are plenty of resources available, and there is little competition among individuals.
However, as \(N\) approaches \(K\), resources become scarce, and the growth rate slows. Hence, when \(N\) is close to \(K\), population growth tapers off. This transition from rapid growth to stable growth reflects how populations self-regulate depending on resource availability and environmental constraints.
The key takeaway: Population growth isn't linear; it's a dynamic process affected by both the number of individuals and the environment they live in.

Carrying Capacity

Carrying capacity, denoted by \(K\) in the logistic growth equation, is the maximum population size that an environment can sustain indefinitely. This number depends on the resources available in the environment, such as food, habitat space, and other necessities.
When the population size \(N\) is below the carrying capacity, the environment can support more individuals, and the population grows. But as \(N\) approaches \(K\), resources become limited, and the growth rate slows down until it stops altogether when the population reaches \(K\).
Examples of factors that impact carrying capacity include:

• Food availability
• Habitat space
• Water supply
• Predation and disease

It's important to remember that carrying capacity isn't a fixed number; it can change over time due to environmental conditions and human activities. If the carrying capacity is exceeded, it can result in resource depletion and population decline.

Intrinsic Rate of Increase

The intrinsic rate of increase, represented as \(r\), is a measure of how fast a population grows when it is not limited by resources. It represents the potential growth rate under ideal conditions where resources are unlimited.
The value of \(r\) depends on several factors, including birth rates, death rates, and the life span of the organism. A higher \(r\) means that the population has a greater potential to grow quickly.
When \(N\) is much smaller than \(K\), the population can grow near the intrinsic rate of increase since there is little competition for resources. However, as the population size \(N\) increases and approaches the carrying capacity \(K\), the effective growth rate decreases due to limited resources.
Key points to remember about \(r\):

• It indicates the potential growth rate under optimal conditions.
• Higher \(r\) values mean faster potential growth.
• As \(N\) approaches \(K\), the growth rate slows, regardless of the value of \(r\).

Understanding \(r\) helps in predicting how quickly a population can grow under ideal conditions and how it will adjust as resource limitations kick in.