Problem 5 - Campbell Biology Concepts & Connections

Question

The term \((K-N) / K\) a. is the carrying capacity for a population. b. is greatest when \(K\) is very large. c. is zero when population size equals carrying capacity. d. increases in value as \(N\) approaches \(K\).

Step-by-Step Solution

Answer

c. The expression is zero when population size equals carrying capacity.

Step 1: Understand the Given Term

The term given is \( \frac{K-N}{K} \). Here, \(K\) represents the carrying capacity of a population, and \(N\) represents the population size.

Step 2: Analyze the Expression at Different Values of N

Consider the term \( \frac{K-N}{K} \) when \(N = K\). Substitute \(N\) with \(K\): \( \frac{K-K}{K} = \frac{0}{K} = 0 \). Hence, the expression is zero when population size equals carrying capacity.

Step 3: Consider Other Statements

Analyze other answer choices: \( \frac{K-N}{K} \) is not always the carrying capacity (eliminates a). The value is not greatest when \(K\) is very large, but when \(N\) is very small or zero (eliminates b). \( \frac{K-N}{K} \) decreases as \(N\) approaches \(K\), not increases (eliminates d).

Step 4: Confirm the Correct Answer

The accurate statement is that \( \frac{K-N}{K} = 0 \) when \(N = K\). Hence, the correct answer is c.

Key Concepts

carrying capacity

Carrying capacity, represented by the symbol \(K\), is a fundamental concept in population ecology. It refers to the maximum number of individuals in a species that an environment can support indefinitely without degrading the environment.
Various factors influence carrying capacity, including:

• Availability of resources like food, water, and shelter
• Environmental conditions such as climate and terrain
• Interactions with other species, including competition and predation

When a population exceeds its carrying capacity, resources become limited, leading to increased competition and often resulting in a population decrease.

population size

Population size, denoted as \(N\), represents the number of individuals in a given population. It is a crucial metric for understanding the dynamics of a population over time.
Several factors can influence population size:

• Birth rates and death rates
• Immigration and emigration
• Environmental changes and availability of resources

Monitoring population size helps ecologists understand the health and viability of species in an environment. When population size \(N\) equals the carrying capacity \(K\), the environment is supporting the maximum number of individuals it can sustain.

logistic growth model

The logistic growth model describes how populations grow in an environment with limited resources. It is represented by the equation: \[\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)\]
In this model:

• \(dN/dt\) is the change in population size over time
• \(r\) is the intrinsic growth rate
• \(N\) is the current population size
• \(K\) is the carrying capacity

The logistic growth model shows that population growth slows as the population size \(N\) approaches the carrying capacity \(K\). When the population size is small, growth is rapid. As it nears \(K\), growth decelerates and eventually stabilizes.

population ecology

Population ecology explores the factors that influence the size and structure of populations over time. Key study areas in population ecology include:

• Population density and dispersion patterns
• Age structure and sex ratio
• Birth and death rates
• Life history strategies

Understanding population ecology helps in conservation efforts, managing wildlife resources, and studying environmental impacts on species. For instance, ecologists use population models to predict how a species might respond to changes in habitat, climate, or human activities, helping to ensure the sustainability of ecosystems.