Question

What is meant by the carrying capacity of a population? Describe where it appears, and why, in: (i) S-shaped population growth; (ii) the logistic equation; and (iii) dome-shaped net recruitment curves.

Short answer

Carrying capacity is the maximum sustainable population size and appears as a plateau in S-shaped growth, as \(K\) in the logistic equation, and as a peak in dome-shaped recruitment curves.

Step-by-step solution

Understanding Carrying Capacity

The carrying capacity of a population is the maximum number of individuals that an environment can sustainably support without being degraded over time. It depends on factors such as availability of food, habitat space, and other environmental conditions that provide support for the population.

Carrying Capacity in S-shaped Population Growth

In an S-shaped, or sigmoid, population growth model, the carrying capacity appears as the plateau part of the curve. Initially, the population grows exponentially, but as it reaches the carrying capacity, the growth rate slows down and stabilizes due to resource limitations. This is a typical pattern for biological populations that grow and stabilize over time.

Carrying Capacity in the Logistic Equation

The logistic equation \[ \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \]describes how a population grows in a limited environment. Here, \(N\) is the population size, \(r\) is the intrinsic rate of increase, and \(K\) is the carrying capacity. The term \(1 - \frac{N}{K}\) illustrates how the growth rate decreases as the population nears the carrying capacity, \(K\).

Carrying Capacity in Dome-Shaped Net Recruitment Curves

In dome-shaped net recruitment curves, carrying capacity is observed as the peak of the recruitment curve. Initially, recruitment increases with population size, but after a peak point, it starts to decline as crowding and limited resources decrease the number of individuals that can successfully be added to the population. This illustrates how overpopulation can reduce the net recruitment rate.

Key concepts

S-shaped Population Growth

S-shaped population growth, also known as sigmoid growth, describes how populations grow in an environment where resources are initially abundant but become limited over time. This type of growth is depicted by an S-shaped curve. The curve begins with a period of slow growth, followed by rapid exponential growth, and eventually steadying out as the population reaches its carrying capacity.

The carrying capacity is the point on the curve where the population size stabilizes, as the available resources can no longer support further growth. Resources such as food, space, and water play significant roles in determining this capacity. When populations reach this point, the environment can no longer sustain additional individuals without deterioration.

This model is representative of real-life biological populations, especially when they initially grow unrestrained but eventually face resource limitations. Understanding this growth pattern is crucial for ecological management and conservation efforts.

Logistic Equation

The logistic equation is a mathematical model that describes how populations grow in a restricted environment. It is given by: \[ \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \]This equation is vital for understanding population dynamics, as it captures how growth slows as a population nears its carrying capacity, denoted by \( K \).

Here's a breakdown of the elements:

• \( N \): The population size at any given time.
• \( r \): The intrinsic rate of increase, which is the potential for growth in unrestricted conditions.
• \( K \): The carrying capacity, which limits growth as resources become scarce.
• \( 1 - \frac{N}{K} \): The fraction representing how far the population is from reaching its carrying capacity.

As \( N \) approaches \( K \), the growth rate approaches zero, indicating that the population is stabilizing. The logistic equation is a powerful tool used in ecology to predict changes in population sizes and to assess the impacts of variables like conservation measures and resource management on ecosystems.

Net Recruitment Curves

Net recruitment curves provide insights into how populations change over time by focusing on the number of new individuals that join the population. These curves can take on different shapes, with the dome-shaped curve offering significant understanding in relation to carrying capacity.

In a dome-shaped curve, net recruitment - which is the number of new individuals minus deaths - increases as the population size grows, reaching a peak before declining again. This peak represents the optimal recruitment zone, where conditions are ideal for adding new members to the population.

However, as the population continues to grow beyond this peak, overpopulation begins to occur, leading to increased competition and insufficient resources. Consequently, the net recruitment declines because new individuals struggle to survive and thrive.

Understanding these curves is essential for managing population sizes effectively, ensuring that environments can continue to sustain healthy populations without the adverse effects of overpopulation.