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A catfish farm: Catfish in a commercial pond can be expected to exhibit logistic population growth. Consider a pond with a carrying capacity of \(K=\) 4000 catfish. Take the \(r\) value for catfish in this pond to be \(r=0.06\). a. Write the equation of change for logistic growth of the catfish population. (Hint: If you have difficulty here, refer to Example 6.10.) b. Make a graph of \(\frac{d N}{d t}\) versus \(N\). c. For what values of \(N\) would the catfish population be expected to increase? d. For what values of \(N\) would the catfish population be expected to decrease? e. Recall from Section \(5.1\) that the maximum sustainable yield model says that a renewable resource should be maintained at a level where its growth rate is at a maximum, since this allows the population to replenish itself quickly. According to this model, at what level should the catfish population be maintained?

Step by step solution

01

Write the Equation of Change

The logistic growth model is given by the equation \( \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \), where \( N \) is the population size, \( r \) is the growth rate, and \( K \) is the carrying capacity. For this particular pond, the equation becomes \( \frac{dN}{dt} = 0.06N \left(1 - \frac{N}{4000}\right) \).

02

Graph the Rate of Change

To graph \( \frac{dN}{dt} \) versus \( N \), plot the equation \( \frac{dN}{dt} = 0.06N \left(1 - \frac{N}{4000}\right) \). This equation will produce a parabolic curve that starts at \( N = 0 \), rises to a peak, and then returns to zero when \( N = 4000 \). The peak of the curve is at \( N = 2000 \).

03

Determine Population Increase Range

The population increases when \( \frac{dN}{dt} > 0 \). Solve \( 0.06N \left(1 - \frac{N}{4000}\right) > 0 \). Simplifying, the inequality \( 1 - \frac{N}{4000} > 0 \) yields \( N < 4000 \). Thus, the population increases when \( 0 < N < 4000 \).

04

Determine Population Decrease Range

The population decreases when \( \frac{dN}{dt} < 0 \). However, since \( N \) cannot exceed the carrying capacity, the population does not naturally decrease below this level in isolation. Mathematically \( 0 < N < 4000 \) is where growth occurs, \( N > 4000 \) would imply more than the carrying capacity, but in real terms, such a population does not sustain without other factors (like resource insufficiency). Thus, theoretically, decrease happens when external forces push \( N > 4000 \).

05

Calculate Maximum Sustainable Yield Level

The maximum growth rate occurs at \( N = \frac{K}{2} \). For this scenario, \( N = \frac{4000}{2} = 2000 \). Therefore, to maintain the population at the maximum sustainable yield, keep it around \( N = 2000 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carrying Capacity

In ecological terms, the carrying capacity refers to the maximum population size of a species that an environment can sustain indefinitely without being degraded. In the context of a catfish pond, this would mean the number of catfish it can support without hurting future productivity. Here, the pond's carrying capacity for catfish is 4000. It acts as a ceiling that limits growth. As the population grows and approaches 4000, resources become limited, slowing down population growth.

Understanding and setting carrying capacity is crucial in population dynamics because it keeps the ecosystem in balance. Overstepping this number often results in negative consequences such as resource depletion, increased disease spread, and habitat degradation, which can lead to a drop in population. Staying within this limit ensures long-term sustainability.

Growth Rate

The growth rate in a logistic growth model, such as for our catfish population, is represented by the variable \( r \). This rate reflects how quickly the population grows over time under ideal conditions. For the catfish, we have a growth rate \( r = 0.06 \). This value shows the intrinsic potential for the population to increase.

Growth rate plays a key role in predicting how fast the population will approach its carrying capacity. However, unlike exponential growth, logistic growth factors in resource limitations, causing the rate to adjust as the population size nears carrying capacity. In real-world scenarios, managing the growth rate is essential to ensure a balance between the available resources and the population's needs.

Sustainable Yield

The concept of sustainable yield refers to the optimal population level where the growth rate is maximized, allowing for continual renewal of the resource without depletion. For the catfish pond, the maximum sustainable yield is achieved when the population is maintained at \( N = \frac{K}{2} \), which is 2000 catfish.

This level allows the largest output of catfish while ensuring that the pond remains productive over time. By keeping the population at this midpoint, the fish reproduce efficiently, promoting both harvesting and regeneration simultaneously.

• Ensure that harvesting does not exceed the renewal rate.
• Avoid depleting resources faster than they can replenish.
• Maintain ecological balance for continuous productivity.

Population Dynamics

Population dynamics explores how and why populations change over time, taking into account birth rates, death rates, and migration, alongside factors like carrying capacity and environment. For the catfish pond, these dynamics are captured through the logistic growth model.

This model signifies that populations tend to increase rapidly at first when resources are ample, then slow as they approach the carrying capacity, stabilizing eventually. It presents a realistic portrayal of how populations grow in most natural environments.

• Initially, growth is exponential due to abundant resources.
• Growth slows near carrying capacity due to increasing competition for resources.
• Stabilization occurs as the birth and death rates equalize at carrying capacity.

Understanding population dynamics aids in making informed decisions regarding wildlife management and conservation efforts.