Understanding Maximum Sustainable Yield in Population Dynamics

The question asks about the maximum sustainable yield (MSY) for a fish population following a logistic growth model. The logistic growth equation describes how a population's growth rate changes as it approaches its carrying capacity, where resources become limited.

The given equation for the population growth rate (\(dN/dt\)) is:

\( \frac{dN}{dt} = 0.1 N - 0.001 N^2 \)

The standard form of the logistic growth equation is:

\( \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \)

Where:

• \(r\) is the intrinsic rate of increase (maximum potential growth rate per individual).
• \(K\) is the carrying capacity (the maximum population size the environment can sustain).

To find \(r\) and \(K\) from the given equation, we need to rewrite it in the standard form. We can factor out \(N\) from the given equation:

\( \frac{dN}{dt} = N (0.1 - 0.001 N) \)

Now, to get the term \( \left(1 - \frac{N}{K}\right) \), we factor out the constant \(0.1\) from the parenthesis:

\( \frac{dN}{dt} = 0.1 N \left(1 - \frac{0.001 N}{0.1}\right) \)

Simplify the term inside the parenthesis:

\( \frac{dN}{dt} = 0.1 N \left(1 - \frac{N}{100}\right) \)

Comparing this to the standard logistic equation \( \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \), we can identify the parameters:

\(r = 0.1\)

\(K = 100\)

The carrying capacity (\(K\)) for this fish population is 100.

Calculating Maximum Sustainable Yield (MSY) Population Size

Maximum sustainable yield (MSY) is the largest yield (or catch) that can be taken from a species' stock over an indefinite period. In the logistic growth model, the population growth rate (\(dN/dt\)) is maximum when the population size (\(N\)) is exactly half of the carrying capacity (\(K\)). This is the point where the population is growing at its fastest rate, allowing for the largest sustainable harvest.

The population size at which MSY occurs is given by:

\( N_{MSY} = \frac{K}{2} \)

Using the carrying capacity we found (\(K = 100\)):

\( N_{MSY} = \frac{100}{2} \)

\( N_{MSY} = 50 \)

Thus, the maximum sustainable yield occurs when the fish population size is 50.

The question asks "What is the maximum sustainable yield?". Given the options are population sizes, it is most likely asking for the population size at which the maximum sustainable yield is achieved.

Summary of Parameters

The calculation shows that the population size corresponding to the maximum sustainable yield is 50.

Revision Table: Key Concepts

Additional Information: Significance of MSY

Understanding the maximum sustainable yield is crucial in resource management, particularly in fisheries and wildlife management. Harvesting at the MSY level aims to maximize the yield while ensuring the population can replenish itself and remain sustainable in the long term. Harvesting above the MSY can lead to population decline and potential collapse, while harvesting below it means not fully utilizing the potential resource. However, calculating and implementing MSY in real-world scenarios is complex and often debated due to environmental variability, species interactions, and uncertainties in data.

The population size at which MSY occurs (\(K/2\)) corresponds to the point of inflection on the logistic growth curve, where the rate of increase transitions from accelerating to decelerating.