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Chapter 5: Problem 15

Studies to fit a logistic model to the Eastern Pacific yellowfin tuna population have yielded $$ N=\frac{148}{1+3.6 e^{-2.61 t}}, $$ where \(t\) is measured in years and \(N\) is measured in thousands of tons of fish. \({ }^{7}\) a. What is the \(r\) value for the Eastern Pacific yellowfin tuna? b. What is the carrying capacity \(K\) for the Eastern Pacific yellowfin tuna? c. What is the optimum yield level? d. Use your calculator to graph \(N\) against \(t\). e. At what time was the population growing the most rapidly?

Short Answer

Expert verified
a) Growth rate \(r = 2.61\). b) Carrying capacity \(K = 148\). c) Optimum yield: 74,000 tons. d) Sigmoidal graph. e) Max growth at \(t \approx 0.405\) years.

Step by step solution

01

Identify logistic growth model parameters

The given logistic model is \[ N = \frac{148}{1 + 3.6 e^{-2.61 t}} \]where the logistic equation takes the form \[ N = \frac{K}{1 + A e^{-rt}} \]By comparing, we identify the parameters:- \( K = 148 \) (Carrying capacity)- \( A = 3.6 \)- \( r = 2.61 \) (Growth rate)
02

Extract growth rate r

From the comparison of the given logistic equation with the standard form, we note that the growth rate \( r \) is 2.61 per year.
03

Determine carrying capacity K

The carrying capacity \( K \) is the maximum population size that the environment can sustain indefinitely. From the logistic model, we have \( K = 148 \) (in thousands of tons of fish).
04

Calculate optimum yield level

The optimum yield level is generally half of the carrying capacity, which represents the population size at which the growth rate is maximized.\[ OYL = \frac{K}{2} = \frac{148}{2} = 74 \text{ (thousands of tons)} \]
05

Graph the logistic function

Plot the function \( N = \frac{148}{1 + 3.6 e^{-2.61 t}} \) using a graphing calculator or software. The graph typically shows a sigmoidal curve starting near zero, rising sharply, and then leveling off as it approaches the carrying capacity of 148.
06

Determine time of maximum growth rate

In a logistic model, the population growth is fastest when the population reaches half of the carrying capacity. So, set\[ N = \frac{K}{2} = 74 \]Set this in the equation to solve for \( t \):\[ 74 = \frac{148}{1 + 3.6 e^{-2.61 t}} \]Solving this, we find \( t \approx 0.405 \) years.

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Key Concepts

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

Growth Rate
The growth rate in a logistic growth model represents how quickly a population increases in size over time. Specifically, it denotes the rate at which a population grows when conditions are ideal and resources are unlimited. In the context of the Eastern Pacific yellowfin tuna, the growth rate is given as 2.61 per year. This means that under optimal conditions, the population of yellowfin tuna can increase at this rate annually.
The logistic equation is formulated as:
  • \[ N = \frac{K}{1 + A e^{-rt}} \]
Where:
  • \( K \) is the carrying capacity
  • \( A \) is a constant related to the initial population size
  • \( r \) is the growth rate
  • \( t \) is the time
As time progresses, the exponential term \( e^{-rt} \) becomes smaller, causing the population growth to slow down when close to carrying capacity. Understanding this helps you predict how quickly a population can grow at various stages.
Carrying Capacity
Carrying capacity is a fundamental concept in population dynamics. It refers to the maximum number of individuals that a particular environment can support sustainably without undergoing degradation over time. In the logistic growth model for the Eastern Pacific yellowfin tuna, the carrying capacity is set at 148,000 tons of fish. This number is crucial because it represents the limit beyond which the tuna population cannot grow due to resource constraints like food, space, or other environmental factors. As the population nears this threshold, the available resources per individual decrease, slowing down growth rates. This is why the logistic growth model exhibits a sigmoidal or "S-shaped" curve: fast growth occurs when the population size is low relative to the carrying capacity, while growth slows as the population nears the carrying capacity.
Understanding carrying capacity helps in managing populations to maintain ecological balance and sustainability.
Optimum Yield Level
The optimum yield level (OYL) is an important concept in fisheries management and sustainable harvesting. It is the population size at which the growth rate is at its maximum, allowing for the most efficient use of the population without risking depletion. For the yellowfin tuna population, the OYL is calculated as half of the carrying capacity, which is 74,000 tons in this case. This level is significant because it is the point where the population can replenish itself most effectively. By maintaining the population at or near this level, one can ensure that the tuna population remains stable while still allowing for harvesting. This is a key strategy in achieving sustainability and ensuring long-term benefits from fishery resources.
Population Dynamics
Population dynamics involves studying the changes in population sizes and composition over time, including births, deaths, and migration patterns. The logistic growth model provides a valuable framework for understanding these dynamics, especially when resources are limited.
In the logistic equation for the yellowfin tuna population:
  • Initially, population growth is exponential because resources are abundant relative to the population size.
  • As the population size increases, competition for resources becomes more intense, leading to a slowdown in growth.
  • Finally, growth approaches zero as the population size nears its carrying capacity.
This knowledge helps in making informed decisions about conservation, management, and development initiatives, ensuring that populations maintain a healthy balance.

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