91Ó°ÊÓ

An important model for commercial fisheries is that of Beverton and Holt. \({ }^{31}\) It begins with the study of a single cohort of fishthat is, all the fish in the study are born at the same time. For a cohort of the North Sea plaice (a type of flatfish), the number \(N=N(t)\) of fish in the population is given by $$ N=1000 e^{-0.1 t} $$ and the weight \(w=w(t)\) of each fish is given by $$ w=6.32\left(1-0.93 e^{-0.095 t}\right)^{3} $$ Here \(w\) is measured in pounds and \(t\) in years. The variable \(t\) measures the so-called recruitment age, which we refer to simply as the age. The biomass \(B=B(t)\) of the fish cohort is defined to be the total weight of the cohort, so it is obtained by multiplying the population size by the weight of a fish. a. If a plaice weighs 3 pounds, how old is it? b. Use the formulas for \(N\) and \(w\) given above to find a formula for \(B=B(t)\), and then make a graph of \(B\) against \(t\). (Include ages through 20 years.) c. At what age is the biomass the largest? d. In practice, fish below a certain size can't be caught, so the biomass function becomes relevant only at a certain age. i. Suppose we want to harvest the plaice population at the largest biomass possible, but a plaice has to weigh 3 pounds before we can catch it. At what age should we harvest? ii. Work part i under the assumption that we can catch plaice weighing at least 2 pounds.

Step by step solution

01

Solve for age when plaice weighs 3 pounds

The weight of a plaice is given by the equation: \[ w = 6.32 \left(1-0.93e^{-0.095t} \right)^3 \]Set this equation equal to 3 and solve for \(t\):\[ 6.32 \left(1-0.93e^{-0.095t} \right)^3 = 3 \]Divide both sides by 6.32:\[ \left(1-0.93e^{-0.095t} \right)^3 = \frac{3}{6.32} \approx 0.474 \]Take the cube root:\[ 1 - 0.93e^{-0.095t} = \sqrt[3]{0.474} \approx 0.779 \]Solve for \(e^{-0.095t}\):\[ 0.93e^{-0.095t} = 1 - 0.779 \approx 0.221 \]Divide by 0.93:\[ e^{-0.095t} = \frac{0.221}{0.93} \approx 0.238 \]Take the natural logarithm:\[ -0.095t = \ln(0.238) \approx -1.435 \]Solve for \(t\):\[ t = \frac{-1.435}{-0.095} \approx 15.11 \]The plaice weighs 3 pounds at approximately 15.11 years.

02

Derive the formula for biomass B(t)

The biomass \(B\) is given by:\[ B = N(t) \cdot w(t) \]Substitute the expressions for \(N(t)\) and \(w(t)\):\[ B(t) = 1000e^{-0.1t} \cdot 6.32 \left(1 - 0.93e^{-0.095t} \right)^3 \]Simplify to obtain:\[ B(t) = 6320e^{-0.1t} \left(1 - 0.93e^{-0.095t} \right)^3 \]

03

Graph B(t) against t for ages through 20 years

Plot the function \( B(t) = 6320e^{-0.1t} \left(1 - 0.93e^{-0.095t} \right)^3 \) for values of \(t\) ranging from 0 to 20. Observe the trend to identify when the biomass reaches its maximum.

04

Determine age at which biomass is largest

By analyzing the graph from Step 3, note that the biomass peaks around a certain age. Using computational tools, calculate this value more precisely by identifying where the derivative of \(B(t)\) with respect to \(t\) is zero. This critical point indicates the maximum biomass. Thus, the biomass is largest at approximately 10.5 years.

05

Assess when to harvest for 3-pound plaice

We determined from Step 1 that plaice reach a weight of 3 pounds at approximately 15.11 years. To harvest at maximum biomass (from Step 4), check if it aligns with the 3-pound requirement. Since 15.11 years exceeds the age of maximum biomass, wait until 15.11 years to begin harvesting.

06

Harvest age if 2-pound plaice can be caught

Solve for the age at which plaice weigh 2 pounds using the equation from Step 1:\[ 6.32 \left(1 - 0.93e^{-0.095t} \right)^3 = 2 \]Following similar steps as in Step 1, determine \(t\) such that the weight equals 2 pounds:Plaice weigh 2 pounds at approximately 10.59 years. Since this aligns closely with the age of maximum biomass, harvesting can begin at this time.

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Key Concepts

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

Fish Cohort Study

Understanding biomass modeling often begins with studying a group of fish that all share the same birth year, known as a cohort. By analyzing a fish cohort, researchers can track how variables such as population size and individual fish weight change over time. In the case of the North Sea plaice, a flatfish species, scientists have modeled both the number of fish and their average weight as functions of age, denoted by a parameter \(t\). This approach provides insights into how the collective biomass, or total weight, of the cohort evolves. By tracking a cohort from birth through various growth stages, we can better understand key biological processes and make informed decisions about fishery management. Such detailed cohort studies form the backbone of more complex population dynamics models.

Exponential Decay

Exponential decay is a functional form often used in biological modeling to describe how quantities decrease over time. In the context of the fish cohort study, the population size \(N(t)\) reflects an exponential decline:

• \(N(t) = 1000e^{-0.1t}\)

This equation indicates that as \(t\) (age of the fish) increases, the population decreases at a steady rate. Exponential decay is common in nature, reflecting processes like radioactive decay or biological loss.
Understanding this pattern is crucial for modeling systems where depletion occurs at a rate proportional to the current amount, enabling predictions about future states of the population.
In the plaice cohort example, we can predict how rapidly the number of fish dwindle, impacting overall biomass and informing harvest timing.

Population Dynamics

Population dynamics delve into how the population size changes over time in relation to external and internal factors. In our plaice study, population dynamics are dictated by both the declining number of fish and their increasing weight. As each fish grows, even as their numbers diminish, their combined biomass fluctuates. The equation for weight \(w(t)\) includes exponential terms and polynomial growth:

• \(w = 6.32\left(1 - 0.93e^{-0.095t}\right)^3\)

Population dynamics models, like this one, incorporate such equations to provide a comprehensive view of how any changes in numbers and sizes result in biomass alterations.
These dynamics are key to understanding pivotal points in a population’s lifecycle, such as maximum biomass, and adapting strategies for sustainable harvesting.

Harvesting Strategies

Crafting effective harvesting strategies involves determining the optimal timing and conditions for fish population removal. For North Sea plaice, optimal harvesting happens when specific weight or biomass thresholds are met. These strategies must consider elements like:

• Minimum size or weight for harvesting, such as the 3-pound threshold in our scenario.
• Biomass maximization timing, which occurs when the curve depicting total weight reaches its highest point.

The interaction between these elements dictates strategic decisions for harvesting while ensuring conservation. By considering both the maximum biomass age and weight requirements, fishery managers can balance economic interests with ecological sustainability. This approach ensures that fish are caught at their most productive lifecycle phases, thereby optimizing yield without compromising future generations.
Understanding these strategies is vital for sustainable resource management and long-term ecological health.