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Chapter 2: Problem 16

In fish management it is important to know the relationship between the abundance of the spawners (also called the parent stock) and the abundance of the recruits - that is, those hatchlings surviving to maturity. \({ }^{32}\) According to the Ricker model, the number of recruits \(R\) as a function of the number of spawners \(P\) has the form $$ R=A P e^{-B P} $$ for some positive constants \(A\) and \(B\). This model describes well a phenomenon observed in some fisheries: A large spawning group can actually lead to a small group of recruits. \({ }^{33}\) In a study of the sockeye salmon, it was determined that \(A=4\) and \(B=0.7\). Here we measure \(P\) and \(R\) in thousands of salmon. a. Make a graph of \(R\) against \(P\) for the sockeye salmon. (Assume there are at most 3000 spawners.) b. Find the maximum number of salmon recruits possible. c. If the number of recruits \(R\) is greater than the number of spawners \(P\), then the difference \(R-P\) of the recruits can be removed by fishing, and next season there will once again be \(P\) spawners surviving to renew the cycle. What value of \(P\) gives the maximum value of \(R-P\), the number of fish available for removal by fishing?

Short Answer

Expert verified
The maximum number of recruits \( R \) is reached when \( P \approx 1.429 \) (thousand) leading to \( R \approx 2.858 \) (thousand), and the maximum \( R-P \) occurs when \( P \approx 0.5 \) (thousand).

Step by step solution

01

Equation Setup for R as a function of P

Given the Ricker model for recruits, we have the equation \( R = 4P e^{-0.7P} \). Here, our task is to graph this function for \( P \) ranging from 0 to 3000.
02

Graph R vs P

To graph \( R \) against \( P \), plot the equation \( R = 4P e^{-0.7P} \) over the interval \( 0 \leq P \leq 3000 \), using software tools or a calculator that supports plotting. This graph will show the relationship between spawners and recruits based on the provided model.
03

Determine Maximum R

To find the maximum recruit number \( R \), we set the derivative \( \frac{dR}{dP} \) to zero and solve. Starting with \( R = 4P e^{-0.7P} \), use the product rule to differentiate: \( \frac{dR}{dP} = 4e^{-0.7P} - 2.8Pe^{-0.7P} \). Set this derivative to zero and solve for \( P \), leading to \( 4 = 2.8P \), which simplifies to \( P = \frac{4}{2.8} \approx 1.429 \). Substitute \( P = 1.429 \) back into the original function to find \( R \approx 2.858 \).
04

Find Maximum R-P

To find the maximum \( R-P \), first note \( R-P = 4P e^{-0.7P} - P \). Differentiating this function with respect to \( P \), using a similar method, find the \( P \) that maximizes \( R-P \). By solving \( \frac{d(R-P)}{dP} = 0 \), the critical point results in \( P \approx 0.5 \). Substitute \( P = 0.5 \) into the original \( R \) function to find the maximum \( R-P \).
05

Verify Solutions

Substitute \( P \approx 0.5 \) into \( R = 4P e^{-0.7P} \) to determine the maximum difference \( R-P \). Calculate to ensure all mathematical steps are conducted accurately for both \( R \) maximization and \( R-P \) maximization.

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Key Concepts

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

Fish Population Dynamics
Understanding fish population dynamics is essential for effective fishery management. It involves studying how fish populations change over time and the factors influencing these changes. One key aspect is the interaction between the number of spawners, which are the adult fish capable of reproduction, and the number of recruits, which are the young fish that survive to adulthood. The Ricker model is a popular mathematical approach used in this context because it helps predict fish population sizes and supports sustainable fishing practices.

The model uses a function to describe the number of recruits based on the number of spawners, considering that while more spawners can lead to more recruits, there can also be a point where too many spawners result in too much competition for resources, thus reducing the number of recruits. This dynamic helps illustrate the balance required for a healthy fish population.
Differential Equations
In the study of fish populations, differential equations are vital tools for understanding change over time. They are mathematical equations that relate a function with its derivatives, essentially describing how a particular variable changes. Specifically, in the Ricker model, differential equations help us understand the rate of change of the recruit population with respect to changes in the spawner population.

By taking the derivative of the Ricker model equation, we can find key points where changes occur most significantly, such as local maxima or minima. This process involves using calculus techniques like differentiation and solving for critical points. Understanding these critical points allows fisheries scientists to predict population dynamics more accurately and make informed decisions about fishery management.
Mathematical Modeling
Mathematical modeling serves as a powerful tool for simulating real-world systems in fisheries management. With the Ricker model, a mathematical equation is used to capture the relationship between spawners and recruits. The equation incorporates factors such as growth rates and carrying capacity to create a realistic representation of fish population dynamics.

This model allows for experimentation by changing variables and observing potential outcomes. For example, by adjusting the constants in the Ricker model, fisheries managers can predict how fish populations will respond to different levels of fishing pressure or environmental changes. This predictive power is crucial for establishing policies that ensure sustainability and prevent overfishing.
Graphing Functions
Graphing functions is an essential method for visualizing relationships described by mathematical models like the Ricker model. By plotting the function for recruits against spawners, we can see how the population dynamics of fish change visually. It helps to identify patterns and critical points, such as maximum or minimum recruitment levels.

For students and researchers, graphing the function provides insights into the model's behavior. Using software tools or graphing calculators, we can plot the curve of the Ricker model equation over a specified range. Graphical representation not only aids in understanding the theoretical model but also in applying it practically for fishery management. It makes it easier to communicate findings and predictions to policymakers and stakeholders who may not have a background in mathematics.

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Most popular questions from this chapter

The length \(L\), in inches, of a certain flatfish is given by the formula $$ L=15-19 \times 0.6^{t} \text {, } $$ and its weight \(W\), in pounds, is given by the formula $$ W=\left(1-1.3 \times 0.6^{t}\right)^{3} $$ Here \(t\) is the age of the fish, in years, and both formulas are valid from the age of 1 year. a. Make a graph of the length of the fish against its age, covering ages 1 to 8 . b. To what limiting length does the fish grow? At what age does it reach \(90 \%\) of this length? c. Make a graph of the weight of the fish against its age, covering ages 1 to 8 . d. To what limiting weight does the fish grow? At what age does it reach \(90 \%\) of this weight? e. One of the graphs you made in parts a and c should have an inflection point, whereas the other is always concave down. Identify which is which, and explain in practical terms what this means. Include in your explanation the approximate location of the inflection point.

The factorial function occurs often in probability and statistics. For a non- negative integer \(n\), the factorial is denoted \(n\) ! (which is read " \(n\) factorial") and is defined as follows: First, 0! is defined to be 1. Next, if \(n\) is 1 or larger, then \(n\) ! means \(n(n-1)(n-2) \cdots 3 \times\) \(2 \times 1\). Thus \(3 !=3 \times 2 \times 1=6\). Consult the Tech nology Guide to see how to enter the factorial operation on the calculator. In some counting situations, order makes a difference. For example, if we arrange people into a line (first to last), then each different ordering is considered a different arrangement. The number of ways in which you can arrange \(n\) individuals in a line is \(n !\). a. In how many ways can you arrange 5 people in a line? b. How many people will result in more than 1000 possible arrangements for a line? c. Suppose you remember that your four-digit bank card PIN number uses \(7,5,3\), and 1 , but you can't remember in which order they come. How many guesses would you need to ensure that you got the right PIN number? d. There are 52 cards in an ordinary deck of playing cards. How many possible shufflings are there of a deck of cards?

Competition between populations: In this exercise we consider the question of competition between two populations that vie for resources but do not prey on each other. Let \(m\) be the size of the first population and \(n\) the size of the second (both measured in thousands of animals), and assume that the populations coexist eventually. Here is an example of one common model for the interaction: Per capita growth rate for \(m=5(1-m-n)\), Per capita growth rate for \(n=6(1-0.7 m-1.2 n)\). a. An isocline is formed by the points at which the per capita growth rate for \(m\) is zero. These are the solutions of the equation \(5(1-m-n)=0\). Find a formula for \(n\) in terms of \(m\) that describes this isocline. b. The points at which the per capita growth rate for \(n\) is zero form another isocline. Find a formula for \(n\) in terms of \(m\) that describes this isocline. c. At an equilibrium point the per capita growth rates for \(m\) and for \(n\) are both zero. If the populations reach such a point, they will remain there indefinitely. Use your answers to parts \(a\) and \(b\) to find the equilibrium point.

One interesting problem in the study of dinosaurs is to determine from their tracks how fast they ran. The scientist R. McNeill Alexander developed a formula giving the velocity of any running animal in terms of its stride length and the height of its hip above the ground. 7 The stride length of a dinosaur can be measured from successive prints of the same foot, and the hip height (roughly the leg length) can be estimated on the basis of the size of a footprint, so Alexander's formula gives a way of estimating from dinosaur tracks how fast the dinosaur was running. See Figure \(2.57 .\) If the velocity \(v\) is measured in meters per second, and the stride length \(s\) and hip height \(h\) are measured in meters, then Alexander's formula is $$ v=0.78 s^{1.67} h^{-1.17} . $$ (For comparison, a length of 1 meter is \(39.37\) inches, and a velocity of 1 meter per second is about \(2.2\) miles per hour.) a. First we study animals with varying stride lengths but all with a hip height of 2 meters (so \(h=2\) ). i. Find a formula for the velocity \(v\) as a function of the stride length \(s\). ii. Make a graph of \(v\) versus \(s\). Include stride lengths from 2 to 10 meters. iii. What happens to the velocity as the stride length increases? Explain your answer in practical terms. iv. Some dinosaur tracks show a stride length of 3 meters, and a scientist estimates that the hip height of the dinosaur was 2 meters. How fast was the dinosaur running? b. Now we study animals with varying hip heights but all with a stride length of 3 meters (so \(s=3\) ). i. Find a formula for the velocity \(v\) as a function of the hip height \(h\). ii. Make a graph of \(v\) versus \(h\). Include hip heights from \(0.5\) to 3 meters. iii. What happens to the velocity as the hip height increases? Explain your answer in practical terms.

One class of models for population growth rates in marine fisheries assumes that the harvest from fishing is proportional to the population size. For one such model, we have $$ G=0.3 n\left(1-\frac{n}{2}\right)-0.1 n \text {. } $$ Here \(G\) is the growth rate of the population, in millions of tons of fish per year, and \(n\) is the population size, in millions of tons of fish. a. Make a graph of \(G\) versus \(n\). Include values of \(n\) up to \(1.5\) million tons. b. Use functional notation to express the growth rate if the population size is \(0.24\) million tons, and then calculate that value. c. Calculate \(G(1.42)\) and explain in practical terms what your answer means. d. At what population size is the growth rate the largest?
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