/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 The spruce budworm is an enemy o... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 61

The spruce budworm is an enemy of the balsam fir tree. In one model of the interaction between these organisms, possible long-term populations of the budworm are solutions of the equation \(r(1-x / k)=x /\left(1+x^{2}\right),\) for positive constants \(r\) and \(k\) (see Murray's Mathematical Biology). Find all positive solutions of the equation with \(r=0.5\) and \(k=7\)

Short Answer

Expert verified
The positive solution for the equation with \(r=0.5\) and \(k=7\) is \(x = (\sqrt[3]{14})\)

Step by step solution

01

Understand the given model and substitute the values, \(r=0.5\) and \(k=7\)

The given model is \(r(1-x / k)=x /\left(1+x^{2}\right)\), we substitute 0.5 for \(r\) and 7 for \(k\) to get \[0.5(1-x / 7)=x /\left(1+x^{2}\right)\] or \[1 - x/14 = x /\left(1+x^{2}\right)\]
02

Rearrange the equation

By clearing the denominator and rearranging, the equation can be simplified to \[x (1 - x^2) = 14 - x\] or \[x^3 - x + x -14 = 0\] simplifying further gives: \[x^3 - 14 = 0\]
03

Solve the equation

Given the obtained cubic equation, \(x^3 -14 = 0\), we can solve it for \(x\). When the equation is set to zero and solved for \(x\), we get \(x = (\sqrt[3]{14})\)
04

Ensure the solution is positive

We need to make sure the obtained solution is positive since the populations of the budworm must be a positive number. Since the cubed root of 14 is positive, our solution is valid.

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Key Concepts

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

The Spruce Budworm Model
Understanding the spruce budworm model is essential for analyzing the complex interactions within forest ecosystems. The model represents the population dynamics of the spruce budworm, a pest that affects balsam fir trees, and incorporates factors like birth rates, death rates, and carrying capacity. In our exercise, the model takes the form of mathematical equations that help predict the long-term population of the budworm.

To unravel the equation, we need to know the parameters: 'r' which can be seen as a growth rate, and 'k', which represents the carrying capacity of the environment. A critical aspect of this model is the assumption that the population changes continuously and that the rate of change is affected by the population size itself. This leads to nonlinear behaviors, including the potential for the population to exhibit sudden increases or collapses, depending on the parameter values.

The given equation in the exercise also suggests an inverse density-dependence, where the effects of competition among budworms become more significant as their population increases. This is reflected in the term \(1+x^2\) in the denominator, implying that as the population size (\

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Most popular questions from this chapter

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