/*! 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 25 Population Model In the study of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 25

Population Model In the study of the effect of natural selection on a population, we encounter the differential equation $$ \frac{d q}{d t}=-.0001 q^{2}(1-q) $$ where \(q\) is the frequency of a gene \(a\) and the selection pressure is against the recessive genotype aa. Sketch a solution of this equation when \(q(0)\) is close to but slightly less than 1.

Short Answer

Expert verified
The frequency \( q \) will start at 0.99, decrease very slowly over time, and approach 0 asymptotically.

Step by step solution

01

- Understanding the Differential Equation

The given differential equation is \ \( \frac{d q}{d t} = -0.0001 q^{2}(1-q) \ \). Here, \( q \) is the frequency of the gene, and the rate of change of \( q \) with respect to time is influenced by \( q \) and \( (1 - q) \).
02

- Analyze the Equation Behavior

Observe the factors in the equation: \( q^{2} \) and \( (1-q) \). As \( q \) approaches 1, \( (1-q) \) approaches 0, which will make \( \frac{d q}{d t} \) negative but very small. Thus, \( q \) will decrease very slowly when it is close to 1.
03

- Initial Condition

Given that \( q(0) \) is close to but slightly less than 1, set \( q(0) = 0.99 \) for simplicity.
04

- Direction of Change

Since \( \frac{d q}{d t} \) is negative, \( q \) will decrease over time. But because \( (1-q) \) is very small (near 0), the decrease will be gradual.
05

- Sketch the Solution

On a graph with time \( t \) on the x-axis and \( q \) on the y-axis, start at \( q(0) = 0.99 \). The curve will slightly dip downwards as time increases, showing a very gradual decrease in \( q \).
06

- Long-Term Behavior

As \( t \) goes to infinity, \( q \) will continue to decrease but will approach 0 asymptotically.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Natural Selection
Natural selection is a key mechanism of evolution. It's the process by which certain traits become more common within a population due to the reproductive advantages they confer. In our exercise context, natural selection is working against the recessive genotype \( aa \). When a recessive gene negatively impacts survival or reproduction, it becomes less frequent over time. This is because individuals with the recessive genotype are less likely to survive and reproduce, thus passing fewer copies of the gene onto the next generation.
Differential Equations
Differential equations are a powerful mathematical tool used to model how quantities change over time. In the exercise, the differential equation \( \frac{d q}{d t} = -0.0001 q^{2}(1-q) \) helps us understand how the frequency of a specific gene in a population changes due to natural selection. The equation captures the rate of change of gene frequency, factoring in both the current frequency \( q \) of the gene and the likelihood of it being paired with itself or its counterpart (1-q).
Gene Frequency
Gene frequency, or allele frequency, refers to how often a particular gene variant occurs in a population. In our equation, \( q \) denotes the frequency of gene \( a \). Initially, we are considering when \( q(0) = 0.99 \), meaning almost all individuals in the population carry gene \( a \). Over time, natural selection affecting the recessive genotype leads to changes in this frequency. Because \( q \) is close to 1, the selection pressure, represented by the factor \( (1-q) \), is very minimal initially.
Selection Pressure
Selection pressure is an environmental factor that can influence which individuals are more likely to survive and reproduce. In this problem, the selection pressure negatively impacts individuals with the recessive genotype \( aa \), leading to a decrease in the frequency \( q \) of gene \( a \) over time. The term \( (1-q) \) in the differential equation signifies that as the gene becomes more prevalent (closer to 1), selection pressure on it weakens, resulting in a very slow decrease.
Long-Term Behavior
The long-term behavior of the gene frequency is influenced by the dynamics captured in the differential equation. As time progresses towards infinity, the equation indicates that \( q(t) \) will continuously decrease, though at a diminishing rate, because the term \( (1-q) \) becomes very small. As a result, the frequency \( q \) will asymptotically approach 0, meaning the gene \( a \) will become extremely rare in the population in the long run.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A person took out a loan of \(\$ 100,000\) from a bank that charges \(7.5 \%\) interest compounded continuously. What should be the annual rate of payments if the loan is to be paid in full in exactly 10 years? (Assume that the payments are made continuously throughout the year.)

Solve the given equation using an integrating factor. Take \(t>0\). $$y^{\prime}=e^{-i}(y+1)$$

Solve the following differential equations with the given initial conditions. $$\frac{d y}{d x}=\frac{\ln x}{\sqrt{x y}}, y(1)=4$$

Rod When a red-hot steel rod is plunged in a bath of water that is kept at a constant temperature \(10^{\circ} \mathrm{C},\) the temperature of the rod at time \(t, f(t),\) satisfies the differential equation $$y^{\prime}=k[10-y]$$ where \(k>0\) is a constant of proportionality. Determine \(f(t)\) if the initial temperature of the rod is \(f(0)=350^{\circ} \mathrm{C}\) and \(k=.1.\)

Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let \(f(t)\) be the fraction of manufacturers that have complied with the regulations within \(t\) months. Note that \(0 \leq f(t) \leq 1 .\) Suppose that the rate at which new companies comply with the regulations is proportional to the fraction of companies who have not yet complied, with constant of proportionality \(k=.1\). (a) Construct a differential equation satisfied by \(f(t).\) (b) Use Euler's method with \(n=3\) to estimate the fraction of companies that comply with the regulations within the first 3 months. (c) Solve the differential equation in part (a) and compute \(f(3).\) (d) Compare the answers in parts (b) and (c) and approximate the error in using Euler's method.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.