Direction Fields for Population Growth

Section 3.15 Direction Fields for Population Growth

Subsection 3.15.1 Recommended Tutorial

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

Differential Equations

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Subsection 3.15.2 Introduction

In this activity, you will use direction fields to predict the population dynamics for a population of rabbits. You will need to include the DETools package in your Maple worksheet to use the DEplot() command.

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Suppose that you have a population of rabbits and \(P(t)\) is the number of rabbits at time \(t\text{.}\) A basic model for the population of rabbits could be given as

\begin{equation} \frac{dP}{dt}=\alpha P - \beta P\text{,}\tag{3.13} \end{equation}

where \(\alpha\) is the birth rate and \(\beta\) is the death rate.

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However, equation (3.13) does not consider limitations due to habitat and food supply. If you wish to use a more accurate model, then you may consider the logistic growth model

\begin{equation} \frac{dP}{dt}= kP \left(1-\frac{P}{M}\right)\text{,}\tag{3.14} \end{equation}

where \(k\) is the relative growth rate and \(M\) is the carrying capacity (the maximum population that is sustainable).

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The death rate of the rabbits (due to hunting) can be added to this logistic model to obtain the differential equation

\begin{equation} \frac{dP}{dt}=kP \left(1-\frac{P}{M}\right)-bP\text{,}\tag{3.15} \end{equation}

where \(b\) is the hunting rate.

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In the following exercises, you will examine solutions to equations (3.13)–(3.15) using two different initial conditions: \(P(0)=1\) and \(P(0)=50\text{.}\)

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Exercises 3.15.3 Exercises

1.

Consider the basic population model in equation (3.13).

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(a)

Draw the direction field using \(\alpha=2\) and \(\beta=1\text{,}\) including both initial conditions given.

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Hint.

Remember to use \(P(t)\) and not just \(P\) in your differential equation.

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(b)

Draw the direction field using \(\alpha=1\) and \(\beta=2\text{,}\) including both initial conditions given.

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(c)

In a new paragraph in your worksheet, describe what you can conclude about the importance of the death to birth rate comparison.

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2.

Consider the logistic growth model in equation (3.14).

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(a)

Draw the direction field using \(k=2\) and \(M=30\text{,}\) including both of the initial conditions given.

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(b)

In a new paragraph, describe what you observe about the solutions.

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3.

Consider the logistic growth model in equation (3.15).

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(a)

Draw the direction field using \(k=2\text{,}\) \(M=30\text{,}\) and \(b=1\text{,}\) including both of the initial conditions given.

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(b)

In a new paragraph in your worksheet, describe what changed from exercise 3.15.3.2 by adding the death rate to the differential equation.

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4.

Most mammal population growth is dependent upon other species in the region, via an interconnected food web. One simple predator-prey model is the Lotka-Volterra model

\begin{align} \frac{dx}{dt} \amp = \alpha x-\beta xy\tag{3.16}\\ \frac{dy}{dt} \amp = \gamma xy-\delta y,\nonumber\notag \end{align}

where \(x(t)\) is the population of prey and \(y(t)\) is the population of predators. In this equation, the prey grow and are eaten by predators. The predators’ growth depends on eating the prey and the predators have a death rate.

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(a)

Open the DE Plots tutor, listed under Differential Equations. Select the Lotka-Volterra Model and click DEPlot. Try changing the parameters and the initial conditions to get a sense of how the prey population \(x(t)\) and predator population \(y(t)\) are connected. Then click quit to display a plot on your Maple worksheet.

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(b)

In a new paragraph, explain what the prey and predator populations do on this direction field. Notice that the prey is on the \(x\)-axis and the predator is on the \(y\)-axis of the direction field.

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