Assignment A: Instructions.

Working together

You may do this assignment in groups of up to three people. Only one assignment per group is to be

handed in, with all your names included. The assignment must be your own group’s work.

Marking

This assignment is worth 3% of your final grade and will be marked out of 6. Two of these marks

are for presentation, that is, your introduction and conclusion, and your ability to communicate your

ideas clearly and concisely. Marks may be deducted if your layout and grammar is poor. The other 4

marks for the body of your assignment will be awarded on the following basis:

• 4: Complete.

• 3: Good but some elements missing.

• 2: OK, got the main points but quite a bit missing.

• 1: Got some points but too many missing.

Presentation

Read the General Instructions in Learn on writing assignments for guidelines and further details on

presentation. Also look at the sample assignment, The Hyperbolic Functions.

• You may present your assignment as a Word document, or you may handwrite your assignment.

• Use a computational package to produce the graphs even if you handwrite.

• You do not have to present all steps when solving a DE, a few key steps will suffice.

• Include an introduction and a conclusion. A few sentences for each is enough.

• Your assignment should be about 3 pages long, and no more than 4. (Here a ‘page’ means a

single ‘side’ of A4.)

Graphs

• Graphs must be large enough to be clear to the reader. Use a suitable font.

• Axes must be labelled with the name of the variable and the units (if applicable).

• If there is more than one graph on a figure, there must be a legend.

• To transfer a Maple graph, click on the figure, then copy and paste it into the document.

• If you use a screenshot to transfer graphs from Maple GeoGebra Desmos etc. to your assignment

make sure they are clear. Grey backgrounds are not acceptable.

Hand in Details

The deadline is 6pm Friday 31st July.

• Hand in your assignment as a pdf file.

• Local students: hand in at the start of a Thursday class, or upload the pdf file into Learn.

• Distance students: upload the pdf file into Learn.

1

Assignment A: First-order Differential Equations and their Applications.

In the 1830’s the Belgian mathematician P.F Verhulst suggested that population growth depends not

only on the current population but also on the maximum population the environment can support,

the carrying capacity. As the population approaches the carrying capacity, growth slows down. This

idea is modeled by the logistic equation:

dP

dt = kP �1 - PL �

where k and L are constants. L is the carrying capacity, and k defines the growth rate.

The logistic equation assumes that the population growth rate responds immediately to population

changes. For small, quickly reproducing species such as groups of protozoa feeding on bacteria this

might be a fair approximation.

Exercise One: Modeling a Whale Population using the Logistic Equation

Humpback whales had been hunted at an increasing rate in the early 20th century until populations

crashed in the early ’60s to the point that there was probably only about 500 animals left. But as

scientists began to investigate humpback whale recovery in Western Australia back in 1990, they found

that the population was growing at an astonishing rate. Although whales are definitely not a \small,

quickly reproducing species" we can still gain some useful insights about how a population changes

using the logistic model.

Let us suppose that the rate of growth of a certain population of humpback whales can be modeled

by the logistic equation with k = 0:1 and L = 40, where P is measured in units of 1,000 whales and

time is measured in t years from when the population of whales is first studied.

(a) Use separation of variables and partial fractions to find the general solution to the differential

equation.

Key steps: the DE in its separable form; the form of the partial fraction decomposition; the

value of the constants in the partial fraction decomposition; the general solution in terms of

logs; the general solution written explicity as P = P (t) and in terms of negative exponentials.

(b) Find the particular solutions corresponding to the following initial estimates of the size of this

population of whales.

Key steps: the value of the constants of integration; the solutions writen explicitly as P = P (t).

(i) 200 whales (ii) 10,000 whales (iii) 50,000 whales

You might like to tabulate your results.

(c) Sketch the particular solutions corresponding to each initial condition on the direction field of

the differential equation over a suitable domain, say 0 6 t 6 100 years.

(d) Use your graphs to interpret each solution in terms of the population over time. Mention aspects

like the long-term behaviour of the population, factors that may have caused the population

dynamics, whether or not the solutions are physically realistic, and so on. Two or three sentences

for each case will suffice.

(e) There are two constant solutions to this differential equation. What are they?

Explain why these constant solutions makes sense in terms of the model.

2

Exercise Two: Overshoot

For some organisms it is more realistic to account for the time difference between a new member

being born and the point at which they can themselves reproduce - time to reach adulthood and the

gestation period. To account for this, we introduce a lag term, τ. One of the N terms is evaluated at

the current time point t, while the other is evaluated at some earlier point, t - τ:

dN

dt = kN(t) �1 - N(tL- τ)� :

(Interestingly, this model has been shown to accurately predict lemming populations in the Arctic!)

Now when a population surpasses its carrying capacity it enters a condition known as overshoot.

Because carrying capacity is defined as the maximum population that an environment can maintain

indefinitely, overshoot must by definition be temporary. To explore this phenomena we fix our initial

condition N(0) = 1, L = 5, and τ = 1, and see what happens when we vary our growth constant k.

The particular solutions corresponding to the initial conditions k = 1, k = 1:7 and k = 2:3 are shown

below.

Interpret the solution for each value of k in terms of the population.

What factors may have caused such population dynamics (food sources etc.)?

Are the solutions physically realistic?

Why or why not?

These are general instructions. Check the specific

instructions for your assignment carefully.

Assignments: General Instructions

• Start at least two weeks before the assignment is due.

• Check the marking schedule.

• Check the instructions carefully for specific details such as page limits.

• Check when and how to submit your assignment.

• Check your final pdf carefully before you submit it. All assignment partners are jointly

responsible for what is submitted so check the final version together.

Assignment Formatting

• An assignment should have a title, and the names of all authors.

• Use the example assignment as a guide to standards expected for equation layout. Write

equations down the page rather than across. Avoid multiple equations on a single line.

• Graphs

o Graphs must be large enough to be clear to the reader. Use a suitable font.

o Axes must be labelled with the name of the variable and the units (if applicable).

o If there is more than one graph on a figure, there must be a legend.

o The easiest way to transfer a Maple graph is to click on the figure, then copy and paste it

into the document. Maple commands can also be transferred using the “copy as an

image” option.

o If you use a screenshot to transfer graphs from Maple, Desmos etc. to an assignment

make sure they are clear. Grey backgrounds are not acceptable.

• Use a table for numerical data (do not give lists of numbers embedded within sentences).

• An assignment should have numbered pages.

These are general instructions. Check the specific

instructions for your assignment carefully.

How to write an assignment

• Look carefully at the instructions you are given for the assignment.

• Read any relevant course material such as lectures notes, homework and discussion

exercises before starting in on your assignment.

• Be careful when using resources like Wikipedia, and don’t use concepts and symbols that

you don’t understand.

• Explain your methods and results using logical reasoning. Include enough detail in your

mathematical calculations so that the process is clear.

• Use language that is clear and concise. Avoid overly long and complex sentences. Avoid slang

and colloquial expressions. Use correct grammar and spelling.

• Do not write out the questions (I know what they are) but do incorporate the concepts into

your explanations. Use descriptive headings and subheadings to aid clarity.

A typical assignment will include the following:

Introduction

The introduction should give the background to the assignment and describe the problems to be

solved. It should also give a brief overview of the contents. One paragraph will suffice.

Main Body [sections]

The structure of this depends on the specific requirements of the assignment. Use an organised

structure with section headings according to the requirements of the assignment. Keep your layout

clear and uncluttered. Results should be clearly presented using tables, graphs and diagrams that are

linked back to the text.

Conclusion

The conclusion will briefly review the purpose of the report and summarise the results obtained,

answering the questions or problems which the report is designed to address. The conclusions

should map directly to, and answer exactly, the questions in the scope. One paragraph will suffice.