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Course code and title

MATH3070

Natural Resource Mathematics

Semester Semester 2, 2021

Type Online, non-invigilated assignment, under ‘take home exam’ conditions.

Technology File upload to Blackboard Assignment

Date and time

Your assignment will begin at the time specified by your course coordinator. You have a

fixed 101-hour window from this time in which it must be completed. You can access

and submit your paper at any time within the 101-hours. Even though you have the

entire 101-hours to complete and submit this assessment, the expectation is that it will

take students with a strong command of the material around 8 hours to complete.

Note that you must leave sufficient time to submit and upload your answers.

Permitted materials

This assignment is open book – all official course materials are permitted. Some

materials outside of the course, including published books, journal articles, Wikipedia

pages, computer algebra programs, etc. are permitted, but must be cited in your solution

where they are used.

Materials which include answers, discussion, or any other form of communication directly

related to the questions in this assignment are NOT permitted, and the use of such

materials is considered cheating.

Recommended

materials

Ensure the following materials are available during the available time:

R or similar programming language; bilingual dictionary; phone/camera/scanner;

A computer algebra tool, such as wolfram alpha, Mathematica, Maple, Maxima, or

MATLAB symbolic solver package may be useful, but is not required.

Instructions

You will need to download the question paper under the assessment section of

blackboard. Once you have completed the assignment, upload a single pdf file with your

answers to the Blackboard assignment submission link. You may submit multiple times,

but only the last uploaded pdf file will be graded. Any computer code must also be

submitted as an executable file, e.g. ‘.R’ file.

You can print the question paper and write on that paper or write your answers on blank

paper (clearly label your solutions so that it is clear which problem it is a solution to) or

annotate an electronic file on a suitable device.

Who to contact

Given the nature of this assessment, responding to student queries and/or relaying

corrections during the allowed time may not be feasible.

If you have any concerns or queries about a particular question or need to make any

assumptions to answer the question, state these at the start of your solution to that

question. You may also include queries you may have made with respect to a particular

question, should you have been able to ‘raise your hand’ in an examination-type setting.

If you experience any interruptions during the allowed time, please collect evidence of

the interruption (e.g. photographs, screenshots or emails).

Page 2 of 6

If you experience any technical difficulties during the assignment, contact the Course

Coordinator . Note that this is for technical difficulties only.

Late or incomplete

submissions

In the event of a late submission, you will be required to submit evidence that you

completed the assessment in the time allowed. This will also apply if there is an error in

your submission (e.g. corrupt file, missing pages, poor quality scan). We strongly

recommend you use a phone camera to take time-stamped photos (or a video) of every

page of your paper during the time allowed (even if you submit on time).

If you submit your paper after the due time, then you should send details to SMP Exams

(exams.smp@uq.edu.au) as soon as possible after the end of the time allowed. Include

an explanation of why you submitted late (with any evidence of technical issues) AND

time-stamped images of every page of your paper (eg screen shot from your phone

showing both the image and the time at which it was taken).

Further important

information

Academic integrity is a core value of the UQ community and as such the highest

standards of academic integrity apply to assessment, whether undertaken in-person or

online.

This means:

You are permitted to refer to the allowed resources for this assignment, and you

must not use any instances of work that has been submitted previously

elsewhere.

You are not permitted to consult any other person – whether directly, online, or

through any other means – about any aspect of this assignment during the

period that it is available.

If it is found that you have given or sought outside assistance with this

assignment, then that will be deemed to be cheating.

If you submit your answers after the end of allowed time, the following penalties will be

applied to the total mark available for the assessment:

Less than 5 minutes – 5% penalty

From 5 minutes to less than 15 minutes – 20% penalty

More than 15 minutes – 100% penalty

These penalties will be applied unless there is sufficient evidence of problems with

the system and/or process that were beyond your control.

Undertaking this online assignment deems your commitment to UQ’s academic integrity

pledge as summarised in the following declaration:

“I certify that I have completed this assignment in an honest, fair and trustworthy

manner, that my submitted answers are entirely my own work, and that I have neither

given nor received any unauthorised assistance on this assignment”.

Semester Two Final Assessment, 2021 MATH3070 Natural Resource Mathematics

Q1. [25 points] Consider a modification of the Beverton-Holt stock recruitment model,

Xt+1 = sXt + F (Xt),

with,

F (X) =

aX

1 + bX

,

where Xt is the number of adult fish in year t, s is the proportion of adults that survive to

the following year, and F (Xt) is the number of offspring that survive to become adults in the

following year. The parameters a and b are positive real numbers.

(a) [2 points] Derive an expression for ‘proliferation’ in the model.

(b) [3 points] Derive expressions for all biologically meaningful equilibria in this model.

(c) [5 points] Derive conditions for when each equilibrium is stable and unstable.

(d) [5 points] Consider data, {X0, X1, ..., Xn}, for adult population size in years 0 to n, and

assume the values of a and b are known. Derive an expression for the least squares estimate

of s.

(e) [1 point] Consider the model where adults are fished after reproduction, where γ is the

proportion of adult fish that survive fishing, namely,

Xt+1 = γsXt + F (Xt).

That is, we assume fishing mortality proportion, h, and let γ = 1 ? h. Write down

expressions for the equilibria and their stability, given this modified version of the model.

(f) [9 points] What is the optimal value of γ that should be used to achieve maximum sustain-

able yield, at equilibrium, given the model from part (e)?

Page 3 of 6

Semester Two Final Assessment, 2021 MATH3070 Natural Resource Mathematics

Q2. [25 points] In 1995, the government of a hypothetical country banned the fishing of a

large predatory fish species, on a small reef, after the population collapsed. Since then, there

are anecdotal reports that the population recovered. The government would like to reopen the

fishery.

Prior biological knowledge: The fish live many years, and biologists think (and expect you to

assume) that all adult fish have an equal chance of survival, no matter their age. The reproduction

of this fish is well studied. At infinitesimally low adult population densities, every pair of adult

fish produce two offspring that survive to adulthood the following year. At infinitely high adult

population densities, only five total offspring survive to adulthood the following year (despite

infinitely many potential adult parents). Only adult fish can be harvested.

The exact value of the survival probability of adult fish for this species is unknown. However, the

government acquired a data set from a nearby aquarium for a related species tracking 50 adults

over one year. The biologists think the two species should have the same survival probability. A

value of ‘1’ in the file, TankStudy.csv, represents a live fish at the end of the aquarium study,

and a value of ‘0’ represents a dead fish. Each row corresponds to a unique fish, out of the 50.

Government data: The reef underwent long-term monitoring from the closure in 1995 until

the present. Provided with the assignment is a time series of population size data in observed

numbers of fish from 1995 to 2020. See GovernmentData.csv.

Write a summary/report (≤ half a page of text) stating and justifying a recommendation for a

limit on the yearly quota (total catch) in this fishery for when it reopens. Attach supporting

calculations, code, and figures (not counted against your half page limit) and label and refer to the

figures and calculations in your report. You can use the population dynamic model in question

one (or propose a different model that satisfies the assumptions stated in the question), estimate

the survival parameter given the data, and its associated uncertainty, and discuss optimal harvest

rules given uncertainty in the system. You will be graded on the clarity and logical flow of your

report and the completeness and correctness of your supporting calculations, code, and figures.

You can use the week 6 practical to help you think through your supporting calculations.

You are welcome to use any programming language you like. If you use R you may modify any

scripts provided to you in this course to help you answer the question. Submit your code as a

separate .R file along with your solutions to this assignment. If you use a different programming

language, please email the course coordinator m.holden1@uq.edu.au about the best way to

submit your code.

Page 4 of 6

Semester Two Final Assessment, 2021 MATH3070 Natural Resource Mathematics

Q3. [25 points] Consider the following predator-prey model,

where N is the prey density, P is the predator density, r is the intrinsic growth rate, K is the

carrying capacity, a is the maximum predation rate, b is the predator interference parameter, c

is the conversion efficiency, and d is the mortality rate of the predator.

(a) [4 points] Conduct nondimensionalisation of the model.

(b) [3 points] What are nullclines of N and P in the model of (a)?

(c) [3 points] What are equilibria of N and P in the model of (a)?

(d) [4 points] Show the Jacobian matrix of the model of (a).

(e) [3 points] Show the Jacobian matrix of the model of (a) at an internal equilibrium (i.e.,

Nˉ , Pˉ > 0).

(f) [4 points] Show the parameter condition in which the internal equilibrium is stable.

(g) [4 points] What is the main difference between the above model and the Rosenzweig-

MacArthur model (Q1 of Assignment 3)? How does it affect stability of the internal

equilibrium?

Page 5 of 6

Page 6 of 6

Semester Two Final Assessment, 2021 MATH3070 Natural Resource Mathematics

Q4. [25 points] The biomass spectrum (Sheldon spectrum) is often represented as a power

function of the form:

() = ,

where,

() = Biomass (units: mass),

= Weight (units: mass),

= Coefficient (units: ?)

= Exponent (units: ?)

(a) [2 points] What are the units of and for the biomass size spectrum?

(b) [4 points] The biomass spectrum is usually plotted as a log-log plot.

How would you interpret and from the log-log plot of the biomass spectrum,

both mathematically and ecologically?

(c) [10 points] For the biomass spectrum for the ocean globally, typically ≈ 0.

Based on the power function for the biomass spectrum, derive the approximate

slopes of the unnormalised number spectrum and the normalised number

spectrum.

(d) [4 points] To estimate the total number of individuals in an ecosystem, would you

integrate the biomass spectrum, unnormalised number spectrum or normalised

number spectrum? Justify your answer by performing a dimensional analysis.

(e) [5 points] Although on average ≈ 0 across the ocean, individual log-log plots of

the biomass spectrum at different locations can be steeper or flatter. Explain how

the presence of different zooplankton groups could change the slope of the

biomass spectrum, and how this could influence the number of fish. Hint:

Consider the Predator Prey Mass Ratio of different zooplankton groups.

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