MATH1050: Mathematics Toolbox for Science Semester 1, 2023 Assignment 2
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Assignment 2
MATH1050: Mathematics Toolbox for Science
Semester 1, 2023
1. Pen and paper skills
Use pen and paper to answer the following questions. Show all your working and reason- ing. Set your work out neatly. You may use a calculator to do numerical calculations.
(a) Write N = N0 (0.973)t in the form N = N0 ekt .
(b) Make S the subject of the following equation:
V = 10(S3/4T) .
(c) Write eiT/2 + eiT in the form a + ib and plot your answer on the complex plane.
(d) Let u = ┌ 2_5┐ , w = ┌ ┐1(3) and M = ┐ . Find the following:
(i) Mu
(ii) u MT
(iii) u . w
(iv) MT + uwT
(e) Let A = ┐ .
Find det(A) and hence A尸1 . Use A尸1 to solve the linear simultaneous equations
3s _ t = 4
_2s + 2t = 4.
(f) Let B = ┐ .
Find the eigenvalues of B—there are two of these—and their corresponding eigen- vectors.
(g) If f(x) = ex + x2 find
(i) f\ (x) 8
(ii) f(x) dx
尸2
d x
dx 1
(h) If J = (α2 + 1) cos(2πq) find and
2. Mathematica (or other software) skills Use Mathematica or another suitable soft- ware package to solve the following problems. Include all necessary explanation and a screen shot of your Mathematica (or other software) working.
(a) Plot G = 0.07k2 + 3.78k _ 2.12 against k for k = 1, 1.5, 2, 2.5, 3, 3.5, and 4.
(b) Express the following simultaneous equations in matrix form (ie as Ax = b). (You will need to use pen and paper to do this. Find A尸1 and hence solve the equations.
_47.36x + 440.5y _ 46.45z = _901.4
37.81x + 416.9y _ 3.736z = 2274.8
4.03x + 157.87y + 6.73z = 904.6
(c) Find (vT Q)T _ Pv where
┌ ┐ ┌┐ ┌┐
Q = '(')32.2 10 .0 25 .1 54 .2'(') P = '(') 0 6.2 7 .8 8 .2'(') v = '(') 4.5 '(')
' 8.2 94.8 6.4 94.6' ' 0 2.3 5.1 9.3' ' 1.0 '
(d) Find the eigenvalues and the corresponding eigenvectors of the matrix:
┌┐
(Note: please interpret your Mathematica output and present the answer clearly for the reader.)
(e) Use Mathematica to find the derivative with respect to x and the indefinite integral
a
with respect to x of
(f) Plot f(x, y) = y cos x using
(i) a contour plot;
(ii) a 3D (surface) plot.
You will need to find and specify sensible ranges for x and y . Find where = 0
and explain this result with reference to your plots.
3. Iosoma’s swift, Apus iosomae, is a hypothetical bird that is critically endangered. In sum- mer, it nests on the limestone cliffs near Yarangobilly caves in the Kosciuszko National Park. During the winter, it migrates north and forages for insects in coastal heathland areas in the Sydney basin. Because of the development of Sydney over the last two hun- dred years, its foraging areas are now restricted to the West Head region of Ku-ring-gai Chase National Park and the coastal plateau south of Bundeena in the Royal National Park.
Two different hypothetical research groups are monitoring the population of Iosoma’s swift. One group is based in Sydney and the other in Canberra.
Swifts forage by catching insects on the wing. (You will sometimes see swallows doing this on campus.) The Sydney group is able to estimate the population of Iosoma’s swifts in winter by using methods for estimating population numbers from surveys of the number of birds in the air each day over a period of weeks.
The Canberra research group has a number of skilled rock climbers who abseil down the cliffs where the swifts nest. They do their surveys during the time between when the eggs hatch and when the young birds leave the nest. They are able to observe not only the number of nestlings (baby birds still in the nest), but also the adult birds. In particular, because pairs of Iosoma’s swifts return to the same nest year after year, the researchers can observe when new breeding pairs are established and build their nest for the first time as well as the survival of adult birds in general.
After 15 years, the Sydney group has a record of the annual winter populations of Iosoma’s swift observed in heathland in the Sydney basin. These data are given in Table 1 below.
During this time, the Canberra group has been able to gather detailed population data from the nest sites and determine fecundities and survival probabilities in a stage-based model. The life cycle graph for this model is illustrated in Figure 1. In this graph, stage 1 are nestlings, stage 2 are young adult birds nesting for the first time and stage 3 are adult birds nesting for the second or subsequent time.
(Please note that in real life, there would be measures of uncertainty such as standard deviations, included with the data. MATH1050 is not a unit that deals with statis- tics or uncertainty, so they are not included here. You will learn more about these in DATA1001.)
2
0.2
0.65
Figure 1: A life cycle graph for Iosoma’s swift. Stage 1 are nestlings observed in the nest. Stage 2 are birds nesting for the first time. Stage 3 are adult birds nesting for a second or subsequent time.
Year |
Estimate swift population |
2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 |
499 499 488 489 484 485 484 485 478 471 473 469 467 471 459 |
Table 1: Estimated population of Iosoma’s swifts in the years 2007 to 2021. Populations are combined data from observations in Ku-ring-gai Chase and the Royal National Park.
(a) Assume that the population of Iosoma’s swifts is growing or declining exponen-
tially. (This assumption underlies the use of matrix population models.) Using the Sydney data, fit an exponential model of the form N = N0 bt where t is the number of years after 2007. What can you infer about the long-term fate of Iosoma’s swift from the Sydney data, and specifically from the value of b in the fitted model?
(b) Using the life cycle graph, survival probabilities and fecundities illustrated in Fig-
ure 1, construct a matrix population model for Iosoma’s swift. For your matrix, find the largest positive eigenvalue which has no imaginary parts. What informa- tion does this give you about the long-term fate of the Iosoma’s swift population according to this model?
(c) Use the eigenvector that corresponds to the largest positive eigenvalue, that you found in the previous part, to predict the proportion of the population in each stage when they are observed by the Canberra research team each year. Assuming that there are about 500 adult birds (the sum of the number of first-time and subsequent nesters), approximately how many nestlings do the Canberra research team observe each year?
(d) Assume that there were exactly 500 adult Iosoma’s swifts in 2006. How many adult birds are there predicted to be in 2023 according to the model derived from the Sydney data? How many adults are predicted according to the Canberra team’s matrix model? (Remember that there are also nestlings in 2006 and that the adult population in this model is divided into first-time nesters and others. But, if you think carefully, you will see that you do not have to run the entire matrix model to answer this question.) What is the difference in the number of adult birds predicted by these two models? Can you think of any possible explanations as to why this should be so?
(e) Professor Nanguid from the Canberra team and Dr Coldmarsh from the Sydney
team plan to give a joint presentation about the Iosoma’s swift population at the Australian Ecological Society’s annual conference. What do you think should be the main points that they cover in their presentation?
4. This question is designed to be challenging. It assesses your ability to use software creatively and to apply critical thinking to software use and the results you obtain. You will also need pen and paper, mathematical ideas from Topics 7 to 9, and a copy of the paper by Grover 。t dl| which you can download from the Assignment 2 Canvas page.
You will not be told specifically how to answer all the parts in this question. You will need to use your own ingenuity and creativity to find answers and present your solutions clearly. This may involve, for example, looking up Mathematica techniques and code on the internet. Please show all your working, both pen-and-paper working and Mathematica (or other software) code.
The populations of algae in reservoirs are of interest to microbiologists and engineers who are concerned with water quality. Early this century, Grover and coworkers carried out a study on the population dynamics of the alga Prynn。siun pdrvun in the Lake Granbury reservoir in Texas (Grover 。t dl|, 2010). Their aim was to critically assess existing differential equation models and use their data to produce improved models, inspired and validated by field observations. (We will cover differential equations in Topics 10 and 11, but you do not need to know anything about them to complete this question.)
In this question we examine one part of their model, the maximal growth rate of P| pdrvus in Lake Granbury.
The maximal growth rate µmax of P| pdrvun is given in equation (10) of the paper. Grover and coworkers obtained this model for growth by fitting equations to data. The growth rate µmax is essentially the rate that new daughter cells are produced by a single cell of algae. The parameters in µmax are given in Table 2, just below equation (10). In its full form, as given in equation (10), µmax (measured in units 1/day) depends on temperature T (一 C), salinity σ (psu which are ”practical salinity units”) and irradiance I (µmol/m2 /day). Irradiance, in this context, is a measure how much energy from light penetrates the water.
As you can see, µmax = µmax (T, σ, I) is a function of three variables. Temperature, T, is most important input variable for µmax , followed by σ and then I . In this question we will explore the properties of µmax using plots, contour maps and calculus.
Mathematically, the expression f(x) = max{f1 (x), f2 (x)} means that f(x) is equal to the larger of f1 (x) and f2 (x) for any given value of x. For example, if f(x) = max{1, e北 } then f(x) = 1 when x < 0 and e北 < 1, and f(x) = e北 when x > 1 and e北 > 1. Mathematica has a function Max[ ] which will find the maximum automatically for you.
(a) We consider the function of one variable .
µ 1 (T) = µmax (T, σref , Iref ),
where we focus only on the temperature dependence of µmax .
(i) Write down the algebraic expression for µ 1 (T).
(ii) Using Mathematica or other software, plot µ 1 (T) as a function of T for
5 < T < 35.
(iii) Find the temperature where the maximal growth rate has its maximum if the salinity is at the reference salinity σref and and the irradiance is at the reference irradiance Iref . What is the value of µ 1 at this temperature?
(iv) Grover and his coworkers measured the temperature each month at Lake
Granbury. They found that the average daily temperature varied in a si- nusoidal fashion between 7.6 一 C and 30.9 一 C over the course of the year. Write down a function that models the temperature variation over one year. Explain what your independent variable is and what its units are. (Remem- ber that µ 1 has units 1/day and choose units for your independent variable accordingly.) Plot your function for temperature variation over one year. (It should look something like Figure 1(d) in Grover 。t dl| (2010).)
(v) You now have a function which gives temperature as a function of time over the course of a year. We can write T = T (t), for example. From here we can write µ 1 (T) = µ 1 (T (t)) = µ 1 (t). Write down the algebraic expression for µ1 (t).
(vi) We know that µ1 (t) is a rdt。of growth. Use integration to find approximately
how many new algal cells will be produced by a single cell over a year with a typical temperature profile.
(b) We introduce salinity into the maximal growth rate equation now. Define the
function
µ2 (T, σ) = µmax (T, σ, Iref ).
(i) Write down an expression for µ2 (T, σ).
(ii) Plot µ2 as a function of T and σ for 5 < T < 35 and 0 < σ < 7, both as a
contour plot and as a surface plot. Estimate the maximum value of µ2 and the values of T and σ where this occurs.
(iii) Using appropriate software, reproduce Figure 2(a) in Grover 。t dl|, (2010)
and add an extra curve for when σ is at the reference value σref .
(c) Consider now, the function of three variables µmax (T, σ, I).
(i) Draw a surface plot and/or a contour map for µmax when T is set to be constant at 25 一 C, for 0 < I < 600. Approximately what is the maximum value of µmax on this plot. At approximately what values of σ and I does it occur?
(ii) Plot a surface plot and/or a contour map for µmax when σ is set to be
constant at σ = σref . Approximately what is the maximum value of µmax on this plot. At approximately what values of T and I does it occur?
(iii) Draw a plot, similar to Figure 2(a) in Grover 。t dl|, 2010, but with each
curve representing µmax as a function of T for a different value of I when σ = σref .
(iv) Mathematica has the capability to plot level surfaces for functions of three
variables, using the command ContourPlot3D. Plot the level surfaces of µmax . How can you check whether your plot is correct?
(d) The alga P| pdrvun produces harmful algal blooms (HABs) in reservoirs. These are driven, in part, by rapid growth of the alga. What does your analysis suggest about the factors that can contribute to, or mitigate, P| pdrvun HABs?
Reference
JP Grover, JW Baker, DL Roelke & BW Brooks 2010 Current status of mathematical models for population dynamics of Prynn。siun pdrvun in a Texas reservoir. |ounrdl o# t土。│n。ricdn wdt。r R。sourc。s │ssocidtion, 46:92– 107.
2023-05-12