Model 1: Drug Transport into the Heart 40%]

Figure 1. Schematic of blood flow in the heart. 

Nutrients and therapeutics are transported throughout our body and into organs via the circulation of blood. Suppose that blood flowing in and out of the heart with rate r [cm3 min- ] carries a drug at a concentraaon c [mg cm-3]. The volume of blood in the heart is also given by Pg [cm3]. The average human heart can pump 4 litres of blood per minute and has an average volume of 280 cm*.

a) [30 marks] Assume that r is constant and obtain an equahon for the quanaty of therapeutics in the heart as a function of time yp(I) [mg]. You will  need  to write and   solve   an   ordinary   differential   equation   that   models   the   mass   rate  of {mg min-`] at any time the heart. At I = 0 there is no drug in the heart, but the concentration in the bloodstream is r.

Although r was assumed to be constant in queshon la, the eoncentrahon of drugs injected in the bloodstream is known to decay as an exponential function of time given by:

In this expression, D [mgJ is the dose of drug injected, d = 1.3 R !8"* [ “1 is

constantforthe drugand tisthetñnein {mi"]

TheerxpBieenB1 •  `=• ol d()[ â   ]* 8 the   * ” an•‹ied,o,Nspapw;nE«a(xL)andMATLA8(magfonnatusethammm 

urruznu4auoninihehean,oGeoneobtaIneJing«rsHo› JswiGacomtarab1oo•J

Table 1. Experimental measurements for drug concentration in blood plasma.

Model 2: Conservation of Endangered Species [io• ]

Figure 2. An African wild dog in the sun. 

The African wild dog is an endangered canine species native to sub-Saharan Africa. Their scientific name L ycaon pictus means “painted wolf", referring to their irregular coat with red, black, brown, and yellow fur  patches. These  carnivore  wild dogs are known for living in packs and being highly social. Their lifespan goes up to

11 years, and estimates say that about 6,600 adults currently live in about 39 subpopulations.

The shrinking space to roam and their susceptibility to diseases like rabies and canine distemper threaten the survival of this species. These dogs are also often hunted by farmers who are trying to protect their livestock. This means that the total number of African wild dogs is in an irreversible decline. Suppose that there are three populations of African wild dogs that we are interested in for a research project: pups, yearlings, and adults

• Pups: Represented by the variable P , where i =  1, 2, 3, ... denotes the number of years that have passed since the initial measurement of the population. Pups are born from adults or yearlings reproducing. If pups survive, they will grow into yearlings. Pups have a fixed survival probability Up. Pups cannot  reproduce. Every year all pups are lost because they either die or become yearlings.

• Yearlings: Represented by the variable Y,  yearlings are pups that have grown.  If yearlings survive, they will grow into adults. The survival probability of a yearling is St. Yearlings can reproduce, originating pups. The  fixed reproduction probability of a yearling is fip.

• Adults: Represented by the variable  A  , adults are yearlings  that  have grown or adults that have survived another  year. The survival probability of an  adult  is Sq. Adults can reproduce, originating pups. The fixed  reproduction probability of an adult is fig.

** *`o. *t 

and     Ap represent the populations of pups, yearlings, and adults at th

time of the first measurement, the survival of this populahon after one year (i = 1) is given by:

 Similarly, the survival of this population after two years is given by

2  —  RpY   -t- R.A

a)     [30 marks] In 2010, 2011 and 2012, the three subpopulations of African wild dog were measured for a conservation study in a particular city. This data can be found in Table 2. Use the data in Table 2 to estimate the survival and reproduction probabilities St, Sq,Sz, Rq and Bq.

Table 2. Measurements of the three subpopulations of African Wild Oog.

b)    [30 marks] You are asked to create a mathematical model that can predict the number of wild dogs in each of these three subpopulations at any year in the

c)    (40 marks] You are asked to design a conservation strategy for the African wild

dog for the next 50 years in another counhy. A good conservation strategy will maintain the total populahon of African wild dog around 6fD0 or even raise it. Your probability parameters for this project are given in Table 3.

• You will have a +5X survival probability credit that you can use to  increase survival probabilities Sp, Sp and Sz. You should spread this 5% nedit over at least two survival probabilities.
  • You will aleo have a +1% reproduction probability credit that you can use to increase one reproduction probability.

How would you distribute your probability credits to ensure a population of at

least 6000 African wild dogs every year?

Try to think of real-life policies that could cam this survival/reproduction increase, for example: creating sanctuary areas for pups, helping farmers protect their livestock so that they will not hunt adults as much, and so forth.

Support your strategy with MATLAB simulations, figures, and a discussion of

your methodology in designing it.

Table 3. Survival probabilities.

                 IiHdVdue    

                                                          0.665

0.45 

Model 3: Potential FlOW [20%]

Complex numbers are a valuable tool in modelling two-dimensional fluid flow. With a complex potential F(z) we can apply the properties of complex numbers to link a point (r, y) on the cartesian plane R² to a point z = x -F jy in the Gauss plane C. This process, called conformal mapping, is achieved by expressing F(r) as a function of z and y through z = z + Jy, where y = . The complex  potential  F then takes  the  form

F = 8(›.Y) +/t(x.  ).

The real component of F is called the pofextisfJxcfioii Q(z, y) and the imaginary component of F is called the stre«iti fimctioii J(z, y). The potential function Q(z, y) represents a physical quantity that induces flow between points {x, y) with different values of Q. Mechanical pressure, chemical potential, temperature gradients and voltage differences are everyday examples of potentials.

The velocity field v resulhng from a potential function  Q(z, y)  is a  vector v(z, y)  =          containing  the derivatives  of  Q  with respect  to space.  The x-component of the velocity fieldr is given by the partial derivative of Q in ther direction Qg = a‹fi

Similarly,  the y-component  ofr          is the partial derivative of Q in y, given by Qp = dy

As an example, a uniform source of flow is modelled by the complex potential

F{z 1 — In z. Tlus function has a velocity field p(A# J)

r° y+y°

lCh can l;›e visualise

as the arrows in Figure 4.

•        •        •        •        •         G        n        *        t         £        $        *        *        *        •         •        •        •        •

•

0         - - -           - - -

—1               —0.5                 0

Direction x

0.5

Figure 4. Vector field for F(z) = In z. The length of the arrows represents their magnitude. You can find the MATLAB code used to generate this Figure at the end of the question.

Figure 4 was generated in MATLAB using the following code:

% Defining the rdnge of variables x dnd y

x = -1:e. i: i;

Z C re at 1rug a roe s hgr Id

[X,Y]=meshgr1d(x,y);

â Ca1c u1at ing I he x - c ompon ent oT vecton v vx = X./(X."2 + Y. ^2),

% C a1c ula t i ng t he y - component u+ vecto i• v vy = Y./(X."2 + Y. ^2) ;

% Plotting the vector field with components vx and vy in the coordinates X

and Y

qu1ver(X, Y, vx, vy, k’ , a utos c a 1 efact o r’ , 1. 15, 1inewi dt h * , 1. 25 )

x1abe1( D i i e c t i on  x ’ ) y1abe1( D i i e c 1 1on y ’ ) xllm( [ -1 1] )

y1lm( [ -1  1] )

a) Ozs wks] Imagine the potentials £  (z) =  z and f  (r)  = J ln r.  Calculater   for each of them and plot their vector field in MATLAB in order to demonstrate graphically Writ F is the complex potential for uniform horizontal flow, and ^2 is the potential for flow around a vortex.

b)     [25 marks] Discuss the relationship and differences between a uniform source (Figure 4) and a vortex as obtained in question 1s. In order to do that, you should anal yse their complex potentials mathematically and compare their vector fields graphically. How is the direction of the vectors related to the Gauss plane? Use this knowledge to derive the complex potential for uniform vertical flow.

Another valuable property of complex potentials is that they allow us to translate velocity fields. The transformation F{z— •o). where °o` °  +  jb Will translate a potential F tz 1 that is centred initially at the point (0,0) to the point (o, b), represented by  •o = •  -F  jb.  Complex   potentials  also  allow   us  to  superimpose  different flow patterns by summing them. As an example, the velocity field resulting from the interaction of two potentials F(z) = ln(z — 1) + In(z — 2) is plotted in Figure 5. 

0.5                      1                1.s

Direction x

2.5

Figure S. The potential flow resulting from two competing sources, one centred at (1,0) given by In (z — 1) and the other centred at (2,0), given by ln(z — 2). Black arrows indicate velocity components of r(z, y), which are perpendicular to the contour levels of Q(z, y).

Notice that the region between the two siurces in Figure 5 does not have any arrows representing the velocity field. This region is called a sfngnnfioii poiitt and has

r| = 0. Stagnation points are also the points where = 0.

C) [25 marks] Apply the Cauchy-Riemann equations to prove that if the stagnation

point  of  a  potential  f(r) is defined  by any  points z’  such  that ^dP

dz

• = 0, then