1. LetN(t) denote the population of a certain species in its habitat; let?andN0denote, respectively, the difference between birth and death rates, and the carrying capacity of the habitat. The...

1. LetN(t) denote the population of a certain species in its habitat; let?andN0denote, respectively, the difference between birth and death rates, and the carrying capacity of the habitat. The differential equation modelling the population in time is

???N?? N=?N1-N.

0

Now suppose that the population is harvested proportionally to the population density with the rateK, where Kis a positive constant such thatK

1. 
   

   (a) Show that the modified differential equation has two equilibrium points. Draw a phase flow diagram and indicate which equilibrium points are stable or unstable. Discuss the behaviour of the solution ast? 8: how does the behaviour depend upon the value ofN(0)?
   

   
   


2. 
   

   (b) Solve the differential equation, and show that your solution has the features already discovered in part (a).
   

   
   

First downloaded: 28/8/2017 at 15:45::4

1. 
   

   Newton’s law of cooling, as discussed in lectures, assumes that air at room temperature is blown past the cooling body (”forced cooling”). For cooling instillair (”natural cooling”) a better model is to assume that the rate of temperature decrease of the cooling body is directly proportional to the 3/2 th power of the difference between the temperature of the body and the temperature of the surrounding air.
   

   
   
   
   
   1. 
      

      (a) Introduce appropriate notation and thus write the law for natural cooling as a differential equation. Is it linear?
      

      
      
   
   
   2. 
      

      (b) By solving the differential equation, show that the temperatureTof the cooling body at timetis given by
      

      
      

      ??-1?t??-2 T=Ts+ (T0-Ts)2+2
      

      
      

      whereT0denotes the temperature at timet= 0, andTsdenotes the temperature of the surrounding air and?is a constant. Note that it is assumed thatT0> Ts.
      

      
      
   
   
   
   


2. 
   

   A predator-prey model for the populations of prey fishx(t) and predator fishy(t) in a lake at timetis the system of differential equations
   

   
   

   x ? =ax-cxy,(1) y ? =-by+dxy,(2)
   

   
   

   wherea, b, canddare positive constants.
   

   
   
   
   
   1. 
      

      (a) Explain the meaning and significance of the constantsaanddfor the model.
      

      
      
   
   
   2. 
      

      (b) In the absence of predators (that is, wheny(t)=0) what happens to the number of prey ast? 8?
      

      
      
   
   
   3. 
      

      (c) Show that there is exactly one equilibrium point (x0,y0) in the system (1)–(2) in which bothx0andy0 are positive.
      

      
      
   
   
   4. 
      

      (d) Suppose that prey fish are added to the lake at a rateKx, that is, at a rate proportional to the number of prey, whereKis a positive constant. Modify the system (1)–(2) to incorporate the adding of the prey fish, and determine the new equilibrium point (x1,y1) corresponding to the one determined in (iii). Comment on the effect of adding prey fish on the equilibrium levels of each species.
      

      
      
   
   
   5. 
      

      (e) Let (x1, y1) be the equilibrium solution obtained in part (iv). Setu(t) =x(t)-x1andv(t) =y(t)-y1. Determine the system of differential equations satisfied byuandv.
      

      
      
   
   
   6. 
      

      (f) Now suppose thatuandvare small (so thatxandyare near the equilibrium solution). Consider the system obtained by neglecting all terms that are small compared touorv. Show that the functionu(t) then satisfies a second order linear differential equation. By solving this equation, show that the functions uandvare periodic, and determine their periods.
      

      
      

      Part B (Fourier Analysis).
      

      
      
   
   
   
   

1. LetV=PC[-p,p] be the infinite linear inner product space of piecewise continuous on [-p,p] functions f: [-p, p]?Cwith the standard inner product

p

??

-p

(f, g) =

w(x)f(x)g(x)dx

and the norm defined via inner product

1. 
   

   (a) Show that the setS=??einx??8n=-8is orthogonal set (with respect to the weighting functionw(x) = 1)
   

   
   

   on the interval-p=x=p.
   

   
   


2. 
   

   (b) Calculate the norm????einx????of each element in this set.
   

   
   


3. 
   

   (c) Letf(x)?PC[-p,p]. Assume thatf(x) =??8n=-8cneinx. Derive the formulas for Fourier coefficients cn.
   

   
   

?f?=??(f,f).

2. Sketch the following 2p-periodic functionsf(x), which for-p

1. 
   

   (a) Find the trigonometric Fourier series of the above functions.
   

   
   


2. 
   

   (b) Find the exponential Fourier series of the above functions.
   

   
   


3. 
   

   (c) Assume that at the points of continuity offon the intervalx?[-p,p] the series converge tof(x). To what value does each series converge atx= 14p/3?14.66 ? Sketch the graphs of the sums of the series on the interval [-5p,5p].