Because of high mortality and low reproductive success, some fish species experience exponential decline over many years. Atlantic Salmon in Lake Ontario, for example, declined by 80% in the 20year period leading up to 1896. If xt denotes the number of Atlantic Salmon per square kilometer in Lake Ontario in year t, where year 0 is 1876, then we can model the decline in the number of fish per square kilometer with a DTDS of the form xt+1=rxt. Given the data above, calculate the value of r. Give your answer with an accuracy of three decimal places. Answer: r= Due to various factors, the value of r has since changed. The population is now less at risk, but the major reason for the recovery of Atlantic Salmon is a massive restocking program. The number of fish per square kilometer can now be described by the DTDS xt+1=0.3xt+c, where c is the number of fish per square kilometer restocked every year. a) What is the updating function of this DTDS ? Answer: f(x)= b) What is the equilibrium point p of this DTDS ? Answer: p= c) Determine the stability of the equilibrium point p . Answer: The equilibrium point is because stable The slope m of the updating function satisfies |m|<1. d) If we assume that there are c=85 fish per square kilometer restocked annually, find the general solution formula for the DTDS. Use x0=x because Maple TA doesn't accept subscripts. Answer: xt= e) Using the number of fish per square kilometer restocked annually in (d), draw the cobweb, for at least 4 steps, starting with x0=70 fish per square kilometer. The file you upload will be humangraded. Use a ruler, label the axes and label the points that you computed above. Illegible graphs and unreadable files will be marked as 0. Allowed Extensions: jpg pdf png f) How many fish per square kilometer need to be restocked every year to ensure an equilibrium population of 140 fish per square kilometer? Give your answer with an accuracy of two decimal places. Answer: A group of patients is given a certain dose of a drug once. The patients eliminate the drug at a steady rate. Two measurements of the drug concentration in the blood are taken 24 hours apart to determine the rate at which the drug is removed from the blood stream. The measurements are given below. patient initial measurement (t=0) final measurement 1 0.2 0.04 2 0.4 0.08 3 0.8 0.16 4 0.9 0.18 a) Find the value of a that will give a DTDS of the form xt+1=axt for the drug removal, where t is in days. Express a as a simple fraction. Answer: a= b) For patient 4, how long will it take until the drug concentration is below or equal to 0.01? Give your answer with an accuracy of two decimal points. Answer: days c) How long does it take for the initial concentration to decrease by 50%? Give your answer with an accuracy of two decimal points. Answer: days d) Suppose patients are given a dose every 24 hours; namely, we have the DTDS xt+1=axt+b, where a is the value found in part (a) above. How much of the drug has to be given every 24 hours for the steady state concentration to be 3/5? Give an exact answer. Answer: b= In order to keep the songbirds in the back yard happy, Sara puts out 25 g of seeds at the end of each week. During the week, the birds find and eat 2/3 of the available seeds. a) We set up a DTDS St+1=f(St) for the amount of seeds in the back yard, where t is measured in weeks and seeds are counted just after a new supply is provided. What is the updating function of the DTDS? Answer: f(S)= b) Find the fixed point of the DTDS if there is one. Give an exact answer. Answer: c) Find the general solution formula for the DTDS. Use S to denote the initial value S0 because Maple TA doesn't understand subscripts. Answer: St= Graph the updating function and draw the cobwebbing, starting from S0=5 g for at least 4 steps. Compare your graph with the solutions after the due date. A patient receives a daily dose d=6 mg of the drug FilGud. In the course of 24 hours, 55 % of the drug is absorbed and a fraction of r=9/20 remains in the blood. The DTDS modelling the daily concentration Mt of FilGud in the blood immediately after administering the dose is Mt+1=rMt+d=(9/20)Mt+6 . a) Find the updating function of the DTDS. Answer: f(M)= b) Find the equilibrium point of the DTDS. Give an exact answer. Answer: c) Give the general solution for the DTDS with general initial condition M0=M. Answer: Mt= d) Calculate M5 if M0=0. Give your answer with an accuracy of two decimal places. Answer: M5= e) Graph the updating function and draw the cobweb diagram of the DTDS, for at least 4 steps, starting from M0=0. Use your graph to help you determine the stability of the equilibrium point p. (Compare your cobweb with the solutions.) Answer: The equilibrium point is because . f) Due to sudden complications, the patient now also needs to take the drug WelSun . This drug inhibits the uptake of FilGud so that only 50% will be absorbed and a fraction of r~=0.5 will remain in the blood. Calculate the new daily dose d~ of FilGudneeded to maintain the equilibrium concentration of that drug at the same level as before. Give your answer with an accuracy of two decimal places. Answer: d~= Consider the nonlinear DTDS xt+1=rxt1+0.4xt where r>0 is some parameter. a) Calculate the fixed point(s). If there are multiple, separate them with a semicolon (;). (Note: the answer may contain the parameter r.) Answer: b) Set r=2. Calculate x1, x2, x3, x4 starting from x0=1 with an accuracy of two decimal places. x1= , x2= , x3= . and x4= . c) Still with r=2, calculate x1, x2, x3, x4 starting from x0=4.5 with an accuracy of two decimal places. x1= , x2= , x3= and x4= . d) Do you observe a trend in these values? If so, what is it? (You may need to compute more points on the orbits in (b) and (c) above to be able to reach a conclusion.) Unless the initial condition x0>0 is an equilibrium point, all the orbits x0,x1,x2,... travel randomly on the real line without converging. Independently of the initial condition x0>0, all the orbits x0,x1,x2,... coverge toward the same equilibrium point. Unless the initial condition x0>0 is an equilibrium point, all the orbits x0,x1,x2,... diverge towards infinity. Unless the initial condition x0>0 is an equilibrium point, all the orbits x0,x1,x2,... either converge to 0 or diverge towards infinity. Consider the discretetime dynamical system Nt+1=rNtNt4, where r is a parameter. Find all values (if they exist) of the parameter r for which each of the following statements are true. If there isn't any value of r that makes it true, write "none". i) The system has no equilibria. Answer: r= ii) The system has only one equilibrium. Answer: r= iii) The system has a positive equilibrium. Answer: r a, where a=