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 20-year period leading up to 1896.

The population is now less at risk, but the major reason for the recovery of Atlantic Salmon is a massive restocking program. For our simplified model here, let us say that the number of fish per square kilometer can now be described by the DTDS

xt+1=0.25xt+c,

where c is the number of fish per square kilometer restocked every year, and xt denotes the number of Atlantic Salmon per square kilometer in Lake Ontario in year t (measured since the start of this restocking program).

a) What is the updating function of this DTDS ?

Answer: f(x)= __________

b) What is the equilibrium point p of this DTDS ?

FORMATTNG: Give the exact answer, not a decimal approximation. Your formula may include c.

Answer: p= __________

c) Determine the stability of the equilibrium point p .

Answer: The equilibrium point is__________because__________

d) If we assume that there are c=100 fish per square kilometer restocked annually, find the general solution formula for the DTDS, for any initial condition x.

FORMATTNG:Use the letter x for the initial condition x0 because Mobius 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 withx0=65 fish per square kilometer. Use a ruler, label the axes and label the points that you computed above.Compare your answer with the published solutions, and with your conclusions above.

f) How many fish per square kilometer need to be restocked every year to ensure an equilibrium population of 140 fish per square kilometer?