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 fishper square kilometer can now be described by the DTDS

xt+1=0.3xt+c,xt+1=0.3xt+c,

whereccis the number of fishper square kilometer restocked every year, andxtxtdenotes the number of Atlantic Salmonper square kilometer in Lake Ontario in yeartt(measured since the start of this restocking program).

a)What is the updating function of this DTDS ?

Answer:f(x)=f(x)=__________

b)What is the equilibrium pointppof this DTDS ?

Answer:p=p=__________

c)Determine the stability of the equilibrium pointpp.

Answer: The equilibrium point is__________because__________

d)If we assume that there arec=85c=85fish per square kilometer restocked annually, find the general solution formula for the DTDS, for any initial conditionxx.

Answer:xt=xt=__________

e)Using the number of fishper square kilometer restocked annually in (d), draw the cobweb, for at least 4 steps, starting with x0=60x0=60fishper 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 fishper square kilometer need to be restocked every year to ensure an equilibrium population of160160fishper square kilometer?

Answer:____________