Ing 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 = 0.3xt +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) = 0.3*x+c 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 = 1.428571429*c 固助 c) Determine the stability of the equilibrium point p . Answer: The equilibrium point is stable because The slope m of the updating function satisfies |m| < 1. d) If we assume that there are c= 80 fish per square kilometer restocked annually, find the general solution formula for the DTDS, for any initial condition x. FORMATTNG: Use the letterx for the initial condition an 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 with xo = 50 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 160 fish per square kilometer? FORMATTNG: Give your answer with an accuracy of two decimal places.
Ing 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 = 0.3xt +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) = 0.3*x+c 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 = 1.428571429*c 固助 c) Determine the stability of the equilibrium point p . Answer: The equilibrium point is stable because The slope m of the updating function satisfies |m| < 1. d) If we assume that there are c= 80 fish per square kilometer restocked annually, find the general solution formula for the DTDS, for any initial condition x. FORMATTNG: Use the letterx for the initial condition an 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 with xo = 50 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 160 fish per square kilometer? FORMATTNG: Give your answer with an accuracy of two decimal places.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Ing
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 = 0.3xt +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) = 0.3*x+c
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 =
1.428571429*c
固助
c) Determine the stability of the equilibrium point p .
Answer: The equilibrium point is
stable
because
The slope m of the updating function satisfies |m| < 1.
d) If we assume that there are c= 80 fish per square kilometer restocked annually, find the general
solution formula for the DTDS, for any initial condition x.
FORMATTNG: Use the letterx for the initial condition an 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 with xo = 50 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 160 fish per square kilometer?
FORMATTNG: Give your answer with an accuracy of two decimal places.
Answer:
112
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