Question

Instructor Questions #10 . Directions: Print this document (2 pages). Do your rough work on a scraph, upload h te (unsubmitted) sheet. work on a separa Write your solutions directly on this page and the next page. crowamark.com IQ10 1. 110 points total Le ). t N(t) be the population of fish (measured in millions of fish) in a lake at time t (measured in years Suppose that the lake has carrying capacity K (in millions of fish), and that H million fish are harves from the lake each year. Then N satisfies is the following modification of the logistic equa tion, for some constant r dN For all eight parts of this problem, we use the numbers K 4 andr (a) [2 pts) Find the steady states of N when H-3 (b) [1 pt] On the grid provided, sketch the state space diagram for N when H- (c) [1 pt] Which of the steady states that you found part (a) (if any) are stable steady states when H 3? (d) 12 pta] Roug three plots of Nt) versus t, one for each of the initial values N(o)-0.3, N(O)-11, and N(o)- hly sketch the slope field for N(t) on the following grid. On the same grid, roughly sketch N 2 (e) 1 pt] Now suppose that rate of fish harvest is increased from H 3 to H- 3.75 15/4 (million fish per year). state space diagram for H 15/4 on the grid at right. Hint: Your sketch will be similar to your answer to part (b). -2 (D [1 pt] Before allowing the harvest of 3, 750, 000 fish per must check that there are enough fish in the lake, so that the fishing will not cause year, the Department of Fisheries and Oceans the fish popuia- tion to collapse. How many fish should be in the lake before allowing the harvest of 3, 750, 000 6 per year? (g) [1 pt] As H continues to increase, the fish population is at risk of collapsing. This is what ha to Atlantic Cod in Eastern Canada. What is the smallest value of H that population will collapse, regardless of the initial population of fish? (h) [1 pt] Sketch the state space diagram for N when H equals the value that you found in part (g)

Answer 1

o) Given hab de (N)- 0 To Stealy Sates

Givun that FN) at 2

Nる t) 1-1 N- 3 N-l 毛

dN dt )Given tat Fishes tobe Per Yeasz

dt H:

END