Question

Q4 When there is no fishing, the population P(t) of the fish is governed by the differential equation dp = 2P – 0.01P2 P(O) = 150. (a) Find P(t). (b) Find the population of the fish after 15 years. (C) Compute lim P(t) (the population of the fish after many and many years or t approaches infinity). (d) Find the critical solution(s) and classify it (them) into stable or unstable.

Answer 1

1. 
   - dP dt 2P - 0.010² ; Plo)= 150 - at (Variable seperable) 21-0.01p2 - at de - (0.01) 2 P-P 001 looxdP - at 2000 - p² completing the square of 200p-p² was - [ p²_200p =- [ p² – 2x100 + (1003² - (100) = [ (P-100) -(1003] - (100)2 – (P-100) - 100 xdP - dt (100) – (P-100² Let P-100= u dp= du
   
2. 
   Integrating both sides loox du Cat 100x I enju + 100 = t + c 2x100 100-u = t + c too en | 100+ (P-100) | I ldo -(P-100) P = ttc. I eu I > a 1 200-P to calculate constant c, we use P(O)= 150 Now - I eu I - ofc 1501 800-150 150 otc -loo 50 I en 11501 - ote I en (3) - c 3 ml Pelt + 1 en 13) 900-P This is the solution
   
3. 
   Finding Plt) from above. en bo plat + ln (3) A taking exponential both sides 2t + ln(3) e et en 13) = - е - 200-P . е 3 2t e goo-p a = atbl. a-b l Applying componendo dividendo il P+(200-P) P- (200-P) 3e24 Jet 200 27 Be tl QP-200 100 2t е +) 302t - P-100 Be 8-100 = 100x
   
4. 
   t Ple) = 100 100 ) Plt) os Thus, Plt) = 600.0 2t 3e2t +1 xe 20 + Population of fuh rafter 15 years 30 Plis) = 600x e o - = 200 sxe + (c) um Plt) - lim 600 e tto X 302t + (as to It becomes o form, thus to avoid this we take out et common from numerator and denominator). to
   
5. 
   - lim tto 0 - 200 - lim 600 to 3te 2t - 2t limē - lim ! to é t ttoo Thus, lim Plt) = 900 to or equilibrium points (d) The critical are where points dp – 0 3P - 0.02 -0 p2 godl=0 . ) P/ P-200)=0 - These il P=0,200 are the critical points dp = (-0.01) P(P-200)
   
6. 
   For PLO dP dt = (-0.01) P (P-200) Lo -ve -ve - ve for OLP < 200 so = (-0.01) P(P-200) -ve tve - ve for P 900 dl – (-0.01) P(P-200) Lo -ve the the The phase portrait is I o crepeling) ; P=0 is unstable DILE 900 Cattracting) ; P=200 This is casymptotically stable. Plt) A OLP2200) (17200) --- - (P=900) > + (P=O) СР<0)