Question

3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable.

Answer 1

y=0, 3y + 2z=7 and The y nullelines aree z-null elines aree, z=0, Y + 2z=5. Freom the above figure, that Z-ncellclines and ynulleines interesect at (9=1, Z=2) wheree, y, 270. it is cleare So, the Creitical point loy. Ze) is (0,1,2), where, ye zero รา Z Phase - poretreait in yz - plane with direction arowws.
figure of nullclines in yz-plane, is drawn below

2.U 39,732 = 0 9:10+ = -0. 9. 0 8. E O 7. lo 5- 4. 3. 2+. 1.0 1. 0.0 0.0 0.0 1 2 2.53 3.5 5 6 X -1 -2-

Y Now put y=0, Z=0 in system 11, then CD changes to dx x 16-2x). Now, dx de =0 > x=0, x=3 The Critical point is 13,0,0) wherce ayo X16-2x) = fwa ( say). Thereeforce, f'(x) = 6-40 At, x=2, f'(3) = – 6 <o. sine, f'(3) 20, then the critical point (3,0,0) is stable.