Question

1. Let x and y represent two animal populations which satisfy 2(9- 2 - 3y) y(-6 + 2x) (a) (5 points) What is the relationship between 2 and y? How does a grow in the absence of y? How does y grow in the absence of x? (b) (5 points) Sketch the nullclines and direction arrows of the system. (c) (4 points) Find the eigenvalues of the interior critical point. (d) (7 points) Sketch the general solution. Be detailed. (e) (4 points) Sketch c(t) and y(t) if (0) = 1, y(0) = 5.

Answer 1

(1) are animal aay y (-6+22) -0 =) -64 30 >/y=0 5 07 22 ） not Vanishes axts as predator b 26=0 dre de x(9-x-3y) = fing) dy y(-6+2x) = g(x,y) du inter-dependent to The each other. when a vanishes, lie a=0) -) 5) y also vanishes While when vanishes xlo-x)=0 9 =) x= q 3-0 may may x acts as basestatus while y prey The nullclines are given by 219-1-2y) = 0 =) x+3y=9 y (-6+20) = 0 =y=0 critical points wie given by putting dy at o A day 7(9-x-3y) -0 A y (-6+2x)=0 =) x+3y = 9 y zo 08 N=3 at x=0 y (-6 +0)-0 => y=0 at :3+3y=9 6 => y=2 Giy) = (0,0) are the critical point is given by alow The jaco bian matrix 9-20-34 fa J(414)= gx A2y 206 = 6 -> 2=3 e a low the dryer n=0 & ०४ . x= 3 - 3y=6 =) s (312) - 3x fy gy -6+2x Jione) 9 o are the eigenualues 7,09,12=-6 given by -6

J(32) - - are -9 given by 3 4 The eigen IT* ualees AI)=0 -3-2 १ ca my 1(3+1) +36-0 2²+32+36=0 23 -379-14 -3 12J 135 2 2 S 2 - -30 ( 3 jis į 2 Its general by solution.is given NYLt) @o). XIt)