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Equilibrium points

  1. May 6, 2007 #1

    kreil

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    Gold Member

    1. The problem statement, all variables and given/known data

    Given the system
    [tex]x'(t)=ax(t)-bx(t)y(t)[/tex]
    [tex]y'(t)=-cy(t)+dx(t)y(t)[/tex]

    (a) Find the equilibria of the system.
    (b) Choose any positive a,b,c,d. Start close to each equilibrium and find the solution numerically.




    3. The attempt at a solution

    (a) Solving for x and y in the system, the equilibrium points are x(t)=y(t)=0 and x(t)=c/d, y(t)=a/b.

    (b) Choose a=c=0.6 and b=d=0.3. Then the equilibrium points are (x,y)=(0,0) and (x,y)=(2,2). I wrote a fortran program to solve the equations using euler's method, but the values I got back didn't make sense. I figure this is because I am incorrectly choosing the initial values of x and y.

    When it says "start close to each equilibrium", does this mean I should choose the initial values of x and y to be something like 0.1 and 2.1? In part (c) I need to graph the solutions so any quick help is appreciated!

    Also, this is the program I am using:

    REAL*4 T,X,Y,H
    DIMENSION T(11), X(11), Y(11)
    T(1) = 0
    X(1) = 0.1
    Y(1) = 0.1
    H = 1
    DO 1, I=1,11
    T(I+1) = T(I) + H
    X(I+1) = X(I) + H*(0.6*X(I)-0.3*X(I)*Y(I))
    Y(I+1) = Y(I) + H*(0.3*X(I)*Y(I)-0.6*Y(I))
    1 CONTINUE
    WRITE(*,100) (T(I),X(I),Y(I), I=1,11)
    100 FORMAT(1x,3(D24.14))
    END
     
  2. jcsd
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