# Equilibrium points

1. May 6, 2007

### kreil

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

Given the system
$$x'(t)=ax(t)-bx(t)y(t)$$
$$y'(t)=-cy(t)+dx(t)y(t)$$

(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