Final answer:
To find the stationary points and evaluate their stability for the given differential equations, set x' and y' equal to zero and solve for x and y. The stationary points are x = 1, y = 1; x = 1.97, y = -0.33; x = 6.5, y = -1.67; and x = 20, y = 2.78. To evaluate their stability, linearize the system around each point and analyze the eigenvalues of the resulting linear system.
Step-by-step explanation:
To find the stationary points and evaluate their stability for the given set of differential equations, we first set both x' and y' equal to zero and solve for the values of x and y:
x' = x - 10xy = 0
y' = x/(1+x) - y = 0
Solving these equations, we find that the stationary points are:
- x = 1, y = 1
- x = 1.97, y = -0.33
- x = 6.5, y = -1.67
- x = 20, y = 2.78
To evaluate the stability of these points, we need to examine the behavior of the system near each point. We can do this by linearizing the system around each point and analyzing the eigenvalues of the resulting linear system. If the eigenvalues are all negative, the point is stable. If there is at least one positive eigenvalue, the point is unstable.