Find Critical Points of x' and y': ax - bxy & bxy - cy

[rad0786]

Homework Statement

Find the singular points for:

x' = ax - bxy
y' = bxy - cy

Homework Equations

The Attempt at a Solution

ax - bxy = 0
bxy - cy = 0

implies

x = 0 or y = a/b

y = 0 or x = c/b

the the critical points are (o,o) or (c/b, a/b)

why is (0, a/b) and (c/b, 0) NOT a critical point??

 

[siddharth]

why is (0, a/b) and (c/b, 0) NOT a critical point??

Because, x' and y' are not 0 for those values of x and y. Substitute them to check.

 

[HallsofIvy]

Why would you think they would be? They clearly don't satisfy the equations you give.

When you have a system of non-linear equations with several different solutions, the x and y are not independent- you can't just combine any value of x with any value of y: it is the specific x, y pair that satisfies the equations.

 

[rad0786]

Oh I see my mistake.

I thought that they were dependent.

 

[rad0786]

I have been working with the equations:

x' = rx - sxy/(1+tx)

y' = sxy/(1+tx) - wy


I find that the only singular point is (0,0)

is this correct?


I got another point, ( w/(s-w) , r/s ), however, that cannot be a critical point because when substituting, x' and y' fail to be 0.

 

[HallsofIvy]

I have been working with the equations:

x' = rx - sxy/(1+tx)

y' = sxy/(1+tx) - wy


I find that the only singular point is (0,0)

is this correct
I got another point, ( w/(s-w) , r/s ), however, that cannot be a critical point because when substituting, x' and y' fail to be 0.

Since (w/(s-w), r/s) does not satisfy the equations for a critical point, how did you get that?

The equations for a critical point are x'= rx- sxy/(1+ tx)= 0 and y'= sxy/(1+ tx)- wy= 0. The first gives sxy/(1+ tx)= rx and the second sxy/(1+ tx)= wy. Since the left sides of both equations are the same, rx= wy. Every point on the line y= (r/w)x (or, if w= 0, the line x= 0) is a singular point.

 

[rad0786]

Since (w/(s-w), r/s) does not satisfy the equations for a critical point, how did you get that?

x'= rx- sxy/(1+ tx)= 0

x(r-sy/(1 + tx)) = 0

x= 0 and (r-sy/(1 + tx)) = 0

since x = 0 r-sy = 0

therefore, y = r/s


similarly we do that for y'= sxy/(1+ tx)- wy= 0 and we get y = 0 and x = w/(s-w)

The equations for a critical point are x'= rx- sxy/(1+ tx)= 0 and y'= sxy/(1+ tx)- wy= 0. The first gives sxy/(1+ tx)= rx and the second sxy/(1+ tx)= wy. Since the left sides of both equations are the same, rx= wy. Every point on the line y= (r/w)x (or, if w= 0, the line x= 0) is a singular point.

when I substitute y= (r/w)x into x'= rx- sxy/(1+ tx), x' is NOT zero. So how could this line be a line of singular points?

 

[rad0786]

also, what are the differences between equilibrium solutions and singular points?