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Math Help - Autonomous non-linear systems - predator prey equation

  1. #1
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    Autonomous non-linear systems - predator prey equation

    I need to state the type of each point of equilibrium of the for the following system of ODE:
    y'1= y2*(y2-1)
    y'2= -y1*(y1-1)

    I also need to draw the phase portrait but I understand how to do that once I can figure out how to solve this. I am unsure of how to do this so any help would be appreciated
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  2. #2
    Super Member Deadstar's Avatar
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    Can you just re-check your equation...

    Nearly all pred-prey models I've used would be

    y'_1 = y_1(y_2-1)
    y'_2 = y_2(y_1-1)
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  3. #3
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    Yeah I know, all the ones I have solved are of that form too but this one is different. That is why I am confused
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  4. #4
    Super Member Deadstar's Avatar
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    Quote Originally Posted by redwings6 View Post
    Yeah I know, all the ones I have solved are of that form too but this one is different. That is why I am confused
    I would imagine just solve in the same way.

    set y'_1 and y'_2 = 0 to get your 4 critical points.

    Linearize the system (i.e. find the jacobian of the matrix of the system), sub in the critical points, then find eigenvalues.
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  5. #5
    Super Member Deadstar's Avatar
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    So....

    Do you know what to do? Or would you like me to post a solution...
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  6. #6
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    I understand setting it to zero to get the 4 critical points however I am unsure of how to proceed from there
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  7. #7
    Super Member Deadstar's Avatar
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    Quote Originally Posted by redwings6 View Post
    I understand setting it to zero to get the 4 critical points however I am unsure of how to proceed from there
    Lol all those 'other example' tire you out..?

    You'll get 4 critical points.

    (0,0), (0,1), (1,1), (1,0).

    The Jacobian is the matrix...

    We call your equations...
    y'_1 = f(y_1, y_2)

    y'_2 = g(y_1, y_2)

    <br />
\left( \begin{array}{cc}<br />
\frac{\partial f}{\partial y_1} & \frac{\partial f}{\partial y_2} \\<br />
\frac{\partial g}{\partial y_1} & \frac{\partial g}{\partial y_2}<br />
\end{array} \right)

    Ah hell that keeps coming out tiny, top line is partial derivatives of f with respect to y_1 then y_2.
    bottom line is partial derivatives of g with respect to y_1 then y_2.

    For your system its...
    <br />
\left( \begin{array}{cc}<br />
0 & 2y_2 - 1 \\<br />
-2y_1 + 1 & 0<br />
\end{array} \right)

    Then sub in each critical point one at a time.

    Take (0,0), your matrix becomes...

    <br />
 \left( \begin{array}{cc}<br />
 0 & -1 \\<br />
 1 & 0<br />
 \end{array} \right)

    Now find the eigenvalues of this. After that you can classify what the critical point is depending on what your eigenvalues are. Look that up on the net.
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