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Math Help - numerical analysis of competing species??

  1. #1
    crb
    crb is offline
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    numerical analysis of competing species??

    please have a look at the attached picture ...and help me understand what to do next...

    thank you for whatever help anyone can provide ...

    plz note , i know how to linearize at the equillibrium point which is ( x= m/n , y =a/b )
    Attached Thumbnails Attached Thumbnails numerical analysis of competing species??-modelling-competing-species.jpg  
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  2. #2
    MHF Contributor

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    you can change to the "scaled variables" Y= y/(a/b) and X= x/(m/n) by solving those for x and y: x= (m/n)X and y= (a/b)Y. Then the differential equations become
    \frac{dx}{dt}= \frac{m}{n}\frac{dX}{dt}= (a- b(\frac{a}{b})Y)\frac{m}{n}X
    or
    \frac{dX}{dt}= a(1- Y)X
    and
    \frac{dy}{dt}= \frac{a}{b}\frac{dY}{dt}= (m- n(\frac{m}{n}X)\frac{a}{b}Y
    or
    \frac{dY}{dt}= m(1- X)Y
    which have equilibrium points at (0, 0) and (1, 1).

    You then get \frac{dY}{dX}= \frac{m(1- X)Y}{a(1-Y)X}
    which can be separated as
    a\frac{Y}{1-Y}dY= m\frac{X}{1-X}dX

    Integrate that to get g(Y)= f(X).
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