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Math Help - How to deduce ds/dx+ds/dy=0 from s_x(x)=0

  1. #1
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    Unhappy How to deduce ds/dx+ds/dy=0 from s_x(x)=0

    Hello everyone,

    I am stuck with the following problem and would really appreciate your help/hints/solutions:

    Given that s is smooth and that y is near x, I want to obtain

    \frac{\partial s_x(y)}{\partial x}+\frac{\partial s_x(y)}{\partial y}=0

    given the identity

    s_x(x)=0.

    Here s_x(y) is the so-called invasion exponent, i.e. the long-term population growth rate of a mutant population with trait y under environmental conditions as set by the resident population with trait x. So in particular, the subscript x is not a derivative but rather implies that s_x(y)=s(x,y) (At least I think the latter is right.)

    Apparently this is straightforward but I don't seem to get the right answer. Using the linear approximation

    s_x(y)\approx s_x(x)+\bigl[\frac{\partial s_x(y)}{\partial y}\bigr]_{y=x}(y-x),

    I got to the point where I need to show that
    (y-x)\bigl\{\frac{\partial}{\partial x} [\frac{\partial s_x(y)}{\partial y}]_{y=x}+\frac{\partial}{\partial y}[\frac{\partial s_x(y)}{\partial y}]_{y=x}\bigr\}=0.

    However, I am not sure whether I am on the right `path' or if I am making it awfully complicated. Also, if I am heading in the right direction, how exactly would I express that the {}-expression should be zero?

    Thank you so much for your help. I'm sure this should be really easy but I just don't seem to get it. Any hints are greatly appreciated!

    -Hanna

    P.s.: This problem can also be found in Odo Diekmann's paper http://www.environnement.ens.fr/IMG/...nnersguide.pdf on page 60.
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  2. #2
    Grand Panjandrum
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    Re: How to deduce ds/dx+ds/dy=0 from s_x(x)=0

    Quote Originally Posted by Hanna87 View Post
    Hello everyone,

    I am stuck with the following problem and would really appreciate your help/hints/solutions:

    Given that s is smooth and that y is near x, I want to obtain

    \frac{\partial s_x(y)}{\partial x}+\frac{\partial s_x(y)}{\partial y}=0

    given the identity

    s_x(x)=0.

    Here s_x(y) is the so-called invasion exponent, i.e. the long-term population growth rate of a mutant population with trait y under environmental conditions as set by the resident population with trait x. So in particular, the subscript x is not a derivative but rather implies that s_x(y)=s(x,y) (At least I think the latter is right.)

    Apparently this is straightforward but I don't seem to get the right answer. Using the linear approximation

    s_x(y)\approx s_x(x)+\bigl[\frac{\partial s_x(y)}{\partial y}\bigr]_{y=x}(y-x),

    I got to the point where I need to show that
    (y-x)\bigl\{\frac{\partial}{\partial x} [\frac{\partial s_x(y)}{\partial y}]_{y=x}+\frac{\partial}{\partial y}[\frac{\partial s_x(y)}{\partial y}]_{y=x}\bigr\}=0.

    However, I am not sure whether I am on the right `path' or if I am making it awfully complicated. Also, if I am heading in the right direction, how exactly would I express that the {}-expression should be zero?

    Thank you so much for your help. I'm sure this should be really easy but I just don't seem to get it. Any hints are greatly appreciated!

    -Hanna

    P.s.: This problem can also be found in Odo Diekmann's paper http://www.environnement.ens.fr/IMG/...nnersguide.pdf on page 60.
    Expand s(x,y) about (x_0,y_0):

    s(x,y)=s(x_0,y_0)+\left. \frac{\partial s(x,y)}{\partial y}\right|_{(x_0,y_0)}(y-y_0)+ \left. \frac{\partial s(x,y)}{\partial x}\right|_{(x_0,y_0)}(x-x_0)+..

    We now truncate after the linear terms and put x_0=y_0 to get:

    s(x,y)\approx\left. \frac{\partial s(x,y)}{\partial y}\right|_{(x_0,x_0)}(y-x_0)+ \left. \frac{\partial s(x,y)}{\partial x}\right|_{(x_0,x_0)}(x-x_0)

    so:

    s(x,x)=0\approx\left. \frac{\partial s(x,y)}{\partial y}\right|_{(x_0,x_0)}(x-x_0)+ \left. \frac{\partial s(x,y)}{\partial x}\right|_{(x_0,x_0)}(x-x_0)

    Hence dropping the common (non-zero) factor:

    0\approx \frac{\partial s(x,y)}{\partial y}+  \frac{\partial s(x,y)}{\partial x}

    (If you keep track of the order of the error in the approximations you will find that the \approx can be replaced with equality; the error in the second from last expression is O((x-x_0)^2 if s(x,y) is twice differentiable and a bit more complicated otherwise but the argument will still go through)

    CB
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  3. #3
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    Re: How to deduce ds/dx+ds/dy=0 from s_x(x)=0

    Dear CB,

    thank you so much for your quick reply! I has been very helpful. I understand everything up to

    s(x,x)=0\approx\left. \frac{\partial s(x,y)}{\partial y}\right|_{(x_0,x_0)}(x-x_0)+ \left. \frac{\partial s(x,y)}{\partial x}\right|_{(x_0,x_0)}(x-x_0)

    Hence dropping the common (non-zero) factor:
    However, I can not quite follow why we can drop the `evaluated at (x_0,x_0)' to generally conclude that

    0\approx \frac{\partial s(x,y)}{\partial y}+  \frac{\partial s(x,y)}{\partial x}
    where we can also have `evaluated at (x_0,y_0)' for some x_0\neq y_0?

    Thank you so much for your help though!!

    -Hannah
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  4. #4
    Grand Panjandrum
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    Re: How to deduce ds/dx+ds/dy=0 from s_x(x)=0

    Quote Originally Posted by Hanna87 View Post
    Dear CB,

    thank you so much for your quick reply! I has been very helpful. I understand everything up to



    However, I can not quite follow why we can drop the `evaluated at (x_0,x_0)' to generally conclude that



    where we can also have `evaluated at (x_0,y_0)' for some x_0\neq y_0?

    Thank you so much for your help though!!

    -Hannah
    Because it is a mistake, I will have to look at this again when I get a chance#

    CB
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  5. #5
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    Lightbulb Re: How to deduce ds/dx+ds/dy=0 from s_x(x)=0

    I think I might just have found the answer -finally!! I will post it asap.

    Thank you again for you time!!

    Hanna
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