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

• November 26th 2011, 01:38 PM
Hanna87
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.
• November 26th 2011, 11:31 PM
CaptainBlack
Re: How to deduce ds/dx+ds/dy=0 from s_x(x)=0
Quote:

Originally Posted by Hanna87
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
• November 27th 2011, 02:29 AM
Hanna87
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

Quote:

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

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

Quote:

$0\approx \frac{\partial s(x,y)}{\partial y}+ \frac{\partial s(x,y)}{\partial x}$
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
• November 27th 2011, 05:57 AM
CaptainBlack
Re: How to deduce ds/dx+ds/dy=0 from s_x(x)=0
Quote:

Originally Posted by Hanna87
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
• November 27th 2011, 06:52 AM
Hanna87
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