Hello everyone,

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

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

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

given the identity

$\displaystyle s_x(x)=0$.

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

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

I got to the point where I need to show that

$\displaystyle (y-x)\bigl\{\frac{\partial}{\partial x} [\frac{\partial s_x(y)}{\partial y}]_{y=x}+\frac{\partial}{\partial y}[\frac{\partial s_x(y)}{\partial y}]_{y=x}\bigr\}=0$.

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.