Hello everyone,

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

Given that

is smooth and that

is near

, I want to obtain

given the identity

.

Here

is the so-called invasion exponent, i.e. the long-term population growth rate of a mutant population with trait

under environmental conditions as set by the resident population with trait

. So in particular, the subscript

is not a derivative but rather implies that

(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

,

I got to the point where I need to show that

.

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.