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 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.

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

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.

Expand about :

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

so:

Hence dropping the common (non-zero) factor:

(If you keep track of the order of the error in the approximations you will find that the can be replaced with equality; the error in the second from last expression is if is twice differentiable and a bit more complicated otherwise but the argument will still go through)

CB

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

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

' to generally conclude that

where we can also have `evaluated at

' for some

?

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

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