How to replace a complex term in an equation using a function?

Question

I have recently been working on a few models that look at mosquito predation. Now one of the peers wants me to add the complete equation of my model in the manuscript. I previously had the equation broken down in several parts but the reviewer was not happy with that. So I decided to add the full equations, here is an example:

$$N_{e}=N{}_{0}-\frac{W\left(\frac{a}{1+c(P-1)+abcN_{0}(P-1)}bN_{0}\exp\left(-\frac{a}{1+c(P-1)+abcN_{0}(P-1)}(PT-bN_{0})\right)\right)}{\frac{a}{1+c(P-1)+abcN_{0}(P-1)}b}$$

The editor of the journal now told me to break it down again. This are his exact words: "...there is one complex term that occurs three times. This term could be replaced by a function name, e.g. F(x, y, z, …), and then F(x, y, z, ..) defined in the following line."

I'm not really sure how to write this because I am not very good with mathematical notations, symbols and all that. Is the editor basically asking my to do the following?:

$$N_{e}=N{}_{0}-\frac{W(F(x,y,z)bN_{0}\exp(-(F(x,y,z)(PT-bN_{0})))}{(F(x,y,z)b}$$

and then: $$F(x,y,z)=\frac{a}{1+c(P-1)+abcN_{0}(P-1)}$$

If this is what he is asking I don't see how this differs with my first version in which I had the following: $$N_{e}=N{}_{0}-\frac{W\alpha bN_{0}\exp(-(\alpha(PT-bN_{0})))}{\alpha b}$$ combined with: $$\alpha=\frac{a}{1+c(P-1)+abcN_{0}(P-1)}$$

Answer 1

Say something like this:

For notational convenience, let $$F:=\frac{a}{1+c(P-1)+abcN_{0}(P-1)}.$$ Then our model is $$N_{e}=N{}_{0}-\frac{W(F\,bN_{0}\exp(-(F\,(PT-bN_{0})))}{F\,b}.$$

Here, the $:=$ means "is equal to be definition".

It's not clear to me if variables on the right-hand side of $F$ are indeed independent variables or parameters. If variables, I'd say $$F(a,b,c,P,N_0):=$$ instead of $$F:=$$ but they feel like parameters to me. And if they are indeed parameters, I'd probably choose a different letter than $F$ to represent that quantity.

You're right, this isn't any different than what you had using $\alpha$ unless the point of this is to treat $\alpha$ as a function of those variables on the right-hand side of the defining expression for $\alpha$ (or $F$).

Answer 2

You have same things written three times : it s a function F

the function depends on a,b,c,$N_{0}$, and P so your function can be written $F(a,b,c,N_{0},P)$

$N_{e}=N{}_{0}-\frac{W(F(a,b,c,N_{0},P)bN_{0}exp(-F(a,b,c,N_{0},P)(PT-bN_{0}))}{F(a,b,c,N_{0},P)b}$

or

$N_{e}=N{}_{0}-\frac{W(FbN_{0}exp(-F(PT-bN_{0}))}{Fb}$ where $F=\frac{a}{1+c(P-1)+abcN_{0}(P-1)}$

Note you have on extra ')' at the end