Derivation of Viability Selection Difference Equation

[jp5125]

Hi all,

This probably isn't a super tough question for any of you to answer, but I'm stumped on this and could use some help. I am an evolutionary biologist and my background in mathematics is solid but by no means phenomenal. I am delving deeper into mathematical modeling of evolutionary phenomenon and am working through Richard Mcelreath and Robert Boyd's book "Mathematical Models of Social Evolution". I have taken up to pre-calculus and loads of advanced multivariate statistics courses, but difference and differential equations have been beyond my educational reach thus far.

My question is derived from a screenshot I've attached from McElreath and Boyd's book below. Where I get confused is how do I go about simplifying the equation in the middle of the page [after you multiply the top and bottom of the far right term by pV(A)+(1-p)V(B)]... The book just says "after simplifying, we obtain {the final difference equation}", and I can't seem to figure out they got to this final equation. This might honestly more of a math fundamentals question than calculus level question but given that my question exists within a difference equation I figured someone here should be able to assist.

Thanks in advance for your replies!

 

[Cubist]

This is simpler than you're (probably) thinking. It doesn't involve any calculus. Just think of V(A) and V(B) as simple variables. Notice that the denominators are the same therefore we can write...

[math] \Delta p = \frac{ p\color{red}V(A)\color{black} - p\left(pV(A)+(1-p)V(B)\right) }{pV(A)+(1-p)V(B)} \,[/math]now move the highlighted term inside the following bracket, being careful with +/- signs...

[math] = \frac{ p\left(\color{red}V(A)\color{black} - pV(A) - (1-p)V(B)\right) }{pV(A)+(1-p)V(B)} [/math]
can you continue, hint:- factor!