I'm trying to reproduce the analysis of a simple biological model from an old paper. The model describes the frequency of two organism types in a population as:
$p_{t+1} = \frac{p_t ( s + (1 - s)p_t)}{s + (1 - s)(p^2_t + q^2_t)}$
Where $p$ is the frequency of one type [0-1], and $q = 1 - p$ is the frequency of the other. $s$ is the frequency of selfing, which is also [0-1].
To convert the recursion equation to the corresponding difference equation I substitute the left hand side above into:
$\Delta p = p_{t+1} - p_t$
$\Delta p = \frac{p_t ( s + (1 - s)p_t) - p_t(s + (1 - s)(p^2_t + q^2_t) )}{s + (1 - s)(p^2_t + q^2_t)}$
$\Delta p = \frac{(2 s - 2) p^3 + (3 - 3 s) p^2 + (s - 1) p}{s + (1 - s)(p^2_t + q^2_t)}$ (or many similar variations)
This is where I hit a snag. My answer is correct, but not terribly useful. In the original paper, the difference equation is reported as:
$\Delta p = \frac{2p_t q_t (1-s)(p - \frac{1}{2})}{s + (1 - s)(p^2_t + q^2_t)}$
This is also correct, and more importantly it's a useful answer. It can easily be interpreted biologically: the first two terms of the numerator are always positive, meaning that $p$ increases whenever $p > \frac{1}{2}$; in other words, when $p > q$.
My answer and the published answer are numerically identical, but I can't figure out how they got their equation into the form it's in. Can anyone help me with strategies for manipulating equations to produce intepretable forms?
Thanks!
You have:
$$\Delta p = \frac{(2 s - 2) p^3 + (3 - 3 s) p^2 + (s - 1) p}{s + (1 - s)(p^2_t + q^2_t)}$$
Re-write the numerator as follows:
$$(2 s - 2) p^3 + (3 - 3 s) p^2 + (s - 1) p = p (1-s) (-2p^2 +3p-1)$$
Now factorize $(-2p^2+3p-1)$ as $(2p-1)(1-p) = 2(p-\frac{1}{2}) q $
Put together everything and you get the desired result.