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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?


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up vote 2 down vote accepted

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.

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Thanks! Any advice on how to learn to see that? It looks easy once you point it out. – Tyler Dec 1 '11 at 4:30
Well, you said that your version matches numerically with the one in the paper. So, that suggested to me that you are unlikely to have made error in algebra. The rest is just a matter of working with the expression to see if you can factor it in a meaningful way. I had the advantage of knowing what the final answer is. But, in general, if you have a polynomial then factorization and then re-arranging to get an interpretable form is useful to try. – tards Dec 1 '11 at 4:42

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