solutions of period three for a non-linear difference equation

Abstract

The paper uses the factorisation method to discuss solutions of period three for the difference equation which has been proposed as a simple mathematical model for the effect of frequency dependent selection in genetics. Numerical values are obtained for the critical values of a at which solutions of period three first appear. In addition, the interval in which stable solutions are possible has been determined. Exact solutions are given for the case a = 4 and these have been used to check the results.

Solutions of period four for a non-linear difference equation

“…Another result which arises from equation (5.1) is that 5 = ± 81 for a = 4. It was noted previously [4] that for a = 4, equation (1.1) has a general solution x n = cos(3"4>), where x 0 = cos 4>, and this leads to cyclic solutions for suitable values of <j>. In particular, jc n+4 = x n when (i)<?> = Mr/41, or (ii) < f > = NTT/40, (5.11) for any integer iV.…”

“…The occurrence of a single root for a polynomial of even degree, as recorded for 2 < a < 3.55, looked a little bit strange but in fact it was accompanied by a negative root which we can label Y v For a = 2, Y 1 was -2.68 and it increased smoothly with a, passing through zero around a -3.85 and being identified after that as a small positive root. We can think of Y l and y 4 as a pair of roots which occur throughout the interval [2,4] even although Y t is negative for part of this range and hence of no interest as far as real C4 solutions are concerned. Indeed it turns out that the Y 4 root does not always give real solutions either and it is this property which saves us from having C4 solutions for a < 1 + y/5~.…”

“…Solving this equation directly would be a formidable proposition and the aim of the factorisation method is to split the problem into two simpler steps. As the factorisation method has been applied in earlier papers [2,3,4], it will be summarised fairly briefly in this paper.…”

“…We can take b^jb^ as a nonzero factor (to avoid the equilibrium solution b t = 0) and expand the product on the right-hand side. This gives 1 = a 4 D + a 3 (l -a)C + o 2 (l -a) 2 B + a(l -a) 3 A +(1 -a) 4 . We can obtain a second relationship between fi x and /?…”

“…The present paper is an extension of an earlier one [4] in which the motivation for the work was discussed in the first two paragraphs. This followed May [6,7] who linked the equation *»+i =F (x n ) = axl +(1 -a)x n (1.1)…”