Unequal Crossover Dynamics in Discrete and Continuous Time

Abstract

Abstract. We analyze a class of models for unequal crossover (UC) of sequences containing sections with repeated units that may differ in length. In these, the probability of an 'imperfect' alignment, in which the shorter sequence has d units without a partner in the longer one, scales like q d as compared to 'perfect' alignments where all these copies are paired. The class is parameterized by this penalty factor q. An effectively infinite population size and thus deterministic dynamics is assumed. For the extreme cases q = 0 and q = 1, and any initial distribution whose moments satisfy certain conditions, we prove the convergence to one of the known fixed points, uniquely determined by the mean copy number, in both discrete and continuous time. For the intermediate parameter values, the existence of fixed points is shown.

Erratum and addendum to: The general recombination equation in continuous time and its solution

“…Note that the function b U t generally does not emerge from b S t via marginalisation in the sense of Eq. (22), which is another important difference to the special situation of Section 4 and Remark 10.…”

“…This made the γ-term independent of D, which in turn allowed the last step on the basis of Eq. (22) and another application of Lemma 3. The second-last step shows that the marginalised family indeed satisfies the proper ODE for the subsystem as defined for ∅ = U ⊂ S via Eq.…”

“…for any partition A ∈ P(U ), in complete analogy to Eq. (22). Clearly, one has q U (A) 0, while proper normalisation follows from…”

Freely forming groups: Trying to be rare

“…Consequently, we shall look at the deterministic time evolution of the corresponding (discrete) probability distribution [p, \ 1 < i < M) of the types. This setting is known as the infinite population limit (IPL), compare [6,Chapter 11.2] for background material and [2,10,14] for recent examples. For a finite population, p, = rii/N are the type frequencies.…”

The general recombination equation in continuous time and its solution

“…Note that we assume the sequences to have fixed length. Additional processes that may change this, such as copying blocks, are disregarded here; see [22] and references therein for possible extensions.…”