Signature of a selective sweep in a large population of varying size

In the absence of other factors, an individual with an advantageous version of a gene will generally survive longer and have more offspring. Given enough generations, this can result in the fixation of this advantageous version. This is called a selective sweep, as the gene under selection quickly increases in frequency in a population.
In this talk, I will describe the genetic signature left by a selective sweep in a population described by a multitype birth-and-death process with density dependent competition. I will study the limit behaviour for large $K$, where $K$ scales the population size. I will first focus on two loci: one under selection and one neutral, and distinguish a soft sweep occurring after an environmental change, from a hard sweep occurring after a mutation. I will derive the expression of the neutral proportion variation as a function of the ecological parameters, recombination probability $r_K$, and $K$, and show that for a hard sweep, two recombination regimes appear according to the order of $r_K \log K$.
If I have time, I will also mention the three locus case, when we consider two neutral loci and describe the neutral genealogies during hard sweep.
These results can be used to detect recent selective events in current population genetic data.