I am pretty bad in thinking quantitative genetics models. I am trying to get some basic understanding of modelling the evolution of a quantitative trait. I am therefore asking for help to analyze a very simple model. I welcome any explanation of some other classic model of the evolution of quantitative traits.
Scenario
Consider a haploid population of constant size $N$. The fitness of the individuals is determined exclusively by a single quantitative trait $z$. $l$ loci codes for this trait. The genetic value at each locus adds up to give the quantitative trait $z$. The mutation rate at each locus is $\mu$. A mutation changes the genetic value at any given locus by $+ \Delta z$ with probability $\frac{1}{2}$ and by $- \Delta z$ with probability $\frac{1}{2}$. If you prefer to consider the effects of a mutation to be drawn from a normal distribution with mean $=0$ and variance $= \sigma^2$, please feel free to do so. The fitness $w(z)$ of an individual is given as a gaussian function of its quantitative trait $z$
$$w(z)=\exp\bigl(-Vz^2\bigr),$$
where $V$ is the strength of selection. The optimal phenotype is $0$, that is $w(0)=1$. The larger is $V$, the more severe it is to be at a given distance to the optimal phenotype. If you want to assume that the fitness $w(z)$ is some other function of the trait $z$, such as the even more simple $w(z)=z$, please feel free to do so. For simplicity, we assume no environmental variance on the quantitative trait.
Questions
What is the equilibrium mean fitness $\bar w$ of the population?
What is the equilibrium genetic variance (=phenotypic variance) for fitness $w$ and for phenotype $z$?