I often see the breeders equation written like this:
$R = h^2s = \frac{V_a}{V_z}s = V_a\beta$ or $\Delta \bar z = G\beta$
But is it possible to get something like this:
$\Delta \sigma_z = G\gamma$
In this case, it would be the change in variance of the population due to nonlinear selection.
Basically, it would me that the response of selection is a direct measurement of the change in $V_z$. So to come back to this equation: $R = \frac{V_a}{V_z}s$, it would mean that $\gamma$ is related to $V_z$ somehow. But is there a way to predict what would be the response?
Is it possible to also predict the total response of a non-linear selection gradient if there are multiple traits?
When one wants to calculate the total response to selection, it is needed to account for the correlated effect of selection on other traits (correlational selection): $R_1 = V_{a,1} *z_1 +\text{COV}_{V_{a,12}} * z_2$
$R_2 = \text{COV}_{V_{a,21}} * z_1 +V_{a,2}* z_2$
In matrix form:
$ \begin{bmatrix} R_{1} \\ R_{2} \end{bmatrix} = \begin{bmatrix} V_{a,1} & \text{COV}_{V_{a,12}} \\ \text{COV}_{V_{a,21}} & V_{a,2} \end{bmatrix} * \begin{bmatrix} \beta_{1} \\ \beta_{2} \end{bmatrix} = \boldsymbol{R}=\boldsymbol{G} \boldsymbol{\beta} $
How would that be translated to non-linear selection?
EDIT:
I think this answer is here:
$\Delta \sigma_a^2 = \frac{\sigma_a^4}{2}(\gamma - \beta ^2 )$
But this equation predict a change in genetic additive variance not phenotypic variance. Would it be possible to simply take $h^2=\frac{V_a}{V_z}=\frac{\sigma_a^2}{\sigma_z^2}$ and get the change for phenotypic variance?
This could also represent an answer:
$\boldsymbol{G}^* -\boldsymbol{G}=\boldsymbol{G}(\boldsymbol{\gamma}-\boldsymbol{\beta}\boldsymbol{\beta}^T)\boldsymbol{G}$