# Is it possible to calculate the evolutionary response to nonlinear selection gradient as a change in variance of phenotypes?

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:

$\boldsymbol{G}^* -\boldsymbol{G}=\boldsymbol{G}(\boldsymbol{\gamma}-\boldsymbol{\beta}\boldsymbol{\beta}^T)\boldsymbol{G}$

• By non-linear selection, do you mean cases where fitness of an individual is a non-linear function of its phenotype or do you mean cases where fitness of an individual depends upon the genetic diversity in the population? – Remi.b Aug 22 '18 at 12:37
• That's what I mean: $w = \alpha_0 + \beta X + \frac{1}{2}\gamma(X-\bar{X})^2+ \epsilon$ w: fitness, $\beta$ is the linear selection gradient and $\gamma$ is the nonlinear selection gradient. Like in Lande Arnold 1983. – M. Beausoleil Aug 22 '18 at 12:40
• Now, if you would predict the response of selection on the next generation of phenotypes, you can use the breeders equation as shown above. But what about the nonlinear selection gradient $\gamma$? Can we predict it? – M. Beausoleil Aug 22 '18 at 12:43