okay so in this final segment now we're going to consider how we can expand our univariate breeders equation to take account the complexities of in in the way that selection actually acts okay so selection very very rarely targets single traits so we can account for the complexity when making predictions about evolutionary change by simply expanding the Breeders equation to um to a multivariate form and this is what I've got here this um here is termed the um multivar breeders equation and here Delta Zed um describes the change in the phenotypic value of um of a given trait across one generation of selection okay so in a way this is like our response to selection in the previous one okay however where things get much um more um interesting is when we now instead of thinking about heritability so the addtive genetic variance the trait we think about the Matrix a matrix of additive genetic variances and co-variances which is what we term the G Matrix okay so G denotes a matrix of um variances and covariances for all the traits of interest in our analysis and B as before denotes the strength of selection um occurring on those um uh different um components of um phenotypic variant okay so to account for the genetic variances the multivariate Breeders equation it basically invokes a matrix equation and as I said the G Matrix here is the Matrix of additive genetic variances and co-variances and that here is pictured at the bottom okay I'll go back so um here we've got a effectively vectors for our uh Matrix equation um which we call Delta Z 1 so that's the change in in trait one once we take account of the variances for that trait given as um the diagonals on this Matrix equation in the middle and also the covariances with other traits that might be experiencing selection okay so if we look now uh we can see that if we draw um shade across or shade the diagonals these are the additive genetic variances and the causes of direct selection on our trait of Interest okay so um so to use the example of G11 this effectively is one and one so it's the same trait okay so when we then move on to the off diagonals we can see so g12 are the covariances for um trait zed1 and Z2 and G13 is zed1 and Z3 now these are the additive genetic covariances and they are the causes of course of indirect selection so now we can ask well what factors might constrain evolution responses we're now in a much better and stronger position to consider how selection actually acts once we take account of covariances with other traits well there are a couple of factors that should come to mind first of all a lack of additive genetic variance underlying the trait of course will constrain an evolutionary response we've already dealt with that in the opening lectures of this um of this portion of the unit but what we now need to consider is that there are additive genetic covariances with other traits which are identified in our G Matrix so let's now just consider a very simple case where we've got two traits uh of Interest okay let's imagine two traits that have no genetic correlation at all so here what I've done is I've done this imaginary plot of what do we term breeding values these could be basically the family values for a trait of Interest so these are if you like um the variance across these different families is an indicator of the additive genetic variance under the trait so we've got trait one here plotted against trait two as you can see there's no genetic correlation between these traits so we can just simply fit a circle around this and the um phenotypic mean for our population here is given by where the crosses intersect for both traits okay so now let's consider we have a fitness surface here so this is like a little mountain if you like a landscape where the optimum Fitness is at this Peak at the top okay Okay so we've got these gradients we have our relationship that we have depicted here with a little circle little ellipsoid um for trait one and trait two and as we can see there's obviously no constraint here for phenotypic evolution towards the optimum we have additive genetic variance in both traits and no co-variance so nothing constraining an evolution response can go along any direction in the as in the direction of selection occurring so selection is obviously pushing these populations towards their Optimum phenotypic values for both traits which is depicted by that red dot at the top of the fitness Peak but now let's consider a case where we see two traits with a positive genetic correlation so trait one and trait two positively genetically correlated here okay so if selection acts on trait one trait two would also change now bearing in mind what I've already said it should be fairly clear that if selection occurs exclusively on trait one this is not correlational selection this is exclusive on trait one then trait two moves um according to selection because of indirect selection okay so it's not actually experiencing selection and same would go for trait two so clearly here we could we've got um direct and indirect selection occurring on trait one so okay we can put our ellipsoid around this again and then we can place this little population on the same adaptive Peak and as we can see here we now have a constrained evolutionary response okay our population is obviously going to move in the axis where there's most most genetic variants available for selection and this is not necessarily directly in line with where that population needs to be in terms of its adaptive Fitness Peak so as you can see um even a positive genetic correlation between traits experiencing selection indirect selection can constrain an evolutionary response now clearly negative genetic correlation between traits might constrain a response altogether so if the optimum value for that trait is actually moving even slowly our hypothetical population here might never actually reach its adaptive Peak despite experiencing very strong selection on that on those traits and despite of course there being additive genetic variants underlying them okay so we're going to end our lecture here uh I'm going to suggest some reading here um there's a very good um and readable background to all of the topics that are covered here in the um book a primer on of ecological genetics by Connor and Hart uh again that's been suggested throughout my lectures as a very good and um easy background to some of this material and if you want a little bit more insight into the G Matrix I've included a web link here which will take you to a nice um starting point for your reading thanks very much