 Okay, usually I give this sort of presentation at the beginning when we start talking about spatial structure that is, before I talk about beta diversity, but at this point it will be just fine in any case. My main message here is that spatial structures are very important for the understanding of processes in ecosystems and actually this goes back to the 1980s and early 1990s where the general recommendation of statisticians was to remove spatial structures from data because they created autocorrelation in data and that was bad. Well, I wrote a paper in 1993 I think in ecology where I said, wait a minute, spatial structures are very important for the functioning of ecosystems indeed can you imagine an ecosystem where organisms would be located at random and I described for half a page the consequences, the absurd consequences that this would lead to so I said after that if you remove the spatial structure before analyzing your data you are throwing out the baby with the water of the bat and we should not do that if we want to understand what is really going on in ecosystems we should keep the spatial structures into the analysis and the spatial structures even if we analyze them for their own sake there are a proxy information for processes that are going on in ecosystems and that paper turned out to be one of my most highly cited papers it seems that the message has been heard and now statisticians are very quiet about their previous suggestion of removing spatial structures before doing anything else we have not heard about that so I think that now the new trend is to keep the spatial structures in the analysis following the geographers who had been doing that for a longer time a longer time than ecologists so I will here describe why spatial structures are important and what can be their origin spatial structures can happen in different ways and we will examine some of these ways now this the study of spatial structures is well considered and it is included in the name of two branches a branch of ecology and the branch of genetics called landscape ecology and landscape genetics where we study the spatial variation of species composition or of genetic structures of individuals or local population throughout the landscape and here that variation is called beta diversity and I think that in genetics it can be called also genetic beta diversity now why do we want to understand and model a spatial community structure when we study species assemblages observed at sampling sites and we know that this is it takes a lot of effort planning energy time money to do that why do we do that are we sick in our head to go to all this trouble going to sample the rebated mites in clouds of black flies and so on you know it is a hard job that we are doing why do we do that it's because ecologists firmly believe that species assemblages are the best response available response variable available to us to estimate the impact of changes in the ecosystem especially in our so-called modern times where the impact is produced by man and these impacts can best be studied by asking the species that live in ecosystems in the old days government agency that wanted to know something about changes in ecosystem would ask an ecologist to come along and they would bring him to some area and they would walk around the area and they would say very interesting yes maybe there was some change oh there is a beaver there he must be responsible for the change well we don't do that in this way anymore we send graduate students full of energy to the field to collect mites or insects or plants or look at birds mammals and so on and we ask the organisms that live in the ecosystem what they think about the ecosystem and by redoing the sampling at different times we can measure then the response of the species to the changes in the ecosystem that we would do it in modern days so it is the best response variable available to us it is better than sending a so-called specialist to tell us everything about it the difficulty of course is that species assemblages form multivariate data tables and we have seen in this short course how we can analyze multivariate data tables in different ways by pca but not in any possible way we have to do it in a careful way after transformation of the data and our careful attention that has to be given to the way these analysis are carried out and we tried to give you some information some basic ideas about how to do these analysis in this short course also for testing it is more difficult to test hypotheses from multivariate data tables but we have seen that we can do that very well using canonical analysis in landscape genetics it is the same thing but with genetic data describing local populations observed at georeference sampling sites okay so we talk about beta diversity in general as the variation in species composition among sites we can look along temporal gradient temporal series of sampling the same site and talk about temporal beta diversity and in genetics we can talk about genetic beta diversity this term is becoming to be used to represent the sort of diversity through space or through time okay now we will look at the kind of organization that we can have of the of the communities through space what kind of spatial structures do we have this is a prairie of daisies it was I took this picture in the small island off the coast of britain and that represented for me something close to a random distribution of daisies of the points when I saw that prairie I said yes I must take a picture of that now there is a trick to that picture it looks very homogeneous and random but otherwise homogeneous now when I took that picture I was standing in a small gravel road here and I knew that in the back there was a hedge I did not frame I framed my picture so that you couldn't see that the front on the left there was another dirt road I did not include it in the picture and on the right there was a medieval church that I did not include either so I only took the picture at that at that scale this few maybe four meters four or five square meters of the prairie and that looked random so my message to you is that if I had taken the picture by zooming out or by packing up I would have included these other items of the landscape and you would have seen that it was not a uniform distribution at all it was a patch so a uniform distribution or a patch it depends on the scale of perception so the scale at which we do our observation is very important and this is my first introduction to the idea of scale that we will look at in more detail tomorrow with multi-scale analysis but this is the message of that picture our yeah our perception varies with the scale of observation sometimes we have spatial structures that appear this is a picture that I took at Stonehenge in southern England Stonehenge actually this is a drawing that was on the side of the road the when it was originally constructed about 5000 years ago it was double or triple circle of stones where you have standing stones related by lentils here and these pieces of rock have holes that fit into pins of the of stone left into these carved stones and the whole thing is solid because of this pinning of these stones into that so that was the structure and it is one of the morals of the world to still have access to this structure standing 5000 years after it was built and there are big discussions among archaeologists as to what was the purpose of this circle of stone especially after they found out that each of these stones had been carried from kilometers away you know where each stone comes from from the chemistry of the stone and so why did they do with the did they go through all this trouble to construct this structure so it is very nice but when you visit it this is one of the standing stones and you have a lentil on top of it here it is a bit eroded after 5000 years but what do you see in there you see in this spatial structure hey it's right in front of you in the idea lichen right uh-huh you have lichens here at the top of that stone and nothing at the bottom why is that as soon as we I ask why you will immediately try to imagine reasons that could explain that for instance is it that the cows in the pasture are rubbing against the stone maybe but maybe that was the case in the old days but not anymore because now there is a fence around it the cows are kept away could it be that the top of the stone is protected by the lentil and when there is dry temperature rare in the south of england uh then maybe this dries out but this remains wet because of this cover I don't know maybe or there may be some other reason maybe there is some ghost that comes during the night and rubs off the lichens maybe and we are approaching uh halloween you know it's important uh it's all sorts of reason whatever the reason is that we have a spatial structure and we try to imagine reasons why we have these spatial structure and because we can discuss them endlessly or we can design experiments to be carried out not directly on these stones but on makeshift uh stones outside the stone circle of course and see if we can reproduce the phenomenon under different circumstances or in different countries or what so it's fun it's interesting to do uh and what I will try to describe here is that there are uh that spatial structures in community and indicate that some process has been act towards to create them that's why we are interested in spatial structure it's because there is a process and we can try to discover this process and when we have a hypothesis that we have numerical ways of testing these hypotheses it doesn't make sense that it is this process that created the spatial structures that we observed so spatial structures are a proxy information and basic information that tells us that something has been going on and it is left to us to discover what this process is so it is a game of detective more or less that we are doing here is a picture representing a small area uh in in kebec it is near a large lake large called the two mountain lake and uh here is a small town called Figo along the highway going from Montreal in that way to Ottawa in that way and uh in the small village we could imagine that the graduate student would might want to do a sampling for a master's thesis for instance and here I got this map from google maps and I added a red line on the map that could be sampling transect and but then why would anyone choose this sampling transect if one was studying let's say soil insects or birds uh yeah not vegetation because it is too artificial but let's say soil insect or birds it would be a good choice this transect because it goes into three different environments this is a small mountain with trees so here is in the uh forested portion this portion is in town and this is in the agricultural portion you can see the fields there so uh at the broad scale along this transect you may expect to have variation in the birds that you will observe or the insects that you will find because the environment is very different right so this is simply calling upon the environmental control or environmental filtering model that tells us that species have different optima so we might expect to have different species there there and there indeed when you are designing uh uh field sampling like this you want to be sure that you will have variation variation is our friend imagine if you sample on the transect where there is no variation what sort of a paper are you going to publish at the end nothing so we need the variation so when we plan the sampling we have to make sure that there is variation now you can also expect to have variation within the forested portion within the town and within the section in agricultural fields because we know from experience that community composition is not exactly the same at all points along the transect even if the environment is homogeneous in this portion or that or that we may expect variation here due to other processes it may not be uh environmental differences it may just be the community the dynamics of the community that will generate differences in species composition so we invoke another part of our knowledge in ecology when we we say that we expect to have this sort of variation and I will describe that a bit for more formally here uh and they are maybe forcing by the explanatory variables the environmental variables and this the environmental variables may be responsible for the spatial structure found in species assemblages and we will call that induce spatial dependence meaning that the spatial structure is induced by the dependence of the species on the environmental variables and that this is for the broad scale but then community dynamics may also generate spatial structures creating auto correlation in the response variable true auto correlation is created by the response variables themselves by their dynamics and mechanisms can be the neutral processes that have become very popular since 2001 uh since publication of different models called neutral models in ecology that are derived from the neutral models in genetics by the way so it can be these neutral processes with mechanisms such as random drift and limited dispersal when species reproduce then for plants the seeds and so on may fall close to the parent or be carried some distance away by wind or by animals and so on but there is still limited dispersal that create patches in species distribution same thing for animals they may disperse a short distance away from the parent there may be interactions among species that create also spatial structures so these spatial structures generated by community dynamics are usually at more at finer scale than the spatial structures generated by the environmental variable not always but many cases they are at finer scales here's a summary picture that summarizes the set of processes at least some of the processes that may be going on this is actually a shorter picture from another picture with many more slices that is available at the CTFS site the Smithsonian group that study entirely sampled forest where they have a picture like like that with more slices here you have a regional species pool you can imagine that as the cloud that we are using now with computers where we send data in the cloud and then we get our data back from the cloud if we lose them so you can imagine a cloud above an area with all the species in the cloud and then some of the these species from the regional species pool may have seeds or young animals and so on available that comes into the area and there will be the random sampling filter that is that some of the species will be available and will pass through this filter and other species will remain in the regional species pool and will not make it by random sampling from the regional species pool not all species not if you are working with vegetation not you don't have the seeds of all the species that fall at every one there is random selection from the regional species pool by whatever process you may imagine then after that there is an abiotic filter again with vegetation a seed falls here it may germinate if the conditions are appropriate or it may not germinate at all because the conditions are not appropriate so this would be the abiotic filters some species are stopped there others make it true then there is the biotic interaction filter that makes it possible for some species to survive and others are stopped there so from the regional species pool and in the local species assemblages you end up with only a few species okay so you have these different mechanisms that filter the species step by step that's the general idea uh then uh oh i think i should stop very quickly well there is a sort of simulation here in my uh in the presentation where i have four ponds and in which i have one species y and one environmental variable x very quickly here in the null situation the ponds are not connected and nothing much of interest happens but this is the null situation the null model that serves as a comparison for all the other models you could generate data like that by generating for instance uh random deviates and for species assemblages they can be we can use the exponential distribution to generate data and random exponential distribution to generate data like the species composition for that and here for environmental data you can use random normal distribution for instance to generate data that look like that but then things may differ if you add the idea that the environmental conditions have an influence on the species so here if you have that the basic information is what kind of the values you have in the environmental variable in each of these ponds and that will condition the species assemblage and you can analyze that using a regression equation where you will say that the value y let's say here is a function the linear function of the values of x plus some random component the next model here model 3 does not have this vertical arrow it simply says we open the gates here in the channels connecting the ponds and we say that some water may flow from pond to pond carrying individual if you are looking at zooplankton or phytoplankton some of the plankton from that pond may flow into this one some of the the plankton from this may flow there and so on so that will create autocorrelation in the data by direct flow of individuals from that pond to that pond and then the equation modeling it is very different it simply says that the value here depends on the value there with the regression coefficient plus some random innovation and due to the composition in that pond which is there so this is very different from a regression equation the next model is to say that we have this process of autocorrelation in the x plus transfer of the influence of x into the y this leads to a bit more complex and simulation equation where you will have this sort of equation to to model values of x and then the regression equation that we had at the top to model this influence and finally what we have in real data is a composite of these two last models that is we may have x that is autocorrelated and influence is y but y that is also autocorrelated so generating data like that can be done with these equations but when we have real data and we don't know what are the processes at work we may have to do analysis to separate the influence of x and y from the autocorrelation in the y and we will explain tomorrow how this can be done but it is based on essentially on the idea that in many cases the spatial structures found among the y due to autocorrelation are at finer scale than the influence of the x on y because x is generally and generally has a spatial structure at broader scales so by doing a multi-scale analysis we may be able to distinguish partially the influence of x and y from the influence of the random processes random neutral processes going on in the y so the sort of analysis that we will explain tomorrow allows us to do that for the first time and this is why we will describe them to you it may give you ideas for your own analysis