 And this is a joint work with a number of colleagues. So most important, Friedrich Lenz fit his PhD with me exactly onto this project. So I'm going to tell you about his work. And Thomas Ings and Lars Cytgar are two experimental biologists, colleagues of mine at Queen Mary. So they did all the experiments on bumblebees. And Alexei Chechkin is an expert on stochastic theory. And my expertise is also from a theoretical point of view. So that was a joint collaboration. And as this workshop is quite cross-disciplinary, and I feel perhaps a little bit as an outsider, I thought I'd give you on one slide just a little bit of my scientific background that you know what to expect. So my background is statistical physics. Interfacing was applied mathematics. So I'm in a math department at Queen Mary. And my field is really non-linear dynamics, stochastic processes, and non-equilibrium statistical physics. And more recently, I like to apply this to nanosystems, and especially to biological dynamics. But here I have some pictures, so I'm interested in things like fractals, diffusion and dynamic systems, as well. And as I'm very interested in diffusion, there was a natural cross-link then to biological implications. And that's what I started to study about 10, 15 years ago, and I'm getting more and more into it. And in fact, it will come to exactly this picture to the end. You will see it again. And so you will see a bumblebee that performs a flight. And indeed, this is a trajectory generated from a model. And I will explain to you how we did that, at least on one slide. And now, as we're talking about biological dynamics, I may start with this cartoon to introduce the basic theme of this talk. Because what I'm interested in is really analyzing search patterns, biological search patterns. And as an example, this is from a nature paper by Shupur, Benishu, and others. You may think of an autumn that you're out in the woods searching for mushrooms. And as a certain physicist, I would like to model this problem. And now the most knife approach, well, we have already heard about it, is, OK, you put everything into a quadratic grid in the plane, quadratic grid. And now you put the mushrooms randomly on some nodes of the grid. And then you model the searcher as a point particle that moves with probability 1 quarter to the left to the right, up or down. Fine. OK, now you have a search problem. And that's what me, the physicists, like to play around with. And you can solve this or put it on a computer. Now, if you want to do better, then if you want to go to continuous time in space, you may model the search, say, by a groin in motion. So particles, say, moving in a fluid here in 2D. And again, looking for randomly distributed targets. So far, so good. But I mean, this workshop is on locomotion and navigation. And this has absolutely nothing to do with locomotion and navigation. So it's a point particle moving according to some rules, finding some point target. So this is nice because you can solve this analytically, but I'm not sure whether it tells you very much. And now, if you look at search problems, very quickly they can become very complicated. The problem is that then we can solve them anymore. But they are more real, they are closer to vitality. So to give a flavor, certainly one thing is whether you want to find all of them or only perhaps one of them or a few of them. Finding all of them, for example, was done exactly in this paper. It's called cover times. So you can develop then series of how to efficiently find all of the targets in a certain region. Finding, say, one of them is mathematically a problem of first passage or first arrival. Again, you can solve this analytically in certain situations. But now the question is also, you may wish to do efficient search in terms of optimality. But what do you mean with efficient? Next question. Efficient may mean you want to minimize the search time. And again, this is the problem that you can solve under certain circumstances. But this is not all because now they're coming into play, locomotion, navigation, and perception. So certainly, this is not just a point particle reality. Looking for some targets depends on your locomotion of whether you find them or not. If you move very quickly, you may easily miss them. Then sensory perception. You see this person has a certain range of perception. Of course, it depends on your perception of how you interact with the environment, whether you find targets easily or not. Another problem is, for example, search form depends on whether, here, if you pick a mushroom, it's gone. If you're punishing a target, you have a completely different sort of problem. OK, I stopped here. I could go on, of course. But you see, I'm covering the whole range from extremely abstract mathematical models that we are happy to solve up to models that are of interest to most of you, I suppose, as if you are experimental biologists. And so this is a problem here to bring this together. It's not easy. And one has to watch out here that one established a clear communication because, if you solve the search problem, it may not be what you are interested in. Now to summarize, and I may go through this picture. Four years ago at the Max Planck Institute in Dresden, we had an advanced study group, a very well-together statistic theorists, experimental biologists, and people doing statistical physics. And we stuck our heads together to make sense out of applying search theories to understand biological dynamics. And here you see the title, Statistical Physics Meets Movement Ecology. It's something where I may advertise Rantz talk later today, that you will talk about this in more detail, the discipline of movement ecology. So this is just a very small subset of it in view of search. So the pictures as follows. So this is the game you really need to play if you want to take things seriously. Now we have a frog moving in some environments. And what you can do and be afraid about is you put a sensor onto the frog and you measure the path of the frog of how it moves. And a certain index with the environment in a certain way. So according to locomotion, here the interaction also with respect to a certain perception and of how the environment is structured. And then certainly it also has a brain, so a certain memory. Now all what you record is then the movement ecology data meaning the path. And so this is now the experimental input. And as a theorist, now the statistical physics comes into play. What we want is to analyze the data and to construct a model by which you can reuse this data ideally. So you play the following game. Ideally, you want to capture as many details of the national environment as possible. So we are here. You may put this, for example, on the computer, do a computer simulation, and try to build a very complex model. Then you start here. But now if you want to do really theory, meaning if you want to cable it to something and obtain analytical solutions, you need to simplify. And meaning we go up here. So you see on the lowest level, we keep a lot of model complexity. But we pay a price because we can do very much apart from simulation. Now you may do what a statistical physics is called cross-graining, meaning you neglect details. You throw out information. For example, here you neglect many details of the real environment. You simplify it. And now you see we can go even further. Here we still have perception. But that's too complicated, so we kill it, we kick it out. Then we are here. We only keep, say, memory and interaction with the environment. And here, even further, you see now we have eliminated any memory. This is, in a way, a point particle moving in an environment. And if you rest the arrow, we are back to the random box I started with. Then this would be a point particle that moves according to certain dynamics in some environment. And this is something you can still solve for serious. But as an experimentalist, again, you may not be very happy. But you have this whole range of models. Now you take your model and you either solve it exactly or you do simulations. And you generate, again, synthetic trajectories. Then you analyze them according to some averaging. I don't go into detail here. And you calculate observables, which, again, then you can match to your data. So this is really, in our view, that's what we came up with a game you need to play if you really want to understand the biological search problem. But you see it's quite difficult. But I will exactly illustrate this now in the following. So the general framework is what one may call the physics of foraging. And there's a very nice book with the same title by a certain group of people where they studied this. And my interest is in identifying biologically relevant search strategies by mathematical modeling. And now what I'm going to tell you that was a lengthy introduction, but I just wanted to properly set the scene, is an experiment on bumblebee foraging in a lab on a predation list. And so I will explain to you the experiment, the statistical data analysis that we did, and to the end, the stochastic models we come up with. So that's the goal. OK, so now let's get into business. Bumblebees, if they try to find food, face two very practical problems. So A, if you're out in the wild, you have a very nice meadow. And so they need to find food in a very complex landscape. However, life may not always be easy. For example, there are these crab spiders here. You see them, and they can even camouflage, so they can change their color. So in order to survive, they need to avoid the predators. And that poses the question, OK, what type of motion should they perform in order to optimize their foraging and to avoid the predators? And my colleagues, Lars Schittke and Tom Ings, they did a very nice lab experiment as follows. And it's published here in this paper. So they let Bumblebees foraging in a cube of about 75 centimeter sight length. You see here a picture. And one wall of this cube was equipped with a grid of artificial yellow flowers, so 4 times 4 grid. And each flower was equipped with an artificial nectar source, and the Bumblebees were trained to feed on them. And now they took two cameras. And with the cameras, the position of the Bumblebees could be tracked with a very high frame rate. You see 50 frames per second. And here you see a trajectory of one Bumblebee. So only one was left into the box here at the front side and then performed the foraging. The advantage of this experiment is certainly it's in the lab, so you have full control about everything. And especially here you can vary the environment in a systematic way. This advantage is certainly there's no real free flight of a Bumblebee because I was certainly told by the biologists that they can always see the walls. You see it's a very small box. So we certainly cannot conclude on a free flight of a Bumblebee, but that was not the point. The point was rather here that we could vary the environment as follows. Now Tom put on four out of the 16 artificial flowers, artificial spider models. So you see one here. So it's this white plastic thing that he glued onto four randomly distributed artificial flowers. And now you see actually there is a mechanism here by which you can gently squeeze a Bumblebee. So no Bumblebee was harmed in the experiment I was told. So there are safe flowers without the spider images and dangerous flowers. And indeed there was a training phase where the Bumblebees learned to avoid the dangerous flowers, meaning when they approached the flower, then Tom was sitting there and then squeezing them gently and holding them for two seconds. And of course they didn't like it. So they were trained to avoid the flowers. In detail, this was a very complex experiment. It consisted of seven stages for our analysis. We were only interested in three of them. Spider-fee foraging, so no spider images. Then foraging on a predation list to one-quarter equipped with spider images and a memory test one day later. So after the night, then it was tested whether they still memorized to avoid the dangerous flowers. And about 30 Bumblebees were studied. And we had on average about 6,000 data points per Bumblebee for the data analysis. So this was the experiment. And now as a theorist, we wanted to reanalyze the data in view of the following two questions. First of all, I'm very interested in what type of motion the Bumblebees perform in terms of stochastic dynamics, meaning what I'm interested in is really to build a stochastic model that can reproduce this kind of not of spaghetti, or it looks like it a little bit. But secondly, in this context, especially are there then changes of the dynamics under variation of the environmental conditions? Certainly naturally, you would expect that if the Bumblebees move without spiders and the flowers, I mean, they are quite teppy because there is no risk. But if they perceive that there are spiders there, that they are getting nervous. And that they may change their type of motion. And the question is, how do they change their motion? Can you quantify this? So this was the question that we were posing. OK, so here's our data analysis. The first thing we did was to extract velocity distribution functions. So here you see actually a semi logarithmic plot. So this is the logarithmic axis. And this is linear. And these are the velocities of a Bumblebees parallel to the y-axis, perhaps just to go back. Sorry. So you see y is parallel to the wall, x is perpendicular to the wall, z is vertical. And here, as an example, I just show you the one parallel to the wall. And the thick line with the crosses, this is the data. And the other ones are with certain functions. And you see we try to fit this with a maximum likelihood with a mixture of two Gaussians. That's the red line, exponential, the blue one, power law, the green one, and the single Gaussian, which is the pink one. And you see, just by visual inspection, you can immediately draw two of them, which is not a single Gaussian. And it's not a power law either. So that's very clear. However, whether it's a mixture of two Gaussians or exponential is not so clear. And I mean, this is one of the notorious and difficult questions in this whole business. You extract a PDF from your data. What kind of function is that? It's very difficult to answer this question. But you can do something. And for example, a method that is very convenient is the quanti-quanti plot, meaning you plot regions of equal probability of your probability distribution that you have from the data against your perspective function that you speculate fits the data. And if it matches, you get a straight line. So it's very easy. And I can recommend it if you don't know about it already. So here, you plot the data PDF against your perspective candidate function. And again, you see here, it's very well a straight line. The red lines are actually surrogate data. So they give you the error bars. And the matching is pretty perfect for actually mixture of two Gaussians. And you see it here. Actually, if you take a close look, though we don't have much data here, but you see we have a little peak here, which is the little Gaussian with the smaller variance. And then we have as a background a big Gaussian with a big variance. And so this is our finding that this is the probability distribution that we get. And in fact, this is the best fit. And I mean, we verified this by information criteria. You can do base analysis. And then there are so-called weights, acaika, information criteria, base criteria for the experts. And we really checked also quantitatively that double Gaussian is the best. And this certainly is a very simple biological explanation because there are two different flight modes, spatial flight modes. So one is if the bumblebee is far away from a flower, so then it flies around with a big Gaussian. So with the biggest use of velocities, if it approaches a flower, so then in fact, it reduces the speed. And then it approaches with a small Gaussian on top of it. And so there's really a spatial separation of the different flight modes. And we checked that by eliminating from the data near to the flower. And then we only got the big Gaussian. So it's a spatial variation of the flight. In a way, we physicists call this intermittent dynamics. Intermittent means you sample between two different stochastic processes. And here's a spatial sampling. Now that comes to surprise. We did this certainly for the different phases, especially for the spider phase and the non-spider phase. And I asked my PhD student, well, look at it. There should be a difference. That's what you would nicely expect. That if you have spiders there, you should get a different probability distribution. And I told my PhD student, yeah, I mean, do it. And he came to me and said, well, I don't find any difference. Sorry. After one year, we gave up. Then I told him, OK, I mean, we are not getting, making progress. Let's forget about it. We do something else. We couldn't find any difference. So they were meaning the velocity probability distribution for all the phases were exactly the same within error bars. That was the state of the art. So we couldn't see any difference. Then I told him, OK, look at correlation functions. That's the next best thing to do. And indeed, the answer was there. And that was a big surprise to us. So physicists call this velocity autocorrelation function. In statistics, it's called covariance. And if you don't know about it, I can only highly recommend it for statistical data analysis. OK, I don't have much time. No, a few minutes, right? But if you want, I can certainly explain to you what the correlation function is. The point is, it tells you about memory in the dynamics, meaning how much your dynamics is correlated in time. So meaning, this correlation function is optimal. I mean, the boundary starts with the velocity in a certain direction. And then you check how much your velocity at time t is correlated to this initial velocity. So meaning, if it's the parallel, the correlation function is maximal. You see here. That's why it starts from 1. It's normalized. But then, certainly, the velocity will vary. And then this is a scalar product if you do it via vectors. Meaning, if it's perpendicular, it's 0. If it's anti-parallel, it's negative. So this velocity correlation function gives you very nice information of how your trajectory looks like. Again, here, then first it flies parallel. Then it may mess around. And if it's negative, in fact, it has reversed its direction. And you see that exactly this is going on here. So this is the velocity autocollation function parallel, again, through the feeding plane for these stages. So this is a spider-free predation thread. In memory tests, you see that the two are quite identical. But this means, in the spider-free stage, the bumblebees fly quite persistently in the same direction. They are not worried. So they're approaching the flower very fast. That's what it means. But you see that for the infested stages, you have anti-correlations, meaning it starts here, but then it reverses the direction. So it flies anti-paralleled. So that means the bumblebee flies through the flower, but then spots, oh, there's a problem. There's a spider there, so it makes a turn. And this is exactly why this becomes negative. So they turn. That means the explanation is, observe no change in the PDFs. And if anyone of you has an explanation for that, I would be very happy to hear it. So it seems, perhaps for a biological reason, they don't want to or they can't change their velocity or speed distribution. It's the same. However, what they do change is the topology of their trajectories. So all the information about the interaction with the environment is in the topology, not in the speeds. So they perform more curved trajectories, which, of course, makes a lot of sense, because they are more careful. They're checking out the situation. So that's what we find. And that's what I've said. All the changes regarding the phases are in the velocity correlations, not in the PDFs. So if you do data analysis, do not always focus on the probability distributions. That may not be the whole story. And we did something else a bit more, because we wanted to check for the interaction with the environment as follows. We defined what one may call predator avoidance in a very simple way, because Friedrich looked at the position of probability distributions in front of a flower. So in fact, this is wrong. That should be a set. And why? Sorry for that. But you see, this is the plan exactly in front of a flower. And this is simply the difference of the probability distributions, the spatial ones, between the phase where there's no spider minus no spider. And you see, in fact, here, there's a peak. And here, above the peak, there's a minimum. So this means, in fact, the minimum says it's avoidance. So if there's a flower with a spider, there might be approaches, the flower, but then it sees, ah, there's a problem, so I fly back. So certainly it will not stay there. It will move away. And this means hovering. So this means, in fact, the bummer bee then stays in front of the flower, a little bit below, and checks out the situation. So we can quantify this. This is from the data. And I would say the bummer bee typically approaches the flower from the top, so it goes like this. And then it sees there's a problem. That's why there's a minimum. And then it goes back and hovers almost horizontally in front of the flower. And now we modeled this by a simple so-called Lange-Wang equation, but if you're not familiar with that, it's a stochastic differential equation where this is a kind of Newton law. So this is, in a way, the force, mass we have scaled away. It's on the right-hand side. And this is the following. This is a kind of friction. It's a friction coefficient. And this is a force that models the interaction with the flower. So this is a potential derivative of it. And this is Gaussian white noise. For physicists, this is very familiar. If you're an experimental biologist, perhaps not so much. But this, in a way, models what's called Brownian motion. But here, Brownian motion is an interaction. And this thing, if there is no spider, we switch it off. So we eliminate it. If there is a spider, we switch it on. And it's a repulsive force that models that the bummer bee wants to stay away from the spider. And these are simulation results where you can also solve this analytically. If you have no spider, you have some simple exponential decay. So you don't see that your velocity correlation function becomes negative. It remains positive. However, if there's a spider, and if you think as a physicist about it, it's very trivial effect, then you see that the correlation function becomes negative, as we have seen in the data here. So qualitatively, we use this one that this remains positive. I should say this is the error. So you see this is not an artifact. Really, this effect is beyond any experimental error. And now, if there's a spider, we reproduce qualitatively, at least, that this becomes negative. So we have an emergent, this emerges from the interaction between the bee and the flower. OK, this is a very simplistic model that doesn't tell you very much about the flights. I still have how much? Two minutes or so? OK, great. This is just an advertisement, this one slide, because we did better in terms of modeling the bummer bee flights. But here, I don't go into detail. We have heard already about correlated random walks. So what we did now, we wanted to model the free bummer bee flights, meaning flights far away from a flower, still in the box. And we started with a correlated random walk in 2D, which is a description of the dynamics in the co-moving frame. So meaning you have a picture here, you have the velocities, and you have the angle between two velocities at adjacent time steps. And standard correlated random walk means you assume the speed is constant, and then you simply sample the turning angle, the angle between the two velocities, from a non-uniform distribution. Say you may use a wrapped Gaussian distribution, for example, and this model's persistence, that your bummer bee flies with a certain probability in the same direction. This is a classical correlated random walk. Now we started from that, and we find it. And again, I don't go into detail. I'll just give you the result. We did quite a sophisticated data analysis, and we really extracted this model from the data. So it's not a talk. It's really extracted from real data analysis. And we found that the angle, in fact, is best drawn from a Gaussian noise, but it's correlated. Again, it's a correlation function, and in fact, the correlations are power law. So the angle is not random, but it persists in time with the power law correlations. And now you see here the velocity. We sampled it from a function g. So this is, again, friction. And if you have ever heard of active particles, another hot topic in the field of biophysical modeling, this model's activity of a biological organism. So it's a kind of activity term. And this is a Gaussian noise, but it's anti-correlated. So again, it's complicated correlation decay. So this may be called the generalized Lange-Wang equation again. So you put it on the computer, and what you get here is exactly the trajectory from our stochastic model. And it seems to reproduce the external data very well. We were surprised about that, because quite some assumptions go in. So if you're interested, this is around as a model for modeling more sophisticated trajectories of moving organisms. OK, so I summarize. The bumblebees are obviously quite clever. So they especially adjust their flight modes to the environment. And also, this is the main result. They have a temporal adjustment to the risk of infected or non-infected flowers. However, we only see it in the velocity autocollation functions when the spiders are there. And we have qualitatively have reproduced this from a simple stochastic model. OK, we have two papers about that. So what I've told you about is mainly published in this first letter. And if you're interested in the most sophisticated stochastic model, then this is in the second paper. And do I have one more minute? OK, then because I want to show you this one, this is something entirely different. But as we have seen some looping trajectories, I just included this. And it's a question to you. So this is another project that I am doing again with experimental biologists at Queen Mary. And they are tracking sea turtles. The logger had sea turtles at the coast of South Africa. So you see the islands kept ready. And this is the foraging dynamics. And you see it's not brown in motion, definitely not. So you see there are a lot of loops. This is looping over scales of hundreds of kilometers. And I've seen some loops in some talks. That's why I asked you. And so I'm just curious if anyone knows any other trajectories of animals that look like that. I mean, I would be very happy to discuss. So I'm just trying to learn whether this is special to the sea turtles or whether this is more general and animal foraging. OK, thank you very much.