The fact that the Lotka-Volterra system admits periodic solutions can be proved by exhibiting a conserved quantity. At this point I recall the well-known fact that while conserved quantities and their generalizations, the Lyapunov functions, are very useful when you have them there is no general procedure for finding them. This naturally brings up the question: if I did not know the conserved quantity for the Lotka-Volterra system how could I find it? One method is as follows. First divide the equation for by that for to get a non-autonomous equation for , cheerfully ignoring points where . It then turns out that the resulting equation can be solved by the method of separation of variables and that this leads to the desired conserved quantity.

One undesirable feature of the Lotka-Volterra system is that it has a one-parameter family of periodic solutions and must therefore be suspected to be structurally unstable. In addition, if we consider a solution where predators are initially absent the prey population grows exponentially. The latter feature can be eliminated by replacing the linear growth term in the equation for the prey by a logistic one. A similar term corresponding to higher death rates at high population densities can be added in the equation for the predators but the latter modification has no essential effect. This is a Lotka-Volterra model with intraspecific competition. As discussed in the book ‘Evolutionary Games and Population Dynamics’ by Josef Hofbauer and Karl Sigmund, when this model has a positive steady state that state is globally asymptotically stable. The proof uses the fact that the expression which defines the conserved quantity in the usual Lotka-Volterra model defines a Lyapunov function in the case with intraspecific competition. This is an example of the method of obtaining conserved quantities or Lyapunov functions by perturbing those which are already known in special cases.

It follows from Poincaré-Bendixson theory that the steady states in the Lotka model and the Higgins model are globally asymptotically stable. This raises the question whether we could not find Lyapunov functions for those systems. I do not know how. The method used for Lotka-Volterra fails here because the equation for is not separable.

]]>In a talk of Takashi Hiiragi I learned a number of interesting things about embryology. The specific subject was the mouse embryo but similar things should apply to the human case. On the other hand it is very far away from what happens in Drosophila, for instance. There is a stage where the first eight cells are essentially identical. More precisely there is a lot of random variation in these cells, but no systematic differences. The subsequent divisions of these cells are not temporally correlated. By the time the number of cells has reached thirty-two an important differentiation step has taken place. By that time there are some of the cells which belong to the embryo while the others will be part of the placenta. If the first eight cells are separated then each one is capable of giving rise to a complete mouse. (The speaker did seem to indicate some restriction but did not go into details.) In order to understand the development process better one of these cells is studied in isolation. The cell contains a clock and so it ‘knows’ that it is in the eight cell stage. It then develops into a group of four cells in the same way that the eight cells would normally develop into 32. Differentiation takes place. The key symmetry-breaking step takes place when one end of the cell (at the eight-cell stage) develops an area at one end where actin is absent. This polarization then influences the further motion of the cells. It is interesting that the interaction between the cells in these processes seems to have more to do with mechanical signals then with chemical ones.

There was a talk of Fredric Cohen about cholesterol. His claim was that the concentration of cholesterol as usually measured is not a useful quantity and that the quantity which should be measured is the chemical potential of cholesterol. This has to do with the fact that cholesterol is hardly soluble in water or in hydrophobic liquids. I must say that the term ‘chemical potential’ was something which was always very opaque for me. As a result of this talk I think I am beginning to see the light. The cholesterol in a cell is mainly contained in the cell membrane. However it is not simply dissolved there as single molecules. Instead most of the molecules are interacting with proteins or with other cholesterol molecules. The chemical potential has to do with how many molecules get transferred when the system is connected to a reservoir. Only those molecules which are free are available to be transferred. So the issues seem to be the relationship between the amounts of free and bound molecules and what the real significance of the concentration of free molecules is for understanding a system.

]]>The long-term goal of this project is to understand the evolution of warm-blooded animals in connection with the evolution of birds from dinosaurs. This involves understanding the way in which animals allocate energy to different tasks. How much do they use for generating body heat and how much do they use for other tasks such as maintenance or reproduction? A first step is to find a parametrization of the possible energy allocation strategies. In other words we want to identify suitable variables which could be used to describe the evolutionary process we want to understand. This is the content of the paper which has just appeared. At this point it might be asked how we can find out about the energy consumption of dinosaurs at all. It turns out that there are general relations known between the energy consumption of an animal and its growth rate over its lifetime. Thus the growth curve of a dinosaur gives indirect information about its energy consumption. But how can we get information about the growth curve? This is something I learned in the course of this project. The large bones of dinosaurs exhibit annual growth rings like those known for trees. The rings are of different thicknesses and thus give information on the growth rate in different years.

The paper does not contain a detailed dynamic model of the energy use of an animal over its lifetime. Instead it introduces a set of possible time evolutions depending on a finite number of parameters and then tests (by numerical methods) whether these suffice to reproduce the experimental growth curves of a number of animals with sufficient accuracy. It is also checked that the results obtained are consistent with known facts about the age of sexual maturation of the different species. It turns out that the mathematical model is successful in fitting the experimental constraints. It is found that, as expected, the model predicts that exotherms continue to grow as long as they live while endotherms stop growing at a time comparable to the age of sexual maturity.

]]>From El Calafate we travelled overland and crossed the border into Chile.We had hardly crossed the border when we saw our first Andean Condor. This part of Chile has no road connection within the country to the rest of Chile. All necessary goods are imported by ship and the prices are correspondingly high. We were first in Puerto Natales. There it was convenient to observe the local ducks and cormorants along the waterfront. There is a statue of a giant sloth, a creature whose remains were found in a cave in the region. We travelled to the Torres del Paine, a spectacular mountain range. What you can see there is heavily dependent on the weather and we were quite lucky. Only the very top of the largest of the three ‘Cuernos’ (horns) refused to emerge from the clouds as long as we were there. What sounded like thunder turned out to be an avalanche. From Puerto Natales we travelled to Punta Arenas on the Strait of Magellan. There we visited some reconstructions of famous naval vessels. There is the Beagle, with which Darwin travelled, one of the ships of the expedition of Magellan himself and the modified lifeboat with which Shackleton sailed from Elephant Island to South Georgia and thus saved the lives of the members of his expedition. The text of the famous advertisement with which Shackleton recruited men for this expedition is reproduced there. According to Wikipedia the story of this advertisement is apocryphal but the text is so delicious that I cannot resist repoducing it here: ‘Men wanted for hazardous journey. Low wages, bitter cold, long hours of complete darkness. Safe return doubtful. Honour and recognition in event of success’.

From Punta Arenas we flew to Puerto Montt and travelled from there to Puerto Vargas. Here the attraction was the volcano Osorno. This time our luck with the weather seemed to be at an end. Our guide explained in a quite amusing manner how if the weather had been different the wall of cloud which we saw in a certain direction would have been replaced by a view of the beautiful volcano. In fact it turned out that there was a small window before breakfast the next day where the volcano could be seen from the hotel. The last step of the journey was a flight to Santiago. The city did not make a very good impression at first sight. Compared with Buenos Aires all the signs seemed to be reversed. Later we learned that the city is divided very strictly along economic lines. The rich upper part of the city looks quite different to the rest. We learned about the story of Chile as a model system for testing neoliberal theories. Just now the Chileans, who used to be considered as very backward are proud to be doing better (economically) than their neighbours, the Argentinians. We also had the opportunity to learn from our guides about the politics of Allende and Pinochet, in particular that Pinochet still enjoys considerable popularity in Chile.

We flew back from Santiago to Frankfurt via Madrid, our heads full of many images of Argentina, Chile, their people and their natural environment.

]]>The first time I visited Argentina I also flew to Buenos Aires but I saw almost nothing of the city. I just took a taxi from the international airport to the domestic one and my memory was that it drove around the periphery. Looking at how the airports lie this impression was probably mistaken. During my first trip I had a few hours to wait for my continuing flight to Cordoba and I was able to watch the for me exotic gulls, since the airport is close to the water. This time we were warned on arrival about the dangers of the city and how to behave so as to avoid them. We actually had no problems although we did not pay that much attention to security issues. One member of the group was attacked in the middle of the day close to our hotel. A man jumped on his back and tried to steal his watch. I do not think the watch was of particular value but the thief could not know that. Two brave women came and intervened and the thief ran away without the watch. The victim was left with some nasty-looking bruises on his arm but there was no further damage.

Our impression of Buenos Aires was of a beautiful city with a very pleasant atmosphere. It must be said that the days we were there were holidays, so that we experienced the city in a much more relaxed mode than it would be in on a working day. We discovered that there is a very nice nature reserve within easy walking distance of the city. My experience travelling with groups which are not specialised on birds is that if you go anywhere which might be good for birds you see so little that it is frustrating. The present trip was an exception to this rule. In the reserve area near Buenos Aires one of the most prominent species was the Jacana, whch was numerous. Some other prominent sightings were Roseate Spoonbill, Black-Necked Swan (which we later also saw in many other places), Red-Crested Cardinal and a Hummingbird of an undetermined species. From our guides we learned a bit about the complicated subject of Argentinian politics. One story which stuck in my mind and which I reproduce here without further comment is the following. There was a time when many Argentinians were protesting about the meat prices being too high. The Kirchners banned the export of beef. The result was a situation of overproduction which did lead to a decrease in the prices. This led in turn to many producers going out of business or drastically cutting their stocks. The final results were then shortages (beef had to be imported to Argentina from Uruguay!) and that a key national industry had been damaged in a major and probably irreparable way.

After a couple of days in Buenos Aires we flew to Trelew and spent some time exploring the Valdes peninsula, staying in Puerto Madryn. We learned what dry pampa looks like, a brown very dry landscape which in that area forms huge monotone expanses. At that stage I did not find the landscape attractive although other variants we saw in other areas later looked better. We were able to see some of the standard wildlife: sea elephants, sea lions and a huge colony of Magellanic Penguins. The temperature was around 30 degrees (I mean Celcius, not Fahrenheit) and the penguins were suffering a lot from the heat. I was pleasantly suprised to get a close view of an armadillo. The towns in this area was established by settlers from Wales, which explains the curious names. I would be interested to read about the adventures of these pioneers.

After this we flew to the town mentioned in the title of this post, Ushuaia. Before the trip I felt that Ushuaia was more like a mythical place than a real one. But now I have been there. Because of a last-minute change of plane schedule we had less time in Ushuaia than planned. Despite this we were able to take a trip on the Beagle channel with a catamaran in the late afternoon. The weather was excellent. For me this was the highlight of the whole trip. In the town itself we saw Dolphin Gulls and the first Giant Petrels. We visited some seabird islands with breeding colonies of terns and cormorants. We even landed on one island where there were Great Skuas flying around. In one place I saw a couple of Sheathbills on the beach. What was special is that we came to one place where there was a big concentration of fish. There was a corresponding concentration of seabirds, including several Albatrosses. Afterwards one Black-Browed Albatross followed the ship for quite a long time. At 22.00 we caught a flight to El Calafate.

]]>In a talk of Susanna Manrubia I learned about a class of biological systems I had never heard of before. These are the so-called multipartite viruses. In this type of system a genome is distributed between two or more virus particles. To allow the production of new viruses in a cell it must be infected with all the components. The talk described experiments in which this type of system was made to evolve in the laboratory. They start with foot and mouth virus in a situation where infection takes place with very high frequency. This is followed through many generations. It was found that in this context a bipartite virus could evolve from a normal virus. This did not involve an intermediate situation where (as in influenza) a virus has several segments of genetic material in a single virus particles. When the bipartite virus was propagated in isolation under circumstances where the frequency of infection was much lower it evolved back to the ordinary virus state. There exists a large variety of multipartite viruses in nature. They seem to be most common in plants (where apparently multiple virus infections are very common) but are also found in other organisms, including animals.

I had a poster at the conference on my work on T cell activation with Eduardo Sontag. To my surprise Paul Francois, on whose results this work built, was at the conference and so I had the chance to discuss these things with him. In our work there is a strong focus on the T cell receptor and at the conference there were several other contributions related to the modelling of other receptors. Eugenia Lyaschenko talked about how receptors can sense relative levels of ligand concentration and how endocytosis plays a role in this. Nikolas Schnellbächer had a poster on the theme of how dimerization of receptors can lead to major qualitative changes in their response functions. There are also important differences between homo- and heterodimers. I learned something about the mechanisms which play a role there. Yaron Antebi talked about the dynamical significance of a situation where several related ligands can bind to several related receptors.

Turing instabilities came up repeatedly at the meeting and were getting positive comments. One ‘take-home message’ for me was that the usual story that a Turing instability requires different diffusion constants should be weakened. It is based on the analysis of a system with two components and as soon as there are more than two components no such clear statement can be made. In addition, taking into account cell growth can help to trigger Turing instabilities.

A talk by Pieter Rein ten Wolde deepened my understanding of circadian clocks in cyanobacteria. They have a clock on a post-translational level involving phosphorylations of the KaiABC proteins and also a clock which involves translation. In the talk it was explained how the bacterium needs both of these for its time-keeping. A key point is that the period of the oscillator defining the clock (around 24 hours) can be longer than the period of the cell cycle. Thus this is like a clock which can continue to tick while it is being disassembled into its constituent parts and put together again.

]]>For the first part of the discussion here I follow these notes. If is an (unoriented) graph then a spanning tree is a connected subgraph which includes all vertices of and contains no cycles. The graph Laplacian of is the matrix for which is equal to the number of edges containing the vertex , if and there is an edge joining vertex to vertex and otherwise. The first version of the matrix tree theorem, due to Kirchhoff, says that the number of spanning trees of can be calculated in the following way. Choose any vertex of and remove the th row and th column from to get a matrix . Then compute the determinant of . Surprisingly the value of the determinant is independent of . The first version of the theorem can be obtained as a consequence of a second more sophisticated version proved a hundred years later by Tutte. This concerns directed graphs. A vertex is said to be accessible from if there is a directed path from to . A vertex in a directed graph is called a root if every other vertex is accessible from it. A directed tree is a directed subgraph which contains a root and which, when the orientation is forgotten, is a tree (i.e. it is connected and contains no unoriented cycles). There is then an obvious way to define a spanning rooted tree. To formulate the second version of the theorem we introduce the matrix Laplacian of the directed graph. If the entry is the number of edges with final vertex . if and there is a directed edge from to . if and there is no edge connecting and . The statement of the theorem is that the number of rooted trees with root is equal to the determinant of , a matrix derived from as before. To see that the first version of the theorem follows the second first associate an oriented graph to an unoriented one by putting in oriented edges joining a pair of vertices in both directions whenever they are joined by an edge in the original graph. Then there is a bijection between trees rooted at in the unoriented graph and oriented trees rootes at in the oriented graph. On the other hand the two graphs have the same Laplacian since the number of edges ending at a vertex in the oriented graph is the same as the number of edges having that endpoint in the unoriented graph.

What is the connection of all this with reaction networks? Consider a chemical reaction network with only monomolecular reactions and reaction coefficients all equal to one. Then under mass action kinetics the evolution equations for the concentrations are , where is the Laplacian matrix of the network. There is a conserved quantity, which is the sum of the concentrations of all species. A steady state is an element of the kernel of . The next part of the discussion follows a paper of Gunawardena (PLoS ONE, 7(5), e36321). He allows general reaction constants. The notion of the Laplacian is extended correspondingly. If the network is strongly connected (in the terminology of graph theory) or weakly reversible (in the terminology of chemical reaction network theory) the kernel of the Laplacian matrix is one-dimensional. Thus there is precisely one steady state in each stoichiometric compatibility class. It is moreover possible to compute a vector spanning the kernel of by graph theoretical methods. This also goes back to Tutte. To get the th component first take the product of the reaction constants over a rooted tree and then sum these quantities over the rooted trees with root . More generally the dimension of the kernel of is equal to the number of terminal strong linkage classes. It is also revealed there that the Laplacian corresponds to the matrix considered by Horn and Jackson. These ideas are also related to the King-Altman theorem of enzyme kinetics. I have the impression that I have as yet only scratched the surface of this subject. I hope I will be able to come back and understand more about it at a later date.

]]>Sternberg’s theorem shows that if is and there are no resonances then can be chosen to be . The definition of a resonance in this context is as follows. Let be the eigenvalues of the linearization. A resonance is a relation of the form with positive integers satisfying . For example, if the eigenvalues are and we have the resonance . Consider the two-dimensional case. If there the eigenvalues are real with opposite signs then there can be no resonance and Sternberg’s theorem applies to all saddles in two dimensions. If both eigenvalues are positive then it is known that can be chosen to be but resonances can occur and there are cases where although is analytic cannot be chosen to be . Even when there are no resonances it is in general not the case that if is analytic the mapping can also be chosen analytic. It is necessary to make a further assumption (no small divisors). This says roughly speaking that the expressions should not only be non-zero but that in addition they should not be able to approach zero too fast as the number of summands increases. This is the content of a theorem of Siegel.

All the results discussed up to now concern the hyperbolic case. What happens in the presence of purely imaginary eigenvalues? The Grobman-Hartman theorem has a generalization, the theorem of Shoshitaishvili, also known as the reduction theorem. In words it says that there is a continuous mapping which reduces the system to the product of the flow on any centre manifold and a standard saddle. A standard saddle is linear and so if the centre manifold is trivial this reduces to Grobman-Hartman. The next question is what can be achieved with mappings of higher regularity. A fact which is implicit in the reduction theorem is that linear systems where the origin is a hyperbolic fixed point whose stable and unstable manifolds have the same dimension are topologically equivalent. Evidently they are in general not smoothly equivalent since any smooth mapping preserving the origin will not change the eigenvalues of the linearization. However once this has been taken into account the theorem does generalise under the assumptions that there are no resonances among the eigenvalues with non-zero real parts.More precisly the eigenvalues in the definition of a resonance must have non-zero real parts. By contrast quantity should either be an eigenvalue with non-zero real part or zero. The result being referred to here is Takens’ theorem.

Suppose that for a steady state we introduce coordinates , and corresponding to the stable, unstable and centre manifolds. Then the form achieved by the reduction in Takens theorem is that we get the equations , and for matrices and and a function depending on the central variables . One technical point is that while for a smooth system the reduction can be done by a mapping which is for any finite it cannot in general be done by a mapping which is itself infinitely differentiable. As an example, consider a two-dimensional system and a fixed point where the linearization has one zero and one non-zero eigenvalue. In this case there can be no resonances and the theorem says that there is a coordinate transformation which reduces the system to the form , with , and .

]]>Now Subgroups are being set up within the Society to concentrate on particular subjects. One of these, the Immunobiology and Infection Subgroup had its inaugural meeting this week and of course I went. There I and a number of other people learned a basic immunological fact which we found very surprising. It is well known that the thymus decreases in size with age so that presumably our capacity to produce new T cells is constantly decreasing. The obvious assumption, which I had made, is that this is a fairly passive process related to the fact that many systems in our bodies run down with age. We learned from Johnna Barnaby that the situation may be very different. It may be that the decrease in the size of the thymus is due to active repression by sexual hormones. She is involved in work on therapy for prostate cancer and said that it has been found that in men with prostate cancer who are getting drugs to reduce their testosterone levels it is seen that their thymus increases in size.

There were some recurrent themes at the conference. One was oncolytic viruses. These are genetically engineered viruses intended to destroy cancer cells. In modelling these it is common to use extensions of the fundamental model of virus dynamics which is very familiar to me. For instance Dominik Wodarz talked about some ODE models for oncolytic viruses in vitro where the inclusion of interferon production in the model leads to bistability. (In reponse to a question from me he said that it is a theorem that without the interferon bistability is impossible.) I was pleased to see how, more generally, a lot of people were using small ODE models making real contact to applications. Another recurrent theme was that there are two broad classes of macrophages which may be favourable or unfavourable to tumour growth. I should find out more about that. Naveen Vaidya talked about the idea that macrophages in the brain may be a refuge for HIV. Actually, even after talking to him I am not sure if it should not rather be microglia than macrophages. James Moore talked about the question of how T cells are eliminated in the thymus or become Tregs. His talk was more mathematical than biological but it has underlined once again that I want to understand more about positive and negative selection in the thymus and the related production of Tregs.

On a quite different subject there were two plenary talks related to coral reefs. A theme which is common in the media is that of the damage to coral due to climate change. Of course this is dominated by politics and usually not accompanied by any scientific information on what is going on. The talk of Marissa Blaskett was an excellent antidote to this kind of thing and now I have really understood something about the subject. The other talk, by Mimi Koehl, was less about the reefs themselves but about the way in which the larvae of snails which graze on the coral colonize the reef. I found the presentation very impressive because it started with a subject which seemed impossibly complicated and showed how scientific investigation, in particular mathematical modelling, can lead to understanding. The subject was the interaction of microscopic swimming organisms with the highly turbulent flow of sea water around the reefs. Investigating this involved among other things the following. Measuring the turbulent flow around the reef using Doppler velocimetry. Reconstructing this flow in a wave tunnel containing an artificial reef in order to study the small-scale structure of the transport of chemical substances by the flow. Going out and checking the results by following dye put into the actual reef. And many other things. Last but not least there was the mathematical modelling. The speaker is a biologist and she framed her talk by slides showing how many (most?) biologists hate mathematical modelling and how she loves it.

]]>A talk I found very interesting was by Sebastian Walcher. I already wrote briefly about a talk of his in Copenhagen in a previous post but this time I understood a lot more. The question he was concerned with is how to find interesting small parameters in dynamical systems which allow the application of geometric singular perturbation theory. In GSPT the system written in the slow time (with the smallness parameter included as a variable) contains a whole manifold of steady states, the critical manifold. The most straightforward theory is obtained when the eigenvalues of the linearization of the system transverse to the critical manifold lie away from the imaginary axis. This corresponds to the situation of a transversely hyperbolic manifold of steady states. The first idea of Walcher’s talk is that whenever we have a transversely hyperbolic manifold of steady states in a dynamical system this is an opportunity for identifying a small parameter. This may not sound very useful at first sight because it would seem reasonable that generic dynamical systems would never contain manifolds of steady states of dimension greater than zero. There is a reason why this observation is misleading for systems arising from reaction networks. In these systems the state space is defined by positivity conditions on the concentrations and there are also certain parameters (such as reaction constants and total amounts) which are required to be positive. To have a name let us call the region defined by these positivity conditions the conventional region of the spaces of states and parameters. In the conventional region manifolds of steady states are not to be expected. On the other hand it frequently happens that they arise when we go to the boundary of that region. A familiar example is the passage to the Michaelis-Menten limit in the system describing a reaction catalysed by an enzyme. This takes us from the extended mass action kinetics for substrate, free enzyme and substrate-enzyme complex to Michaelis-Menten kinetics for the substrate alone. Roughly speaking it is the limit where the amount of the enzyme is very small compared to the amount of the substrate. I often wondered whether there could not be a kind of ‘anti-Michaelis-Menten’ limit where the amount of enzyme is very large compared to the amount of the substrate. I asked Walcher whether he knew how to do this and how it fitted into his general scheme. He gave me a positive answer to this question and some references and I must look into this in detail when I get time. The reason for being interested in this is that if we can obtain suitable information about a limiting case on the boundary it may be possible obtain information on the part of the conventional region where a certain parameter is small but non-zero.

There was one talk which did have a connection to population biology in way closer to what I had expected. It happens all the time that ecosystems are damaged by exotic species imported, deliberately or by accident, from other parts of the world. There are also well-known stories of the type that to try to control exotic species number one exotic species number two is introduced and is itself very harmful. It is nice to hear an example where this kind of introduction of an exotic species was very successful. It is the case of the cassava plant which was introduced from South America to Africa and became a staple food there. Then an insect from South America (species number one) called the mealy bug was introduced accidentally and caused enormous damage. Finally an ecologist called Hans Herren introduced a parasitic wasp (species number two) from South America, restoring the food supply and saving numerous lives (often the number 20 million is quoted). More details of this story can be found here.

I want to mention one statement made in the talk of Gheorghe Craciun in Oberwolfach which I found intriguing. I might have heard this before but it did not stick in my mind properly. The statement is that the set of dynamical systems which possess a complex balanced steady state is a variety of codimension , where is the deficiency. There seemed to be some belief in the audience that this variety is actually a smooth manifold. On one afternoon we had something similar to the breakout sessions in Banff. I suggested the topic for one of these, which was Lyapunov functions. The idea was to compare classes of Lyapunov functions which people working on different classes of dynamical systems knew. This certainly did not lead to any breakthrough but I think it did lead to a useful exchange of information. I documented the discussion for my own use and I think I could profit by following some of the leads there.

To finish I want to mention a claim made by Ankit Gupta in his talk. It did not sound very plausible to me but I expect that it at least contains a grain of truth. He said that these days more papers are published on than on all of mathematics.

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