Archive for January, 2009

Electrifying Gowdy

January 27, 2009

In a previous post I wrote about the polarized Gowdy equations. If the condition of polarization is dropped the full Gowdy equations are obtained. They form a system of two semilinear wave equations:

\displaystyle P_{tt}+t^{-1}P_t-P_{\theta\theta}=e^{2P}(Q_t^2-Q_\theta^2),

\displaystyle Q_{tt}+t^{-1}Q_t-Q_{\theta\theta}=-2(P_tQ_t-P_\theta Q_\theta).

These equations are very similar to those for a wave map from two-dimensional Minkowski space to the hyperbolic plane with metric dP^2+e^{2P}dQ^2 with the difference of the singular term involving t^{-1}. (It fact it is known that these equations can be given a formulation in terms of a wave map. The relevant wave map is on the domain with metric -dt^2+d\theta^2+t^2 dx^2 and is asssumed to be independent of the additional coordinate x.) The boundary conditions are as in the polarized case. Given initial data for t=t_0 there exists a unique solution on the time interval (0,\infty), as was proved by Vincent Moncrief in 1981.

The asymptotics for t\to 0 is significantly more complicated than in the polarized case. There is a large class of solutions with rather simple asymptotics. Their asymptotic form in the limit t\to 0 is:

P(t,\theta)=-k(\theta)\log t+\phi(\theta)+\ldots,Q(t,\theta)=q(\theta)+t^{2k(\theta)}\psi(\theta)+\ldots.

In 1998 Satyanad Kichenassamy and I proved the existence of solutions of this type for prescribed functions k, \phi, q and \psi, subject to the condition 0<k(\theta)<1. I will say no more about he positivity condition on k here but instead concentrate on the inequality k<1. The quantity k is sometimes called the asymptotic velocity and the condition k<1 the low velocity condition. It turns out that this is not just a technical restriction. It has been shown by Hans Ringström that the quantity -tP_t(t,\theta) always has a limit as t\to 0. Call its modulus the asymptotic velocity v_\infty (\theta). In the low velocity case described above it coincides with k(\theta). More generally, when it is allowed to exceed one other phenomena occur. They are associated to the phenomenon of spikes. At a low velocity point P_\theta/P converges to a smooth limit as t\to 0. A spike is a value of \theta where P_\theta/P becomes unbounded. In general v_\infty has a discontinuity at a point of this type. These phenomena have been analysed in detail under a genericity assumption by Ringström. He used these results in his proof of strong cosmic censorship for Gowdy spacetimes. There are also subtle phenomena which occur in the limit t\to\infty. (For more information about many of the results mentioned in this post see the papers on Ringström’s web page.)

The Gowdy spacetimes are solutions of the Einstein vacuum equations. They can be generalized by adding an electromagnetic field. Recently Ernesto Nungesser completed his diploma thesis under my supervision on the subject of these electromagnetic generalizations of Gowdy solutions. He and I just produced a paper where the results of his thesis are described and extended somewhat. One key element of this work is that there is a class of polarized ‘electro-Gowdy’ solutions which can be defined by symmetry conditions. Another is that there is a change of variables which transforms the polarized electro-Gowdy equations to the full (i.e. non-polarized) Gowdy equations. This allows the extensive results available on Gowdy solutions to be applied to the electromagnetic case. On the analytical level nothing happens but the geometrical interpretation of the variables is different in the two cases. It turns out that in this way strong cosmic censorship can be proved for polarized electro-Gowdy spacetimes. As discussed in the paper, much less is known in the non-polarized electro-Gowdy case .

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Competitive exclusion

January 17, 2009

The principle of competitive exclusion arose in ecology. Roughly speaking it says that in an ecosystem with only k different niches (or only k different resources) no more than k different species can coexist on a long-term basis. If there are originally n different species with n>k then at least n-k of them must die out. This principle was introduced by Georgii Gause in his 1934 book ‘The struggle for existence’. On a mathematical level it leads to questions about the behaviour of the solutions of certain systems of ODE. In fact many related questions may be formulated.

The ODE systems of relevance here are generalizations of the classical chemostat. This models a vessel containing fluid in which one species of bacteria lives. A nutrient is introduced continuously from a reservoir with fixed concentration in such a way that the rate of inflow is constant. Fluid is removed at the same rate. It is assumed that everything else the bacteria need to live and proliferate is present in sufficient quantities so as to present no limit to the population growth. The unknowns in the system are the concentrations of bacteria and nutrient. Call them y and C respectively. For an introduction to the subject see for instance the book “Mathematical models in population biology and epidemiology” by Brauer and Castillo-Chavez. This kind of system was considered by Novick and Szilard and by Monod in 1950. In the simplest case, which may be called the ‘classical’ chemostat, it is assumed that the proliferation rate of the bacteria is equal to the population density times an uptake function r(C)=\frac {aC}{C+A} where a and A are constants. It can be proved that in this case there is a threshold value of the flow rate such that above the threshold the population dies out while below the threshold it tends to a constant value. It is very helpful that the system is two-dimensional and admits a Dulac function given by C\log y. This rules out periodic solutions. The conclusion also holds for rather general uptake functions r(C).

To investigate competitive exclusion it is necessary to go to models with more than one species relying on the same resource. How can periodic solutions or more complicated kinds of attractors be ruled out in that case? A recent preprint of Bernold Fiedler and Sze-Bi Hsu, available from Fiedler’s web page makes progress on understanding this question. The authors review results in which Lyapunov functions have been applied to this problem.They then introduce a new approach which is a generalization of the use of Dulac functions to systems of higher dimensions. It rules out periodic solutions for a large class of systems. What is not ruled out is orbits which, while not coming back to the same point, come back to points arbitrarily close to where they have been before. The authors appear to have ideas about how to generalize their approach to cover that case.

Competitive exclusion may also have some relevance to immunology. In a paper by De Boer and Perelson (J. Theor. Biol. 169, 375-390) they discuss the idea that T-cell clones may be selected for by competing for the chance to bind to their (suitably presented) antigen and study corresponding dynamical systems. This suggests that the T-cell response may be dependent on a sufficient variety of antigens being present.

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