Stability of cluster solutions in a cooperative consumer chain model

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ Springer-Verlag Berlin Heidelberg 2012.We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order O(1). Further, there are small eigenvalues of order o(1) which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O(1) leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order o(1) needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ε22.RGC of Hong Kon

Rates for branching particle approximations of continuous-discrete filters

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that $t\to X_t$ is a Markov process and we wish to calculate the measure-valued process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted, corrupted, partial observation of $X_{t_k}$. Then, one constructs a particle system with observation-dependent branching and $n$ initial particles whose empirical measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. Each particle evolves independently of the other particles according to the law of the signal between observation times $t_k$, and branches with small probability at an observation time. For filtering problems where $\epsilon$ is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of $\epsilon$. We analyze the algorithm on L\'{e}vy-stable signals and give rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$, where $\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related convergence results.Comment: Published at http://dx.doi.org/10.1214/105051605000000539 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pathways to Rare Baryonic B Decays

We point out new ways to search for charmless baryonic B decays: baryon pair production in association with $\eta^\prime$ is very likely as large as or even a bit larger than two body $K\pi/\pi\pi$ modes. We extend our argument, in weaker form, to $B\to \gamma + X_s$ and $\ell\nu +X$. Although calculations are not reliable, estimates give branching ratios of order $10^{-5}$--$10^{-6}$, where confidence is gained from recent experimental finding that $B \to D^{*} p \bar n$, $D^{*} p \bar p \pi$ are not far below $D^*\pi$ and $D^*\rho$ rates. Observation of charmless baryon modes would help clarify the dynamics of weak decays to baryonic final states, while the self-analyzing prowess of the $\Lambda$-baryon can be helpful in CP- and T-violation studies.Comment: 12 pages, REVTEX, 3 eps figures include