Biological nitrification (that is, NH 3 -NO 2 À -NO 3 À ) is a key reaction in the global nitrogen cycle (Ncycle); however, it is also known anecdotally to be unpredictable and sometimes fails inexplicably. Understanding the basis of unpredictability in nitrification is critical because the loss or impairment of this function might influence the balance of nitrogen in the environment and also has biotechnological implications. One explanation for unpredictability is the presence of chaotic behavior; however, proving such behavior from experimental data is not trivial, especially in a complex microbial community. Here, we show that chaotic behavior is central to stability in nitrification because of a fragile mutualistic relationship between ammonia-oxidizing bacteria (AOB) and nitrite-oxidizing bacteria (NOB), the two major guilds in nitrification. Three parallel chemostats containing mixed microbial communities were fed complex media for 207 days, and nitrification performance, and abundances of AOB, NOB, total bacteria and protozoa were quantified over time. Lyapunov exponent calculations, supported by surrogate data and other tests, showed that all guilds were sensitive to initial conditions, suggesting broad chaotic behavior. However, NOB were most unstable among guilds and displayed a different general pattern of instability. Further, NOB variability was maximized when AOB were most unstable, which resulted in erratic nitrification including significant NO 2 À accumulation. We conclude that nitrification is prone to chaotic behavior because of a fragile AOB-NOB mutualism, which must be considered in all systems that depend on this critical reaction.

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Different definitions of spectra have been proposed over the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell [J. Differential Equations, 7 (1978), pp. 320-358] and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogonal change of variables transforming the system to upper triangular form, and the assumption of integral separation for the diagonal of the new triangular system, we justify how popular numerical methods, the so-called continuous QR and SVD approaches, can be used to approximate these spectra. We further discuss how to verify the property of integral separation, and hence how to a posteriori infer stability of the attained spectral information. Finally, we discuss the algorithms we have used to approximate the Lyapunov and Sacker-Sell spectra and present some numerical results.

Abstract. We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model and Frenkel-Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals.Key words. mixed type functional differential equations, boundary value problems, traveling waves, collocation AMS subject classifications. 65L10, 65L20, 35K57, 74N991. Introduction. Nonlinear spatially discrete diffusion equations occur as models in many areas of science and engineering. When the underlying mathematical models contain difference terms or delays as well as derivative terms, the resulting differential-difference equations present challenging analytical and computational problems. We demonstrate how functional differential boundary value problems with advances and delays arise from such models, and describe a general approach for the numerical computation of solutions. Solutions are approximated for several such problems, and the numerical issues arising in their computation are discussed.Biology, materials science, and solid state physics are three fields in which accurate first principle mathematical models possess difference (both delayed and advanced) terms. In biology (in particular, in physiology) there is the bidomain model for cardiac tissue (defibrillation), ionic conductance in motor nerves of vertebrates (saltatory conduction), tissue filtration, gas exchange in lungs, and calcium dynamics. Material science applications include interface motion in crystalline materials (crystal growth) and grain boundary movement in thin films where spatially discrete diffusion operators allow description of the material being modeled in terms of its underlying crystalline lattice. In solid state physics applications include dislocation in a crystal, adsorbate layers on a crystal surface, ionic conductors, glassy materials, charge density wave transport, chains of coupled Josephson junctions, and sliding friction. In all of these fields the physical system, and the corresponding differential model with delay terms, exhibit propagation failure (crystallographic pinning, a mobility threshold) and directional dependence (lattice anisotropy) in a "natural" way. These phenomena do not occur "naturally" in the models without difference terms commonly used for the above applications, and are often added to such local models in an 'ad hoc' manner. The reason discrete phenomena are modeled with continuous models is the lack of analytical techniques ...

Abstract. In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coecient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques.In this case, the matrix system has a cubic nonlinearity.Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: autoraatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss-Legendre point Runge-Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples.