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.

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We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by "holes") are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.(1.9) with z decreasing from z 0 to 0 and with Z increasing from 0 to Z ∞ . To better understand the crucial relationship between the asymptotic phaseshift Z ∞ and the additive perturbation z, we note that the super-solution residual J = u t − u xx − g(u) is given by(1.10)Close to the interface, the term g(Φ) − g(Φ + z) ∼ −g ′ (Φ)z is negative and must be dominated by the positive termŻΦ ′ . This requires thatŻ dominate z andż. On the other hand, close to the Post-interaction regime -convergence to waveContinuous Setting In this final step of the program, the large transients generated in the interaction regime must be controlled in a frame that moves along with the unobstructed wave. As discussed above, this analysis leads naturally to a large basin nonlinear stability result for planar travelling wave solutions to the unobstructed PDE (1.2) with Ω = R 2 . For presentation purposes, we will focus our discussion here on this unobstructed special case. Indeed, the inclusion of the obstacle merely adds technical complications that do not contribute to the understanding of the differences between the continuous and discrete frameworks. 7The main task is to construct a super-solution for (1.2) with Ω = R 2 of the form( 1.15) which adds transverse effects to the Ansatz (1.9) discussed earlier. As before, the function z decreases from z 0 to 0 while Z increases from 0 to Z ∞ . Both z and Z should be thought of as small terms. By contrast, the new function θ should be allowed to be arbitrarily large at t = 0, provided that it is localized in the sense θ(·, 0) ∈ L 2 and that it decays to zero as t → ∞ uniformly in y. This function controls deformations of the wave interface in the transverse direction. We note that any localized initial perturbation from the wave can be dominated by the initial condition in (1.15) by choosing z 0 positive and as small as we wish, at the cost of a larger value for θ(·, 0) L 2 . Assuming for the moment that Z ∞ scales with z 0 , this freedom implies that we can dominate the transients caused by such a perturbation by a family of super-solutions that have arbitrarily small asymptotic phase offsets Z ∞ . A similar argument with sub-solutions then establishes the convergence to the planar wave without any asymptotic phase shift.The difference in behaviour between one and two spatial dimensions is hence caused by the extra transverse direction, al...

In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reaction–diffusion systems are considered. In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reaction–diffusion systems are discussed.