Fisher waves

2 Fisher waves the invasion of an unstable state by a stable one 

Fisher (1937), proposed an equation combining the logistic growth mechanism of (3.9), (3.23) or (3.66) with diffusion to model the spatial spreading of favorable genes in a population distributed in one dimension  

Careful study of this and of the more general type of equations 

A first step to understand the nature of these fronts is to consider (4.16) on the infinite line and particularize it to the class of solutions (4.18). One obtains a second order ordinary differential equation for 

In the present case (4.19), involving a single unknown function g and only first and second order derivatives, these counting arguments can be made more explicit and intuitive by writing (4.19) as the mechanical equation of motion of a point particle of mass D at 

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In the approximate treatment which follows we consider the two parts of the wave separately and see that the leading front can be considered as a cubic Fisher wave and the recovery front by a quadratic form. 

The rich consequences of adding a diffusive mechanism of transport to chemical or biological activity are described in Chapter 4. Fisher waves and other types of fronts, excitable waves, Turing patterns, and other spatiotemporal phenomena produce striking structures which are observed in chemical and biological media. Understanding them is needed before addressing the additional impact that advection has on these systems. 

Horsthemke, W. Fisher waves in reaction random walks. Phys. Lett. A 263(4-6), 285-292... 

The reader will recall that in Chapter 2 we gave examples of H2 calculations in which the orbitals were restricted to one or the other of the atomic centers and in Chapter 3 the examples used orbitals that range over more than one nuclear center. The genealogies of these two general sorts of wave functions can be traced back to the original Heitler-London approach and the Coulson-Fisher[15] approach, respectively. For the purposes of discussion in this chapter we will say the former approach uses local orbitals and the latter, nonlocal orbitals. One of the principal differences between these approaches revolves around the occurrence of the so-called ionic structures in the local orbital approach. We will describe the two methods in some detail and then return to the question of ionic stmctures in Chapter 8. 

The evaluation of stability for travelling-wave solutions is by no means a simple process and will not even be attempted here. The result that systems governed by quadratic Fisher equations tend to pick up their minimum permitted velocity will be used later. 

The quadratic Fisher result of the previous section is based on the quadratic chemical timescale tch = 1 /kqa0 if we represent the wave velocity measured in these terms as k, defined by... 

Goddard[21] made the earliest important generalization to the Coulson-Fisher method. Goddard s generalized VB (GGVB) wave function is written in terms of orbitals that are linear combinations of the AOs. Using the genealogical set of spin functions in turn and... 

More recently Hiberty et ol[26] proposed the breathing orbital valence bond (BOVB) method, which can perhaps be described as a combination of the Coulson-Fisher method and techniques used in the early calculations of the Weinbaum.[7] The latter are characterized by using differently scaled orbitals in different VB structures. The BOVB does not use direct orbital scaling, of course, but forms linear combinations of AOs to attain the same end. Any desired combination of orbitals restricted to one center or allowed to cover more than one is provided for. These workers suggest that this gives a simple wave function with a simultaneous effective relative accuracy. 

The PT model is frequently used as a minimalistic approximation for more complex models. For instance, it is the mean-field version of the Frenkel Kontorova (FK) model as stressed by D. S. Fisher [29,83] in the context of the motion of charge-density waves. The (mean-field) description of driven, coupled Josephson junctions is also mathematically equivalent to the PT model. This equivalence has been exploited by Baumberger and Carol for a model that, however, was termed the lumped junction model [84] and that attempts to... 

Catastrophes occurring in these models in the presence of diffusion will be examined next. We shall investigate wave phenomena in the Fisher-Kolmogorov model and in the Oregonator as well as dissipative structures in the Brusselator. The studies of systems of reactions with diffusion, both experimental and theoretical, have not led yet to the formulation of a complete theory. Consequently, only fundamental results concerning the phenomena of a loss of stability type in these models will be presented. 

Wave processes described by the Fisher-Kolmogorov equation will be considered first. We shall analyse solutions to this equation of the travelling wave form... 

We will now proceed to the investigation of properties of general solutions, not necessarily of a travelling wave form, to the Fisher-Kolmo-gorov equation, (6.123b). To simplify further equations, we will perform a change of variables in (6.123b)... 

The roots Ali2(1,2) in this model are identical with the eigenvalues, for a = D = 1, in the Fisher-Kolmogorov model. As the second pair of the roots does not lead to a generation of the sensitive states, the nature of a catastrophe for the Oregonator with diffusion is the same as for the Fisher-Kolmogorov model. In other words, the waves with the velocities v centre manifold theorem the sensitive state is associated only with the zlt z2 variables. 

This is exactly the same as the equation for a Fisher wavefront in a reference frame co-moving with the wave speed. It has solutions consistent with the boundary conditions C( —> — oc) = 0, C( —> Too) = 1 for a range of values of ao- But, as discussed in Sect. 4.2, the one with ao = 2 is dynamically preferred. Thus, the asymptotic... 

Electrochemical Measurements. Cyclic voltammetry and alternating current (ac) impedance spectroscopy were performed using an ac impedance system (EG G Princeton Applied Research model 378) that included a potentiostat-galvanostat (model 273), a two-phase lock-in analyzer (model 5208), and an IBM PS/2 computer. For ac impedance measurements, a 5-mV sine wave was superimposed on an applied voltage bias from the potentiostat. The reference electrodes were saturated calomel electrodes (SCE Fisher) for measurements in aqueous solution and silver electrodes... 

Direct measurements of permeabilities in unconsolidated marine sediments are difficult, and only few examples are published. They confine to measurements on discrete samples with a specially developed tool (Lovell 1985), to indirect estimations by resistivity measurements (Lovell 1985), and to consolidation tests on ODP cores using a modified medical tool (Olsen et al. 1985). These measurements are necessary to correct for the elastic rebound (MacKillop et al. 1995) and to determine intrinsic permeabilities at the end of each consolidation step (Fisher et al. 1994). In Section 2.4.2 a numerical modeling and inversion scheme is described which estimates permeabilities from P-wave attenuation and dispersion curves (c.f. also section 3.6). 

Jones RG, Chan ASY, Roper MG, Skegg MP, Shuttleworth IG, Fisher CJ, Jackson GJ, Lee JJ, Woodruff DP, Singh NK, Cowie BCC (2002) X-ray standing waves at surfaces. J Phys-Condens Mat 14 4059-4074... 

Two different solitary wave solutions for eq. (12.148) have been derived in the literature. There is an asymptotic solution developed by Fisher and others [28] and an exact solution derived by Ablowitz and Zeppetella [29]. The asymptotic solution gives an excellent representation of the population wave from the top saturation level to the front edge where the total population tends to zero. The usefulness of the exact solution of Ablowitz and Zeppetella has been questioned in the literature [28] because, although exact, it does not represent all possible solutions and the most relevant solutions may not be represented by it. In the following we consider only the Fisher solution, which is biologically significant [12]. For the application of the Ablowitz and Zeppetella solution,... 

Ablowitz, M. J. Zeppetella, A. Explicit solutions of Fisher s equation for a special wave speed. Bull. Math. Biol. 1979, 41, 835-840. 

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