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Aims:The aim of this work is to report on the tumour control probability (TCP) of a UK cohort of lung stereotactic ablative radiotherapy patients (n = 198) for a range of dose and fractionations common in the UK.Materials and methods:TCP values for 3 (54 Gy), 5 (55 and 60 Gy) and 8 (50 Gy) fraction (#) schemes were calculated with the linear-quadratic Marsden TCP model using the Biosuite software.Results:TCP values of 100% were computed for the 3 # and for 5 # (α/β = 10 Gy) cohorts; reduced to 99% (range 97–100) for the 5 # cohort only when an α/β of 20 Gy was used. The average TCP value for the 50 Gy in 8 # regime was 97% (range 92–99, α/β = 10 Gy) and 64% (range 48–79, α/β = 20 Gy). Statistical significant differences were observed between the α/β of 10 Gy versus 20 Gy groups and between all data grouped by fraction.Conclusion:TCPs achievable with current planning techniques in the UK have been presented. The ultra-conservative 50 Gy in 8 # scheme returns a significantly lower TCP than the other regimes.

Winter is a challenging period for aquatic research—weather is uncomfortable, ice is hazardous, equipment fails, and daylength is short. Consequently, until recently relatively little research on freshwater fishes has included winter. Telemetry methods for tracking fish and observing movement behavior are an obvious solution to working in harsh conditions because much of the data can be collected remotely, and passive methods collect data year-round without winter maintenance. Yet, many telemetry studies do not collect data during winter or, if they do, only report data from the ice-free seasons while the remaining data are unused. Here, we briefly summarize the advantages and limitations of using telemetry methods in winter, including acoustic and radio telemetry and passive integrated transponder technology, then review the range of questions related to fish ecology, behavior, bioenergetics, and habitat use that can be addressed in winter using telemetry. Our goals are to highlight the untapped potential of winter fish biology and to motivate scientists to revisit their four-season telemetry data and incorporate objectives specific to winter biology in future study plans.

Spawning behavior of lake trout, Salvelinus namaycush, is poorly understood, relative to stream-dwelling salmonines. Underwater video records of spawning in a recovering population from the Drummond Island Refuge (Lake Huron) represent the first reported direct observations of lake trout spawning in the Laurentian Great Lakes. These observations provide new insight into lake trout spawning behavior and expand the current conceptual model. Lake trout spawning consisted of at least four distinct behaviors: hovering, traveling, sinking, and gamete release. Hovering is a new courtship behavior that has not been previously described. The apparent concentration of hovering near the margin of the spawning grounds suggests that courtship and mate selection might be isolated from the spawning act (i.e., traveling, sinking, and gamete release). Moreover, we interpret jockeying for position displayed by males during traveling as a unique form of male-male competition that likely evolved in concert with the switch from redd-building to itinerant spawning in lake trout. Unlike previous models, which suggested that intra-sexual competition and mate selection do not occur in lake trout, our model includes both and is therefore consistent with evolutionary theory, given that the sex ratio on spawning grounds is skewed heavily towards males. The model presented in this paper is intended as a working hypothesis, and further revision may become necessary as we gain a more complete understanding of lake trout spawning behavior.

In this paper, we (1) describe field obser- vations of lake charr (trout) reproductive behaviour, focusing on two morphs from Great Bear Lake, N.T; (2) review and synthesize observations from the literature; and (3) summarize uncertainties and pose hypotheses regarding lake trout spawning behaviour. Our description is set within the context of an existing conceptual model that included three sequential stages to spawning behaviour—traveling, sinking, and gam- ete release. Spawning morphs did not overlap in time

The influence of egg predators and physical disturbance on lake trout Salvelinus namaycush egg mortality was investigated in situ in Lake Michigan where recruitment is below detectable levels and egg predator abundance is high. Comparisons were made with Lake Champlain where recruitment is low and egg predator abundance is also low and with Parry Sound (Lake Huron) where recruitment is moderate and egg predators are in low abundance. A multi-density egg seeding method (100 to 5000 eggs m−2) was used to quantify the effect of physical disturbance and egg predation on egg loss. Wind fetch was used as an index of physical disturbance and comparisons across all locations and egg densities suggested that at sites with high wind fetch (>5 km), physical disturbance may be a greater source of egg loss than predation. When analyses were limited to those sites having a wind fetch of

Lake sturgeon (Acipenser fulvescens) are threatened or endangered throughout much of their range. Juvenile sturgeon utilize sandy and silty habitats extensively during their growth. Invasive zebra mussels change the nature of sandy and silty habitats because they settle on and coat the habitat with the shells of living and dead individuals. The potential impacts of this increased habitat complexity on lake sturgeon is unknown. 2. Juvenile lake sturgeon habitat choice was assessed in laboratory experiments, and zebra mussel impact on the foraging success of juvenile lake sturgeon on three different prey species was measured. 3. Sturgeon foraging on chironomids was virtually eliminated by 95% zebra mussel cover of the sand floor of the foraging arena, and 50% cover reduced foraging significantly. Foraging on more mobile prey items (amphipods and isopods) was essentially eliminated by either 95% or 50% zebra mussel cover of the arena floor. In habitat choice experiments, sturgeon avoided the zebra-mussel- covered habitat more than 90% of the time. 4. This combination of zebra mussel avoidance and reduced foraging in the presence of zebra mussels may be detrimental to sturgeon restocking programmes utilizing smaller sturgeon in zebra-mussel-infested waterways. Copyright # 2006 John Wiley & Sons, Ltd.

Summary 1. We examined the effect of zebra mussel colonisation on invertebrate communities inhabiting soft sediments in two bays along the Vermont shoreline of Lake Champlain, U.S.A. 2. In summer 2001, we conducted manipulative experiments (addition and removal of zebra mussel colonies) with respective controls to assess the effect of colonies on invertebrate abundance, richness, and position within sediments (within colony versus underlying colony) and compared these data to comparative studies of natural communities in summer 2002. 3. Split core samples were taken two months after the manipulation and the following year so that we could quantify individuals and species inhabiting zebra mussel colonies separately from those in sediments underlying zebra mussel colonies and adjacent sediments lacking zebra mussels. 4. Zebra mussel-covered sediments supported more abundant and richer invertebrate communities than adjacent sediments lacking zebra mussels. 5. Abundance and richness patterns in zebra mussel-addition and removal treatments closely paralleled those in natural communities. 6. Despite severe oxygen depletion at the interface of underlying sediments and overlying zebra mussel colonies, most infaunal invertebrates responded positively to zebra mussel colonisation either by remaining in sediments underlying zebra mussel colonies or by migrating into zebra mussel colonies.

Adult zebra mussels are generally thought of as sessile animals. However, when detached from their byssus, adult mussels exhibit creeping behavior similar to that of other bivalve species. Our study examined the effects of size, light, water hardness, temperature, and the presence of injured conspecifics on the movement of adult zebra mussels. Mussels were placed in aquaria and their movement was recorded after a 2-h interval. Mussels left visible trails in a thin layer of sand on the aquarium floor that allowed for accurate measurements of distance and direction of movement. The tendency to move and distance moved by the mussels was inversely proportional to size of the individual mussels. There was no significant effect of temperature on movement, though smaller mussels tended to move more at higher temperatures. Although distance traveled was not significantly different in light versus dark conditions, there was a significant effect of light on direction traveled. Zebra mussels were strongly ne...

Treatment plan quality was investigated to determine whether the total monitor units (MUs) could be reduced by using the MU objective function within the EclipseTM planning system (Varian Medical Systems) for stereotactic ablative body radiotherapy (SABR) lung plans. The plans were produced using RapidArc® (Varian's volumetric modulated arc therapy) technique. This study used physical measurements to assess the deliverability and quality of the resultant plans on the treatment machine. Ten patients (dose prescriptions ranging from 6.25 to 18 Gy per fraction) were re-planned retrospectively using objective limit MUs to approximately 50% of the clinical plan (CP) MUs. A number of parameters including dose volume histogram, conformity index, gradient measure, total MUs, treatment time, mean leaf gap, small aperture score (SAS) and gamma (2%/2 mm) were compared. To assess deliverability, plans were measured on a treatment machine using the MatriXX Array and the CompassTM system (IBA Dosimetry). Whilst the quality of plans with and without MU objective function was not statistically different, the number of MUs and overall treatment time was reduced by 23% and 1.71 min, respectively. Mean leaf gap increased by 22.3%, SAS reduced by 49% and normalised MU (MU/cGy) reduced by 22.0%. Importantly, verification and improvements in plan deliverability were measured. The MU objective method showed an overall improvement in plan quality and deliverability; no plans were statistically worse. By reducing the number of MUs for treatment the delivery time is reduced thereby decreasing the time the patient is on the radiotherapy couch. Our institution now uses this technique for clinical lung SABR patients routinely.

Higher doses are delivered over a short duration for stereotactic ablative radiotherapy (SABR) and as a result individual fraction times are significantly higher compared with conventional radiotherapy. Furthermore, many lung SABR patients are elderly with associated co-morbidities and may not be able to retain their treatment position adequately. These patients benefit from faster treatment deliveries which can be achieved by using flattening filter free (FFF) beams. To determine a clinically appropriate FFF energy for accurate delivery, 15 previously delivered flattened 6 MV lung SABR plans were re-planned at 6 FFF and 10 FFF, with organ at risk (OAR) and target dose-volume statistics examined for significance. A two half arc technique, the Monitor Unit Objective Function and the AcurosXB algorithm were employed within the EclipseTM planning system (V11, Varian Medical System). The deliverability of these FFF plans was verified by physical measurement on a TrueBeamTM (V2.5, Varian Medical System) using the CompassTM dosimetry system (V3.1, IBA Dosimetry) in addition to the usual treatment planning system comparisons. Acceptable plans were produced for all beam energies. 6 FFF provided statistically significant OAR sparing compared to 6 FF and 10 FFF. However, absolute dose differences were not clinically significant and doses were well within recommended clinical tolerances. Skin sparing was superior in the 10 FFF plans. Overall, reduction in treatment delivery time of 61% and 55% was found when using 10 FFF and 6 FFF respectively compared to 6 FF. A 15% reduction in the average treatment time was achieved with 10 FFF when compared to 6 FFF. Treatment delivery verification measurements were compared with clinically delivered 6 FF plans and no significant differences in the deliverability were seen between the plans. As a result of this study 10 FFF has been implemented for SABR lung planning locally.

Stocking of juvenile lake trout reared in hatcheries has been one of several management actions used to restore self-sustaining populations of lake trout Salvelinus namaycush in the Great Lakes. Since 1973, Lake Ontario has been stocked predominately with Clearwater, Jenny, Killala, Manitou, Seneca, and Superior strains. Reproductive success of these strains was assessed by using allozyme and mitochondrial DNA (mtDNA) characters with mixed-stock analysis. Parental origins of wild fry were compared over several years at Stony Island reef and between Stony Island reef and Yorkshire Island reef in 1990. Strain origins at Stony Island reef were examined among four year-classes (1986, 1988, 1989, and 1990) of wild fry and among three year-classes (1988, 1989, 1990) of fry reared from eggs collected in late autumn. Allelic frequencies were not different among year-classes within the wild fry and egg collections from Stony Island reef (P ≥ 0.05). Frequencies of mtDNA haplotypes also were not different a...

The introduction of zebra mussels ( Dreissena spp.) to North America has resulted in dramatic changes to the complexity of benthic habitats. Changes in habitat complexity may have profound effects on predator-prey interactions in aquatic communities. Increased habitat complexity may affect prey and predator dynamics by reducing encounter rates and foraging success. Zebra mussels form thick contiguous colonies on both hard and soft substrates. While the colonization of substrata by zebra mussels has generally resulted in an increase in both the abundance and diversity of benthic invertebrate communities, it is not well known how these changes affect the foraging efficiencies of predators that prey on benthic invertebrates. We examined the effect of zebra mussels on the foraging success of four benthic predators with diverse prey-detection modalities that commonly forage in soft substrates: slimy sculpin ( Cottus cognatus), brown bullhead ( Ameirus nebulosus), log perch ( Percina caprodes), and crayfish ( Orconectes propinquus). We conducted laboratory experiments to assess the impact of zebra mussels on the foraging success of predators using a variety of prey species. We also examined habitat use by each predator over different time periods. Zebra mussel colonization of soft sediments significantly reduced the foraging efficiencies of all predators. However, the effect was dependent upon prey type. All four predators spent more time in zebra mussel habitat than in either gravel or bare sand. The overall effect of zebra mussels on benthic-feeding fishes is likely to involve a trade-off between the advantages of increased density of some prey types balanced against the reduction in foraging success resulting from potential refugia offered in the complex habitat created by zebra mussels.

Theoretical treatments of three and four-wave interactions are typically based on the transformation q j = ρ j exp ( i ϕ j ) . Using the Manley-Rowe relations and the Hamiltonian, these systems are integrated in terms of elliptic functions [1]. A new approach, which ultimately leads to the same elliptic function solutions and is also based on the Hamiltonian structure, is introduced. It yields a useful geometric picture of wave interactions and simplifies the understanding and analysis of these processes. In doing so, it facilitates the development and optimization of control strategies for wave interactions such as quasi-phase-matching [1–3].

This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a three-dimensional elastic body converges, in a sense made precise, to that for a shell model described by a one-director Cosserat surface as the thickness goes to zero. We study limiting procedures that give rise to unconstrained as well as constrained Cosserat director models. The case of a rod is also considered and similar convergence results are established, with the limiting model being a geometrically exact director rod model (in the framework developed by Antman, Simo, and coworkers). The resulting model may or may not have constraints, depending on the nature of the constitutive relations and their behavior under the limiting procedure. The closeness of Hamiltonian structures is measured by the closeness of Poisson brackets on certain classes of functions, as well as the Hamiltonians. This provides one way of justifying the dynamic one-director model for shells. Another way of stating the convergence result is that there is an almost-Poisson embedding from the phase space of the shell to the phase space of the 3D elastic body, which implies that, in the sense of Hamiltonian structures, the dynamics of the elastic body is close to that of the shell. The constitutive equations of the 3D model and their behavior as the thickness tends to zero dictates whether the limiting 2D model is a constrained or an unconstrained director model. We apply our theory in the specific case of a 3D Saint Venant-Kirchhoff material andderive the corresponding limiting shell and rod theories. The limiting shell model is an interesting Kirchhoff-like shell model in which the stored energy function is explicitly derived in terms of the shell curvature. For rods, one gets (with an additional inextensibility constraint) a one-director Kirchhoff elastic rod model, which reduces to the well-known Euler elastica if one adds an additional single constraint that the director lines up with the Frenet frame.

Two polymorphic, duplicated loci (fumarase-1,2 and phosphoglucomutase-3,4), previously undescribed in salmonids, were examined in lake trout to determine their electrophoretic expression, mode of inheritance, and linkage associations. Both loci appear to be located on chromosomes which show residual tetrasomy. Neither locus pair shows joint segregation with other loci examined or with each other.

This section shows how the results of Sections 8 and 8A can be used to establish the bifurcation theorem for the Navier-Stokes equations. Alternatively, although conceptually harder (in our opinion) methods are described in Sections 9A, 9B and Sattinger [5] and Joseph and Sattinger [1].

The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations. Historically, the subject had its origins in the works of Poincaré [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term "Poincaré-Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle-Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper. These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and Kirchgässner and Kielhoffer [1] (contributed by G. Childs and O. Ruiz) are given. The contributions of S. Smale, J. Guckenheimer and G. Oster indicate applications to the biological sciences and that of D. Schmidt to Hamiltonian systems. For other applications and related topics, we refer to the monographs of Andronov and Chaiken [1], Minorsky [1] and Thom [1]. The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value. In Hopf's original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation. We have given explicit algorithms for this calculation which are easy to apply in examples. (See Section 4, and Section 5A for comparison with Hopf's formulae). The method of averaging, exposed here by S. Chow and J. Mallet-Paret in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly. In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for t > O. This enables the invariant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]). To aid in determining smoothness in examples we have presented parts of the results of Dorroh-Marsden. [1]. Similar ideas for utilizing smoothness have been introduced independently by other authors, such as D. Henry [1]. Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here. We have selected results of Chafee [1] and Ruelle [3] (the latter is exposed here by S. Schecter) to indicate some generalizations that are possible. The subject is by no means closed. Applications to instabilities in biology (see, e.g. Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf. Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field (cf. Durand [1]) are appearing at a rapid rate. Also, the qualitative theory proposed by Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9). In this direction, the papers of Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important. Stable oscillations in nonlinear waves may be another fruitful area for application; cf. Whitham [1]. We hope these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods. After we completed our stability calculations we were happy to learn that others had found similar difficulty in applying Hopf's result as it had existed in the literature to concrete examples in dimension ≥ 3. They have developed similar formulae to deal with the problem; cf. Hsü and Kazarinoff [1, 2] and Poore [1]. The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type. The new proof, relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations. These notes originated in a seminar given at Berkeley in 1973-4. We wish to thank those who contributed to this volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive. Many other references are contained in the lengthy bibliography in Cesari [1]. We also thank those who have taken an interest in the notes and have contributed valuable comments. These include R. Abraham, D. Aronson, A. Chorin, M. Crandall., R. Cushman, C. Desoer, A. Fischer, L. Glass, J. M. Greenberg, O. Gurel, J. Hale, B. Hassard, S. Hastings, M. Hirsch, E. Hopf, N. D. Kazarinoff, J. P. LaSalle, A. Mees, C. Pugh, D. Ruelle, F. Takens, Y. Wan and A. Weinstein. Special thanks go to J. A. Yorke for informing us of the material in Section 3C and to both he and D. Ruelle for pointing out the example of the Lorentz equations (See Example 4B.8). Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript. Jerrold Marsden Marjorie McCracken

Suppose we are studying a physical system whose state x is governed by an evolution equation \(\frac{{dx}}{{dt}} = X\left( x \right)\) which has unique integral curves. Let x0 be a fixed point of the flow of X; i.e., X(x0) = 0. Imagine that we perform an experiment upon the system at time t = 0 and conclude that it is then in state x0. Are we justified in predicting that the system will remain at x0 for all future time? The mathematical answer to this question is obviously yes, but unfortunately it is probably not the question we really wished to ask. Experiments in real life seldom yield exact answers to our idealized models, so in most cases we will have to ask whether the system will remain near x0 if it started near x0. The answer to the revised question is not always yes, but even so, by examining the evolution equation at hand more minutely, one can sometimes make predictions about the future behavior of a system starting near x0. A trivial example will illustrate some of the problems involved.

The center manifold theorem is used to reduce bifurcation problems to finite dimensional ones as follows.

As we have seen in earlier sections, there are two methods generally available for proving bifurcation theorems. The first is the original method of Hopf, and the second is using invariant manifold techniques to reduce one to the finite (often two) dimensional case.

We shall give here, following Lanford [1], an elementary and straightforward derivation of the canonical form for the mapping Φµ: ℝ2 → ℝ2. Recall that we have already arranged things so that $$ {{\phi }_{\mu }}\left( {\begin{array}{*{20}{c}} x \\ y \\ \end{array} } \right) = \left( {1 + \mu } \right)\;\left( {\begin{array}{*{20}{c}} {\cos \theta \left( \mu \right)} \\ {\sin \theta \left( \mu \right)} \\ \end{array} \quad \begin{array}{*{20}{c}} { - \sin \theta \left( \mu \right)} \\ {\cos \theta \left( \mu \right)} \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x \\ y \\ \end{array} } \right) + 0\left( {{{{\text{r}}}^{2}}} \right) $$ We want to organize the second, third, and fourth degree terms by making further coordinate changes. It will be convenient to identify R2 with the complex plane by writing $$ {\text{z}} = {\text{x}} + {\text{iy}} $$ Then $$ {{\phi }_{\mu }}(z) = \lambda \left( \mu \right)z + 0\left( {{{{\left| z \right|}}^{2}}} \right),\;\lambda \left( \mu \right) = \left( {1 + \mu } \right){{e}^{{{\text{i}}\theta \left( \mu \right)}}} $$ From now on we shall leave y out of our notation as much as possible.