Nonmonotone invariant manifolds in the Nagylaki–Crow model

We use a change of dynamical variables to prove, subject to certain conditions on the parameters, that a nonmonotone invariant manifold exists and is the graph of a convex function for the planar Nagylaki–Crow fertility–mortality model from population genetics with n = 2. Our results are obtained without the common assumption that fertilities or death rates are additive, and are not restricted to the case that the model is competitive in the new coordinates. We also provide numerical examples demonstrating that the manifold need not be the graph of a convex function, smooth, unique or globally attracting, and that the model exhibits a sequence of nonmonotone manifolds similar to those studied by Hirsch for competitive Kolmogorov systems (Hirsch 1988)

Invariant manifolds of models from population genetics

Many models in population genetics feature some form of convergence of the genetic state of the population, typically onto a globally attracting invariant manifold. This allows one to effectively reduce the dynamical system to a problem with fewer dimensions, making it easier to investigate the stability of the steady states in the model, as well as to predict the long-term evolution of the population. Moreover, along this manifold, there is a balance between multiple processes, such as selection and recombination. For some models, restrictive assumptions such as small selection coefficients or additivity of fertilities and mortalities has helped show global contraction of dynamics onto a manifold which is close to the well-known Hardy-Weinberg manifold, and on this `quasiequilibrium’ manifold the dynamics can be written in terms of allele frequencies (which is of more practical interest to geneticists than the genotype frequencies). This thesis focuses on proving the existence of an invariant manifold for two continuous-time models in population genetics: one is proposed by Nagylaki and Crow and features fertilities and mortalities (death rates), while the other is the selection-recombination model. Common themes in both proofs include a change of coordinates such that the dynamical system is monotone with respect to a certain cone. As a result, it is possible to construct an equicontinuous sequence of functions which has a convergent subsequence. We show this limiting function is indeed invariant. In fact, for the latter model, we show the manifold is globally attracting by proving the phase volume is contracting. The conditions obtained from the proofs are less restrictive than the use of parameters that are small or additive, hence our work is more widely applicable. For the former model, numerical examples are also provided in which the manifold need not be smooth, convex, unique or globally attracting

Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine–Gordon type equation which is known to have soliton solutions

JOURNAL OF GLAUCOMA

Purpose: To compare the efficacy of selective laser trabeculoplasty (SLT) in phakic and pseudophakic eyes in open-angle glaucoma and ocular hypertension. Materials and Methods: Charts of 28 pseudophakic eyes and 60 phakic eyes that underwent 360-degree SLT were retrospectively reviewed. Patients were examined at 1, 3, 6, and 12 months. Treatment success was defined as >= 20% intraocular pressure (IOP) reduction, with no additional medications, laser, or glaucoma surgery. Mean IOP change, mean percentage of IOP reduction, and success rates for phakic and pseudophakic eyes were compared. Results: Mean percentage of IOP reduction post-SLT at 1-, 3-, 6-, and 12-month visits were 21.4%, 25.8%, 24.8%, and 23.7%, respectively, in the pseudophakic group and 22.8%, 25.0%, 25.7%, and 21.2%, respectively, in the phakic group. Success rates ranged between 60% and 64% in the pseudophakic group and between 58% and 73% in the phakic group. No statistically significant differences in IOP change, percentage of IOP reduction, and success rate were seen between the groups at any of the post-SLT visits (P > 0.05). Conclusions: Application of 360-degree SLT seems to be an efficient and safe treatment option for the management of phakic and pseudophakic open-angle glaucoma and ocular hypertension

CURRENT EYE RESEARCH

Purpose: To determine if corneal biomechanical properties change during phases of the menstrual cycle. Methods: Twenty-five healthy women of reproductive age with no ocular pathology or systemic diseases were recruited. Corneal hysteresis, corneal resistance factor, Goldman-correlated intraocular pressure, and corneal-compensated intraocular pressure were measured by a Reichert ocular response analyzer at three phases of the menstrual cycle, beginning on days 3 to 5 (follicular phase), again at ovulation (days 14-16, ovulatory phase), and at the end of the cycle (days 25-28, luteal phase). Results: Twenty-one participants completed the study. The mean corneal hysteresis values at follicular, ovulatory, and luteal phases were 10.7, 10.7, and 10.7 mmHg (p > 0.05), and the mean corneal resistance factor values at the same time points were 9.9, 9.9, and 9.8 mmHg (p > 0.05), respectively. Corneal-compensated intraocular pressure and Goldman-correlated intraocular pressure readings were stable during the course (p > 0.05 for all). Conclusions: Corneal biomechanical properties and intraocular pressure as measured by an ocular response analyzer do not change statistically significantly during the different phases of the menstrual cycle. We conclude that measurements of the ocular response analyzer can be safely utilized in healthy female subjects without considering the possible effects of varying hormonal levels during the menstrual cycle

JOURNAL OF GLAUCOMA

Purpose: To compare the efficacy of fixed combinations of dorzolamide-timolol (FCDT) and brimonidine-timolol (FCBT) in patients with intraocular pressure (IOP) elevations after intravitreal triamcinolone acetonide (IVTA) injections. Materials and Methods: This was a prospective, randomized, open-label study. Patients who received IVTA injections due to diffuse diabetic macular edema and who had an IOP of 24 mm Hg or higher after IVTA treatment were included. They were randomized to receive either FCBT or FCDT twice daily. Follow-up visits were scheduled on week 4 and 12 weeks after starting the study medication. At all follow-up visits, IOP was measured with Goldmann applanation tonometry. The primary outcome measure was mean IOP, the secondary outcome was reduction in mean IOP at 4 and 12 weeks compared with postinjection values. Results: Sixty patients were randomized in 1: 1 ratio. The FCBT and FCDT groups were similar in terms of age, sex, and preinjection IOP (P > 0.05 for all). Mean postinjection IOP was 31.95 +/- 7.39 and 29.83 +/- 5.17 mm Hg in FCBT and FCDT groups, respectively (P = 0.239). After 4 weeks, mean IOP was 17.05 +/- 3.61 mm Hg in FCBT and 18.93 +/- 3.30 mm Hg in FCDT groups (P = 0.063). After 12 weeks, mean IOP in the FCBT and FCDT study groups was 16.35 +/- 2.70 and 18.43 +/- 2.82 mm Hg, respectively (P = 0.012). Both fixed combinations significantly reduced IOP in comparison with the postinjection values (P < 0.05). Mean reduction in IOP after 4 weeks were 14.90 +/- 7.28mm Hg in FCBT and 10.90 +/- 4.83 mm Hg in FCDT groups (P = 0.024); after 12 weeks, these values were 15.60 +/- 7.77 and 11.40 +/- 5.89 mm Hg in FCBT and FCDT groups, respectively (P = 0.035). Conclusions: Both FCBT and FCDT are effective in controlling IOP elevations after IVTA injections. The results of this study suggest that FCBT is superior to FCDT in reducing IOP and provides better IOP control after IVTA injections

JOURNAL OF CATARACT AND REFRACTIVE SURGERY

We report the ocular manifestations of Alport syndrome and the surgical results in 4 patients. All 4 patients had anterior lenticonus; 2 also had posterior lenticonus in both eyes, 3 had flecked retina, and 1 had posterior polymorphous dystrophy. In both eyes of the 4 patients, phacoemulsification with intraocular lens (IOL) implantation was performed to treat anterior and posterior lenticonus. The postoperative visual acuity was excellent in all patients. We recommend phacoemulsification with IOL implantation as a safe and effective procedure in patients with lenticonus secondary to Alport syndrome