We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching a limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120 degrees. In contrast to previous studies, we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves with vorticity on water of finite depth. Aside from the existence of highest waves, we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves). Furthermore, we prove several new facts about steady waves, such as a lower bound for the wavelength of Stokes waves, while also eliminating a possibility of the wave breaking for waves with non-negative vorticity.

We give a quite simple approach how to prove the absence of loops in bifurcation branches of water waves in rotational case with arbitrary vorticity distribution. The supporting flow may contain stagnation points and critical layers and water surface is allowed to be overhanging. Monotonicity properties of the free surface are presented. Especially simple criterium of absence of loops is given for bifurcation branches when the bifurcation parameter is the water wave period. We show that there are no loops if you start from a water wave with a positive/negative vertical component of velocity on the positive half period.

We consider a spectral problem associated with steady water waves of extreme form on the free surface of a rotational flow. It is proved that the spectrum of this problem contains arbitrary large negative eigenvalues and they are simple. Moreover, the asymptotics of such eigenvalues is obtained.

The authors consider the Neumann problem for the nonstationary Stokes system in a two-dimensional angle or a three-dimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions.

We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

Untill 1980 one of the main subjects of study in the theory of nonlinear water waves were the Stokes and solitary waves (regular waves). To that time small amplitude regular waves were constructed and the existence of large amplitude water waves of the same type was proved by using branches of water waves starting from a trivial (horizontal) wave and ending at extreme waves. Then in papers Chen & Saffman [7] and Saffman [32] numerical evidence was presented for existence of other type of waves as a result of bifurcations from a branch of ir-rotational Stokes waves on flow of infinite depth. It was demonstrated that the Stokes branch has infinitely many bifurcation points when it approaches the extreme wave and periodic waves with several crests of different height on the period bifurcate from the main branch. The only theoretical works dealing with this phenomenon are Buffoni, Dancer & Toland [4,5] where it was proved the existence of sub-harmonic bifurcations bifurcating from the Stokes branch for the ir-rotational flow of infinite depth approaching the extreme wave.The aim of this paper is to develop new tools and give rigorous proof of existence of subharmonic bifurcations in the case of rotational flows of finite depth. The whole paper is devoted to the proof of this result formulated in Theorem 5.4.

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearityfisL(p)function withp> 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.

The Cauchy problem for Helmholtz equation, for moderate wave number k(2), is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet-Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet-Robin alternating algorithm was given for general elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by similar operator equations and model problems which are used for symmetric differential operators.

In this work we propose an algorithm for computing a stationary flow in a bifurcation tree. Our idea is to divide the tree into smaller basic blocks, each corresponding to one bifurcation, and solve a sequence of flow problems for the individual blocks. Numerical experiments demonstrate that the algorithm works well. We give a criteria for convergence that can be verified numerically and also an analytical convergence proof for an important special case. The application we have in mind is the computation of the time dependent blood flow in the arterial tree of the human body. The work presented here is for a simplified case but we discuss the extension of our work to the realistic cases. Also potential applications are discussed. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k², in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k² in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

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In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R0≈1. We show even more, that for the values R0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate γ¯γ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively. As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.

We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setupand analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependentmortality. Second, dispersal between patches ensures that each patch can be reachedfrom every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniquenessof solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator,we introduce the net reproductive operator and the basic reproduction number R0 fortime-independent and periodical models and establish the permanence dichotomy:if R₀ ≤ 1, extinction on all patches is imminent, and if R₀ > 1, permanence on allpatches is guaranteed. We show that a solution for the general time-dependent problemcan be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistenceof a solution for the general time-dependent problem and describe its asymptoticbehaviour

We exploit a two-dimensional model (Ghosh et al. in Q J Mech Appl Math 71(3):349-367, 2018; Kozlov and Nazarov in Dokl Phys 56(11):560-566, 2011, J Math Sci 207(2):249-269, 2015) describing the elastic behavior of the wall of a flexible blood vessel which takes interaction with surrounding muscle tissue and the 3D fluid flow into account. We study time periodic flows in an infinite cylinder with such intricate boundary conditions. The main result is that solutions of this problem do not depend on the period and they are nothing else but the time independent Poiseuille flow. Similar solutions of the Stokes equations for the rigid wall (the no-slip boundary condition) depend on the period and their profile depends on time.

A Reynolds equation governing the steady flow of a fluid through a curvilinear, narrow tube, with its derivation from Navier-Stokes equations through asymptotic methods is presented. The channel considered may have a rather large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed by artificially imposing appropriate boundary conditions at the inlet and the outlet. A justification for the approximations is provided along with a comparison with a simpler case.

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to "half-solitary" waves (e.g. bores) which decay in only one direction.

In this note we discuss some recent results on extreme steady waves under gravity. They include the existence and regularity theorems for highest waves on finite depth with and without vorticity. Furthermore, we state new results concerning the asymptotic behavior of surface profiles near stagnation points. In particular, we find that the wave profile of an extreme wave is concave near each crest, provided that the vorticity is negative near the surface.

In this paper we study prey-predator relationship in the presence of prey species pool. We describe the set of parameters when the predator will survive and find the threshold when you lose predator survival. This is based on analysis of dependence of locally stable equilibrium points on the parameters of the problem especially on the predator natural mortality rate e . In particular we give an interval for e in terms of other parameters where predators survive. This can allow to control the survival of predators by certain actions to move certain parameters to a safe area. We discover also a more complicated dynamics for small mortality rate. (C) 2021 Published by Elsevier B.V. on behalf of African Institute of Mathematical Sciences / Next Einstein Initiative.

We consider a tritrophic system with one basal and one top species and a large number of primary consumers, and derive upper and lower bounds for the total biomass of the middle trophic level. These estimates do not depend on dynamical regime, holding for fixed point, periodic, or chaotic dynamics. We have two kinds of estimates, depending on whether the predator abundance is zero. All these results are uniform in a self-limitation parameter, which regulates prey diversity in the system. For strong self-limitation, diversity is large; for weak self-limitation, it is small. Diversity depends on the variance of species parameter values. The larger this variance, the lower the diversity, and vice versa. Moreover, variation in the parameters of the Holling type II functional response changes the bifurcation character, with the equilibrium state with nonzero predator abundance losing stability. If that variation is small then the bifurcation can lead to oscillations (the Hopf bifurcation). Under certain conditions, there exists a supercritical Hopf bifurcation. We then find a connection between diversity and Hopf bifurcations. We also show that the system exhibits top-down regulation and a hump-shaped diversity-productivity curve. We then extend the model by allowing species to experience self-regulation. For this extended model, explicit estimates of prey diversity are obtained. We study the dynamics of this system and find the following. First, diversity and system dynamics crucially depend on variation in species parameters. We show that under certain conditions, the system undergoes a supercritical Hopf bifurcation. We also establish a connection between diversity and Hopf bifurcations. For strong self-limitation, diversity is large and complex dynamics are absent. For weak self-limitation, diversity is small and the equilibrium with non-zero predator abundance is unstable.

This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for C-1-domains with C-1-perturbations. We also derive an optimal estimate for the remainder term in the C-1,C-alpha-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid. (C) 2020 Elsevier Masson SAS. All rights reserved.

The paper deals with the Dirichlet problem for the nonstationary Stokes system in a cone. The authors obtain existence and uniqueness results for solutions in weighted Sobolev spaces and study the asymptotics of the solutions at infinity. (C) 2020 Elsevier Inc. All rights reserved.

The authors consider the Dirichlet problem for the nonstationary Stokes system in a threedimensional cone. In a previous work they had proved the existence and uniqueness of solution in weighted L-2 Sobolev spaces, where the weights are powers of the distance from the vertex of the cone. Now they extend these results to weighted L-q,L-p Sobolev spaces.

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Under the assumption that the vorticity is a negative constant whose absolute value is sufficiently large, we construct a solution with the following properties. The corresponding flow is unidirectional at infinity and has a solitary wave of elevation as its upper boundary; under this unidirectional flow, there is a bounded domain adjacent to the bottom, which surrounds an interior stagnation point and is divided into two subdomains with opposite directions of flow by a critical level curve connecting two stagnation points on the bottom.

A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form f (u) with f belonging to the class L-loc(p) and the solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.

The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect.

Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection are important in this case. We formulated an susceptible infected recovered (SIR) model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of carrying capacity. The model describes five classes of a population: susceptible, infected by first virus, infected by second virus, infected by both viruses, and completely immune class. We proved that for any set of parameter values, there exists a globally stable equilibrium point. This guarantees that the disease always persists in the population with a deeper connection between the intensity of infection and carrying capacity of population. Increase in resources in terms of carrying capacity promotes the risk of infection, which may lead to destabilization of the population.

We develop a novel approach to study the global behaviour of large foodwebs for ecosystems where several species share multiple resources. The model extends and generalizes some previous works and takes into account self-limitation. Under certain explicit conditions, we establish the global convergence and persistence of solutions.

An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed, and several threshold conditions are obtained, which describes the establishment of diseases in the population. We prove that, for small carrying capacity K, there exists a globally stable disease-free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, that is, we prove that, (1) for small values of K, there exists a unique globally stable equilibrium point, and (b) it moves continuously as K is growing (while its face type may change). This indicates that the carrying capacity is the crucial parameter and an increase in resources in terms of carrying capacity promotes the risk of infection.

The paper deals with the Dirichlet problem for the nonstationary Stokes system in a threedimensional cone. The authors study the asymptotics of the solutions near the vertex of the cone. They show that the solutions are sums of singular functions and a regular remainder, where the singular functions depend on the eigenvalues of a certain operator pencil.

We consider a bifurcation of an artery. The influence of defects of the vessels wall near the bifurcation point on the pressure drop matrix is analyzed. The elements of this matrix are included in the modified Kirchhoff transmission conditions, which were introduced earlier in [1], [2], and which describe adequately the total pressure loss at the bifurcation point of the flow passed through it.

The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. The reduced system describes all small-amplitude solutions of the problem and, as an application, we give a proof of the existence of non-symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one. (C) 2019 Published by Elsevier Inc.

We study a non-linear operator equation originating from a Cauchy problem for an elliptic equation. The problem appears in applications where surface measurements are used to calculate the temperature below the earth surface. The Cauchy problem is ill-posed and small perturbations to the used data can result in large changes in the solution. Since the problem is non-linear certain assumptions on the coefficients are needed. We reformulate the problem as an non-linear operator equation and show that under suitable assumptions the operator is well-defined. The proof is based on making a change of variables and removing the non-linearity from the leading term of the equation. As a part of the proof we obtain an iterative procedure that is convergent and can be used for evaluating the operator. Numerical results show that the proposed procedure converges faster than a simple fixed point iteration for the equation in the the original variables.

We consider blood flow in a vessel with an attached capillary system. The latter is modelled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.

One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.

In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.

The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.

We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier-Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved.

We present a two dimensional model describing the elastic behaviour of the wall of a curved pipe to model blood vessels in particular. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the vessel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with the surrounding material and the fluid flowing inside into account and is obtained via a dimension reduction procedure. The curvature and twist of the vessel axis as well as the anisotropy of the laminate wallpresent the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of vessels and their walls are supplied with explicit systems of dierential equations at the end.

We present a two-dimensional model describing the elastic behaviour of the wall of a curved flexible pipe. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the channel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with any surrounding or supporting material and the fluid flow into account and is obtained via a dimension reduction procedure. The curvature and twist of the pipes axis as well as the anisotropy of the laminate wall present the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of pipes and their walls are supplied with explicit systems of differential equations at the end.

We study the biodiversity problem for resource competition systems with extinctions and self-limitationeffects. Our main result establishes estimates of biodiversity in terms of the fundamental parameters ofthe model. We also prove the global stability of solutions for systems with extinctions and large turnoverrate. We show that when the extinction threshold is distinct from zero, the large time dynamics of systemis fundamentally non-predictable. In the last part of the paper we obtain explicit analytical estimates ofecosystem robustness with respect to variations of resource supply which support the R* rule for a systemwith random parameters.

Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.

We consider a bifurcation of an artery. The influence of defects of the vessel's wall near the bifurcation point on the pressure drop matrix is analyzed. The elements of this matrix are included in the modified Kirchhoff transmission conditions, which were introduced earlier in [1], [2], and which describe adequately the total pressure loss at the bifurcation point of the flow passed through it.

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

We study scattering on an ultra-low potential in armchair graphene nanorib bon. Using the continuous Dirac model and including a couple of articial waves in the scattering process, described by an augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies, where the the multiplicity of the continuous spectrum changes and show that a trapped mode may appear for energies slightly less than a thresold and its multiplicity does not exceed one. For energies which are higher than a threshold, there are no trapped modes, provided that the potential is suciently small.

Elastic bodies with cuspidal singularities at the surface are known to support wave processes in a finite volume which lead to absorption of elastic and acoustic oscillations (this effect is recognized in the engineering literature as Vibration Black Holes). With a simple argument we will give the complete description of the phenomenon of wave propagation towards the tip of a three-dimensional cusp and provide a formulation of the Mandelstam (energy) radiation conditions based on the calculation of the Umov-Poynting vector of energy transfer. Outside thresholds, in particular, above the lower bound of the continuous spectrum, these conditions coincide with ones due to the traditional Sommerfeld principle but also work at the threshold frequencies where the latter principle cannot indicate the direction of wave propagation. The energy radiation conditions supply the problem with a Fredholm operator of index zero so that a solution is determined up to a trapped mode with a finite elastic energy and exists provided the external loading is orthogonal to these modes. An example of a symmetric cuspidal body is presented which supports an infinite sequence of eigenfrequencies embedded into the continuous spectrum and the corresponding trapped modes with the exponential decay at the cusp tip. We also determine (unitary and symmetric) scattering matrices of two types and derive a criterion for the existence of a trapped mode with the power-low behavior near the tip.

 A false aneurysm is a hematoma, i.e. collection ofblood outside of a blood vessel, that forms due to a hole  in the wall of an artery . This represents a serious medical condition that needs to be monitored and, under certain conditions, treatedurgently. In this work a one-dimensional model of a false aneurysm isproposed. The new model is based on a one-dimensional model of anartery previously presented by the authors and it takes into accountthe interaction between the hematoma  and the surrounding musclematerial. The model equations are derived  using rigorous asymptoticanalysis for the case of a simplified geometry.   Even though the model is simple it still supports a realisticbehavior for the system consisting of the vessel and the  hematoma. Using numerical simulations we illustrate the behavior ofthe model. We also investigate the effect  of changing the size of the hematoma. The simulations show that ourmodel can reproduce realistic solutions. For instance we show thetypical strong pulsation of an aneurysm by blood entering the hematoma during the work phase of the cardiac cycle, and the blood returning tothe vessel during the resting phase. Also we show that the aneurysmgrows  if the pulse rate is increased due to, e.g., a higher work load. 

We study a Stokes flow in a junction of thin channels (of diameter O(h)) for fixed flows of the fluid at the inlet cross-sections and fixed peripheral pressure at the outlet cross-sections. On the basis of the idea of the pressure drop matrix, apart from Neumann conditions (fixed flow) and Dirichlet conditions (fixed pressure) at the outer vertices, the ordinary one-dimensional Reynolds equations on the edges of the graph are equipped with transmission conditions containing a small parameter h at the inner vertices, which are transformed into the classical Kirchhoff conditions as h -amp;gt;+ 0. We establish that the pre-limit transmission conditions ensure an exponentially small error O(e(-rho/h)),. amp;gt; 0, in the calculation of the three-dimensional solution, but the Kirchhoff conditions only give polynomially small error. For the arterial tree, under the assumption that the walls of the blood vessels are rigid, for every bifurcation node a (2x2)-pressure drop matrix appears, and its influence on the transmission conditions is taken into account by means of small variations of the lengths of the graph and by introducing effective lengths of the one-dimensional description of blood vessels whilst keeping the Kirchhoff conditions and exponentially small approximation errors. We discuss concrete forms of arterial bifurcation and available generalizations of the results, in particular, the Navier-Stokes system of equations.

We investigate the formation of stable ecological networks where many species share the same resource. We show that such a stable ecosystem naturally occurs as a result of extinctions. We obtain an analytical relation for the number of coexisting species, and we find a relation describing how many species that may become extinct as a result of a sharp environmental change. We introduce a special parameter that is a combination of species traits and resource characteristics used in the model formulation. This parameter describes the pressure on the system to converge, by extinctions. When that stress parameter is large, we obtain that the species traits are concentrated at certain values. This stress parameter is thereby a parameter that determines the level of final biodiversity of the system. Moreover, we show that the dynamics of this limit system can be described by simple differential equations.