The morphological stability of dendritic growth from the binary alloy melt with an external flow

The morphological stability of dendritic growth from the binary alloy melt with an external flow is studied by means of the matched asymptotic expansion method and multiple variable expansion method. The uniformly valid asymptotic solution is obtained for the case of the large Schmidt number. The analytical result reveals that the stability of dendritic growth depends on a critical stability number above which dendritic growth is stable. The selection condition of dendritic growth determines the Peclet number, tip growth velocity, tip radius and oscillation frequency, which is significantly affected by the external flow. The stability mechanism of dendritic growth in the binary alloy melt with the external flow remains the same as that in pure melt. In the binary alloy melt with the external flow the solute concentration destabilizes the dendritic growth system. The numerical computation for various growth conditions demonstrates the variations of the critical stability number, tip growth velocity, tip radius, and oscillatory frequency with the undercooling, external flow and morphological number.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

On the stability of jetlike magnetohydrodynamic flows

The linear stability problem is under study for steady axisymmetric translational flows of a density-homogeneous nonviscous incompressible ideal conducting fluid with free surface and “frozen-in” poloidal magnetic field. By the direct Lyapunov method, some sufficient conditions are obtained for the stability of these flows under small long-wave perturbations with the same symmetry. These stability conditions have partial converses; and, for unstable stationary flows, an a priori exponential lower bound is constructed on the growth of small perturbations under consideration, while the increment of the appearing exponent serves as an arbitrary positive parameter. An illustrative analytical example is given of steady flows with superimposed small long-wave axisymmetric perturbations growing in time in accordance with the estimate.     

Stability analysis of mathematical models for nonlinear growth kinetics of breast cancer stem cells

Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to help in designing experiments to develop novel therapeutic strategies against cancer. In this paper, by using theory of functional and ordinary differential equations, we study the existence and stability of nonlinear growth kinetics of breast cancer stem cells. First, we provide a sufficient condition for the existence and uniqueness of the solution for nonlinear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a nontrivial steady‐state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions that allow for this criteria to be satisfied. Next, we apply these theorems to a special case of the system of functional differential equations that has been used to model nonlinear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating

This paper investigates the stability of a thin incompressible viscoelastic fluid designated as Walters’ liquid B″ during spin coating. The long-wave perturbation method is proposed to derive a generalized kinematic model of the film flow. The method of normal mode is applied to study the linear stability. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. Using the multiple scales method, the weakly nonlinear stability analysis is studied for the evolution equation of a film flow. The Ginzburg–Landau equation is determined to discuss the threshold conditions of the various critical flow states. The study reveals that the rotation number and the radius of the rotating circular disk generate the destabilizing effects. Moreover, the viscoelastic parameter k indeed plays a more significant role in destabilizing the film flow than a thin Newtonian fluid during spin coating [27].     

比热容非常值对高超声速平板边界层稳定性的影响

在高超声速条件下,边界层中气体的温度可能很高,以致气体的比热容不再是常数而与温度有关．这时边界层中的流动稳定性如何是值得研究的问题．采用线性稳定性理论,考虑比热容与温度有关时高超声速可压缩平板边界层的稳定性,并与假定比热容为常值的情况作比较,发现对第一模态和第二模态波的中性曲线、最大增长率都有影响．因此,在高超声速情况下,比热容随温度变化是研究边界层稳定性时必须考虑的一个因素．     

The von Neumann facet and a global asymptotic stability

We will study a multi-sector discrete-time optimal growth model with neoclassical non-joint technology and show that any path on ann-dimensional flat supported by the optimal steady state price will converge to the optimal steady state and is optimal. Burmeister and Graham have proved a similar result in a continuous-time setting. Although their result is limited, it is a first challenge to generalize the global stability result obtained by Uzawa and Srinivasan in a two-sector optimal growth model. One prominent advantage of our approach is that due to the discrete-time model setting, we can apply the duality approach and introduce the so called "von Neumann facet" intensively studied by McKenzie, which plays a very important role in proving the saddle point stability.     

Determination of unknown functions in a mathematical model of ductal carcinoma in situ

This study deals with the numerical investigation of a mathematical model of breast cancer at the initial growth stage known as ductal carcinoma in situ. This model considered as an inverse problem and the uniqueness of solution of this inverse problem is proved. To solve this problem, a computational approach is developed based on an iterative procedure and space marching and mollification methods. The stability and convergence results are given to support the method theoretically. Two test problems are considered to demonstrate the efficiency and ability of the proposed numerical approach.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

Effects of prey over–undercrowding in predator–prey systems with prey-dependent trophic functions

The local dynamics of a two-trophic chain in the presence of both overcrowding and undercrowding effects on prey growth is investigated. The starting point is given by a general predator–prey system, in which the prey growth rate and the trophic interaction function are defined only by some properties determining their shapes; in particular, the prey growth function is assumed to model a strong Allee effect. A stability analysis of the system using the predation efficiency as bifurcation parameter is performed; conditions for the existence and stability of extinction and coexistence equilibrium states are determined, and peculiar features of the dynamics exhibited by the system are presented, with particular attention to limit cycles and bistability situations. Results are compared with those obtained when overcrowding and undercrowding effects are considered separately.     

Modelling the depletion of forestry resources by population and population pressure augmented industrialization

In this paper, a nonlinear mathematical model is proposed and analysed to study the depletion of forestry resources caused by population and population pressure augmented industrialization. It is shown that the equilibrium density of resource biomass decreases as the equilibrium densities of population and industrialization increase. It is found that even if the growth of population (whether intrinsic or by migration) is only partially dependent on resource, still the resource biomass is doomed to extinction due to large population pressure augmented industrialization. It is noted that for sustained industrialization, control measures on its growth are required to maintain the ecological stability.     

Effect of stochastic perturbation on a two species competitive model

In this paper first we study the stability and bifurcation of a two species competitive model with a delay effect. Next we extend the deterministic model system to a stochastic delay differential system by incorporating multiplicative white noise terms in growth equations of both species. We consider the stochastic stability of a co-existing equilibrium point in terms of mean square stability by constructing a suitable Lyapunov functional. We perform a numerical simulation to validate our analytical findings.     

STABILITY OF SUBHARMONIC SOLUTIONS OF FIRST-ORDER HAMILTONIAN SYSTEMS WITH ANISOTROPIC GROWTH

Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions(that is, all its Floquet multipliers lying on the unit circle on the complex plane C).     

Existence,uniqueness and stability results of random impulsive semilinear differential systems

In this paper, we consider semilinear differential systems with random impulses. We study the existence and uniqueness of the solutions by relaxing the linear growth conditions, sufficient conditions for stability through continuous dependence on initial conditions and the exponential stability of this system have been established.     

A critical case of stability in a free boundary problem

We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole . We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter is eliminated by means of the Hopf-Cole transformation. Received August 18, 2000, accepted September 27, 2000.     

具有饱和项的互惠系统的分歧与稳定性

运用谱分析和分歧理论的方法,在齐次Dirichlet边界条件下,对具有饱和项的互惠系统的非负定态解的分歧及其稳定性进行研究.一方面,分别以生长率作为分歧参数,讨论了发自半平凡解的分歧;另一方面,以两物种的生长率作为分歧参数,利用Liapunov-Schmidt过程,研究了在二重特征值处的分歧;同时判定了这些分歧解的稳定性.     

Analyticity estimates for the Navier–Stokes equations

We study spatial analyticity properties of solutions of the three-dimensional Navier–Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier–Stokes equations with data in Hr, r?1/2, and prove a stability result for the analyticity radius.     

On the three-dimensional stability loss problems of elements of structures of viscoelastic composite materials

Up to now, numerous problems of the stability loss for elements of structures made from composite materials have been investigated in the framework of the three-dimensional linearized theory of stability (TDLTS). It follows from the analysis of these investigations that the TDLTS was mainly applied to the design of elements of structures made from time-independent materials. For the solution of these problems for viscoelastic materials in the framework of the TDLTS, the dynamic investigation method and the critical deformation method are recommended in many references. However, it is known that a very reliable and frequently used approach for viscoelastic materials is the approach based on the study of the growth of insignificant initial imperfections in elements of structures with time. Taking into account the above-mentioned, an approach based on the growth of the initial imperfection for the investigation of the stability loss problems of elements of structures made from viscoelastic composite materials in the framework of TDLTS is proposed in the present paper. The composite material is modeled as an anisotropic, viscoelastic solid with averaged mechanical properties and all investigations are made on the strip simply supported at the ends.Yildiz Technical University, Dept. Math. Eng., 80750, Besiktas-Yildiz, Istanbul, Turkey. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 761–770, November–December, 1998.     

Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.     

Reactive Infiltration Instabilities

When a fluid flow is imposed on a porous medium, the infiltrationflow may interact with the reaction-induced porosity variationswithin the medium and may lead to fingering instabilities. Anonlinear model of such interaction is developed and morphologicalinstability of a planar dissolution front is demonstrated usinga linear stability analysis of a moving-free-boundary problem.The fully nonlinear model is also examined numerically usingfinite-difference methods. The numerical simulations confirmthe predictions of linear stability theory and, more importantly,reveal the growth of dissolution fingers that emerge as a resultof these instabilities