Bifurcation analysis for a free boundary problem modeling tumor growth

In this paper we deal with a free boundary problem modeling the growth of nonnecrotic tumors. The tumor is treated as an incompressible fluid, the tissue elasticity is neglected and no chemical inhibitor species are present. We re-express the mathematical model as an operator equation and by using a bifurcation argument we prove that there exist smooth stationary solutions of the problem which are not radially symmetric.     

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.     

Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation

In this paper, a mathematical model for tumor growth with time delay in proliferation under indirect effect of inhibitor is studied. The delay represents the time taken for cells to undergo mitosis. Nonnegativity of solutions is investigated. The steady-state analysis is presented with respect to the magnitude of the delay. Existence of Hopf bifurcation is proved for some parameter values. Local and global stability of the stationary solutions are proved for other ones. The analysis of the effect of inhibitor's parameters on tumor's growth is presented. The results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding non-retarded problems for some parameter values.     

Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone

This paper is concerned with the stationary problem of a prey-predator cross-diffusion system with a protection zone for the prey. We discuss the existence and non-existence of coexistence states of the two species by using the bifurcation theory. As a result, it is shown that the cross-diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive. We also study the asymptotic behavior of positive stationary solutions as the cross-diffusion coefficient of the prey tends to infinity.     

Stationary solutions of advective Lotka–Volterra models with a weak Allee effect and large diffusion

This paper is devoted to the Neumann problem of a stationary Lotka–Volterra model with diffusion and advection. In the model it is assumed that one population growth rate is described by weak Allee effect. We first obtain some sufficient conditions ensuring the existence of nonconstant solutions by using the Leray–Schauder degree theory. And then we study a limiting system (with nonlocal constraint) which stems from the original model as diffusion and advection of one of the species tend to infinity. Finally, we classify the global bifurcation structure of nonconstant solutions of the simplified 1D case.     

Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.     

Hopf-Bifurkation bei Lasern

The semiclassical equations describing a ring laser show two successive bifurcations, one stationary and one Hopf bifurcation. This phenomenon is analyzed mathematically. The initial value problem for the laser equations and the stability of the stationary solutions are discussed in detail. The transition to ultrashort laser pulses is shown to be a Hopf bifurcation. The direction of the bifurcation is determined for a numerical example. It turns out that it depends on the parameters of the system.     

The Bifurcation Phenomenon for Scalar Conservation Laws with Discontinuous Flux Functions

This paper is devoted to study the bifurcation phenomenon for scalar conservation laws with flux functions involving discontinuous coefficients. In order to deal with it, the special Cauchy initial data are taken and the interactions of stationary wave discontinuities with shock waves and rarefaction waves are considered in detail. The global solutions of this special Cauchy problem are constructed completely when the bifurcation phenomena appear in their solutions.     

Two-dimensional rotating waves in a functional-differential diffusion equation with rotation of spatial arguments and time delay

We consider the Neumann boundary value problem for a parabolic functional-differential equation in a disk. We describe spatially inhomogeneous solutions in the form of rotating waves branching from the homogeneous stationary solution in the case of an Andronov-Hopf bifurcation. By passing to a moving coordinate system and by reducing the original problem to a stationary boundary value problem for a partial differential equation with a deviating argument, we prove the existence of rotating waves appearing in the disk under the Andronov-Hopf bifurcation.     

Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R$R$, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ$\mu$ and the cell-to-cell adhesiveness γ$\gamma$ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ$\mu /\gamma$ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.     

A Hopf bifurcation in a free boundary problem with inhomogeneous media

We shall consider an interfacial problem arising reaction–diffusion models with inhomogeneous media. The purpose of this paper is to analyze the occurrence of Hopf bifurcation in the interfacial problem and to examine the effects of an inhomogeneous media. Conditions for existence of stationary solutions and Hopf bifurcation for a certain class of inhomogeneity are obtained analytically and numerically.     

Stationary solutions to the Falk system on shape memory alloys

We study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physically closed stationary state is formulated as a nonlinear eigenvalue problem with a non‐local term. Then, some results on existence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamically stable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation

In this paper, a free boundary problem modeling tumor growth under the direct effect of an inhibitor with time delays is studied. The delays represent the time taken for cells to undergo mitosis. Nonnegativity of solutions, the existence of the stationary solutions and their asymptomatic behavior are studied. The results show that when the inhibitor is large, and the initial tumor is not too large, the tumor will disappear. If however, the initial tumor is large enough, then it will grow. When the inhibitor is not as large, the growth of the tumor is determined by the size of the nutrients and whether the initial tumor is large or not. When the inhibitor is smaller, the tumor will grow no matter if the initial tumor is large or not.     

Axially Symmetric Boundary Value Problem Describing the Distribution of Charges in Semiconductors

We prove that bifurcation solutions to the stationary problem exist and can be extended with respect to parameter. The problem under consideration has a solution with the so-called interior transition layer phenomenon. In the nonstationary case, we establish the existence and uniqueness of a solution for any t > 0. Under certain assumptions, the nonstationary problem determines a dynamical system in some compact set. Bibliography: 9 titles.     

Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone

This paper is concerned with the positive stationary problem of a Lotka–Volterra cross-diffusive competition model with a protection zone for the weak competitor. The detailed structure of positive stationary solutions for small birth rates and large cross-diffusion is shown. The structure is quite different from that without cross-diffusion, from which we can see that large cross-diffusion has a beneficial effect for the existence of positive stationary solutions. The effect of the spatial heterogeneity caused by protection zones is also examined and is shown to change the shape of the bifurcation curve. Thus the environmental heterogeneity, together with large cross-diffusion, can produce much more complicated stationary patterns. Finally, the asymptotic behavior of positive stationary solutions for any birth rate as the cross-diffusion coefficient tends to infinity is given, and moreover, the structure of positive solutions of the limiting system is analyzed. The result of asymptotic behavior also reveals different phenomena from that of the homogeneous case without protection zones.     

Bifurcations in reaction-diffusion equations and associated dissipative structures

The paper considers semilinear parabolic equations with conditions dependent on a parameter. A stationary solution of the boundary-value problem is constructed in the neighborhood of the bifurcation value of the parameter. The evolution of the solutions of the Cauchy problem to the bifurcation solution—a spatially nonhomogeneous dissipative structure—is examined. Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 15–30, 1998.     

二维Lengyel-Epstein模型的六边形斑图

从局部分支的观点讨论2维Lengyel-Epstein模型的非常数正平衡态问题. 首先, 当区域为矩形时用局部分支定理构造六边形平衡态斑图; 其次, 在分支点附近确定分支方向.     

Bifurcation for a free boundary problem modeling the growth of multi-layer tumors

In this paper we study bifurcations for a free boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. An important feature of this problem is that the surface tension effect of the free boundary is taken into account. By reducing this problem into an abstract bifurcation equation in a Banach space, overcoming some technical difficulties and finally using the Crandall–Rabinowitz bifurcation theorem, we prove that this problem has infinitely many branches of bifurcation solutions bifurcating from the flat solution.     

New beams of global bifurcation points for a reaction-diffusion system with inequalities or inclusions

We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type which is subject to diffusion-driven instability. We show that an obstacle (e.g. a unilateral membrane) modeled either in terms of inequalities or of inclusions, introduces whole beams of new global bifurcation points of spatially non-homogeneous stationary solutions which lie in parameter domains which are excluded as bifurcation points for the problem without the obstacle.     

The Hopf Bifurcation in a Parabolic Free Boundary Problem with Double Layers

We consider a parabolic free boundary problem which has a bifurcation parameter and double interfaces. We investigate the sign change in a real part of eigenvalues and the transversality condition as a bifurcation parameter cross the critical value in order to examine the stability of the stationary solutions. The occurence of a Hopf bifurcation will be shown at a critical value.