We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. It is a nonlinear boundary value problem with total mass constraint. We obtain exact multiplicity results by investigating a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.

From a viewpoint of the pattern formation, the Keller-Segel system with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, diffusion and growth. In a special case when chemotaxis effect is very strong, some numerical experiment in [1],[22] showed static and chaotic patterns. In this paper we consider the logistic source for the growth and a shadow system in the limiting case that a diffusion coefficient and chemotactic intensity grow to infinity. We obtain the global structure of stationary solutions of the shadow system in the one-dimensional case. Our proof is based on the bifurcation, singular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.

This paper is concerned with the Neumann problem of a stationary Lotka-Volterra competition model with diffusion and advection. First we obtain some sufficient conditions of the existence of nonconstant solutions by the Leray-Schauder degree theory. Next we derive a limiting system as diffusion and advection of one of the species tend to infinity. The limiting system can be reduced to a semilinear elliptic equation with nonlocal constraint. In the simplified 1D case, the global bifurcation structure of nonconstant solutions of the limiting system can be classified depending on the coefficients. For example, this structure involves a global bifurcation curve which connects two different singularly perturbed states (boundary layer solutions and internal layer solutions). Our proof employs a levelset analysis for the associate integral mapping. (C) 2014 Elsevier Inc. All rights reserved.

This paper is concerned with the limiting behavior of coexistence steady states of the Lotka-Volterra competition model as a cross-diffusion term tends to infinity. Under the Neumann boundary condition, Lou and Ni [J. Differential Equations, 154 (1999), pp. 157-190] derived a couple of limiting systems, which characterize the limiting behavior of coexistence steady states. One of two limiting systems characterizing the segregation of the competing species has been studied by Lou, Ni, and Yotsutani [Discrete Contin. Dyn. Syst., 10 (2004), pp. 435-458], and their work revealed the detailed bifurcation structure for the one-dimensional (1D) case. This paper focuses on the other limiting system characterizing the shrinkage of the species which is not endowed with the cross-diffusion effect. The bifurcation structure of positive solutions to the limiting system is stated. In particular, for the 1D case, we obtain a global connected set of solutions that bifurcates from a point on the line of constant solutions and blows up where the birth rate of the species is equal to the least positive eigenvalue of -Delta subject to the homogeneous Neumann boundary condition.

This paper is concerned with stationary solutions of a reaction-diffusion- advection system arising in surface chemistry. Hildebrand et al (2003 New J. Phys. 5 61) have constructed stationary stripe (or spot) solutions of the system in the singular perturbation case and shown a numerical result that the set of stripe (or spot) solutions forms a saddle-node bifurcation curve with respect to a diffusion coefficient. In this paper, we introduce a shadow system in the limiting case that another diffusion and an advection coefficient tend to infinity. Furthermore we obtain the bifurcation structure of stationary solutions of the shadow systems in the one-dimensional case. This structure involves saddle-node bifurcation curves which support the above numerical result in Hildebrand et al (2003 New J. Phys. 5 61, figure 9). Our proof is based on the combination of the bifurcation, the singular perturbation and a level set analysis. © 2013 IOP Publishing Ltd &amp
London Mathematical Society.

This paper deals with the following reaction-diffusion system
(SP) {Delta[(1 + alpha nu)u] + u(a - u - cv) = 0
Delta[(1 + beta u)v] + v(b - du - v) = 0
in a bounded domain of R-N with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients alpha and beta. For this purpose, our strategy is to study limiting behavior of positive solutions when alpha or beta goes to infinity and derive the corresponding limit systems. We will obtain a priori estimates of u and v independently of beta (resp. alpha) with small alpha >= 0 (resp. beta >=> 0) in case 1 <= N <= 3 under Neumann boundary conditions, while we will obtain a priori estimates of u and v independently of alpha and beta in case 1 <= N <= 5 under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When alpha = 0 and beta -> infinity, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.

Minima and one of the authors (1996) proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For this model, Tello and Winkler (2007) [22] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. (2008) numerically showed several spatio-temporal patterns in a rectangle. Motivated by their work, we consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints in the present paper. First we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, we exhibit several numerical results for the stationary and oscillating patterns. (C) 2012 Elsevier B.V. All rights reserved.

We are concerned with a reaction-diffusion-advection system proposed by Hildebrand [4]. This system is a phase transition model arising in surface chemistry. For this model, several stationary patterns have been shown by the numerical simulations (e. g., [15]). In the present paper, we obtain sufficient conditions for the existence (or nonexistence) of nonconstant stationary solutions. Our proof is based on the Leray-Schauder degree theory. Some a priori estimates for solutions play an important role in the proof.

This paper discusses the stationary problem for the Lotka-Volterra competition systems with cross-diffusion under homogeneous Dirichlet boundary conditions. Although some sufficient conditions for the existence of positive solutions are known, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we study the effects of large cross-diffusion on the structure of positive solutions and focus on the limiting behaviour of positive solutions by letting one of the cross-diffusion coefficients to infinity. Especially, it will be shown that positive solutions of the competition system converge to a positive solution of a suitable limiting system. We will also derive some satisfactory results on positive solutions for this limiting system. These results give us important information on the structure of positive solutions for the competition system when one of the cross-diffusion coefficients is sufficiently large.

We study a prey-predator model with nonlinear diffusions. In a case when the spatial dimension is less than 5, a universal bound for coexistence steady-states is found. By using the bound and the bifurcation theory, we obtain the bounded continuum of coexistence steady-states. (C) 2009 Elsevier Ltd. All rights reserved.

We consider the following quasilinear elliptic system:
{Delta u + u(a - u - cv) = 0 in Omega, Delta[(1 + gamma/1+beta u)v] + v(b + du - v) = 0 in Omega, u = v = 0 on partial derivative Omega,
where Omega is a bounded domain in R(N). This system is a stationary problem of a prey-predator model with non-linear diffusion Delta(v/1 + beta u) and u (respectively v) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large beta under the restriction b > (1 + gamma)lambda(1), where lambda(1) is the least eigenvalue of -Delta with homogeneous Dirichlet boundary condition. The present paper studies two shadow systems and gives the complete limiting characterization of positive solutions as beta -> infinity without any restriction on b.

This paper is concerned with the following Lotka-Volterra cross-diffusion system
ut = Delta[(1 + k rho(x)v)u] + u(a - u - c(x)v) in Omega x (0, infinity),
tau v(t) = Delta v + v(b + d(x)u - v) in Omega x (0, infinity)
in a bounded domain Omega subset of R(N) with Neumann boundary conditions partial derivative(nu)v = 0 on partial derivative Omega. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch Gamma under a segregation of rho(x) and d(x). In the present paper, we give some criteria on the stability of solutions on Gamma. We prove that the stability of solutions changes only at every turning point of Gamma if tau is large enough. In a different case that c(x) > 0 is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on Gamma.

This paper is concerned with the following Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment
(SP) {Delta[(1+k rho(x)v)u] + u(a - u - c(x)v) = 0 in Omega,
Delta v + v(b + d(x)u - v) = 0 in Omega,
partial derivative(v)u = partial derivative(v)u = 0 on partial derivative Omega.
Here Omega is a bounded domain in R(N)(N <= 3), a and k are positive constants, b is a real constant, c(x) > 0 and d(x) >= 0 are continuous functions and rho(x) > 0 is a smooth function with partial derivative(v)rho = 0 on partial derivative Omega. From a viewpoint of the mathematical ecology, unknown functions u and v, respectively, represent stationary population densities of prey and predator which interact and migrate in Omega. Hence, the set Gamma(rho) of positive solutions (with bifurcation parameter b) forms a bounded line in a spatially homogeneous case that rho, c and d are constant. This paper proves that if a and |b| are small and k is large, a spatial segregation of rho(x) and d(x) causes Gamma(rho) to form a subset of-shaped curve with respect to b. A crucial aspect of the proof involves the solving of a suitable limiting systent as a, |b| -> 0 and k -> infinity by using the bifurcation theory and the Lyapunov-Schmidt reduction. (C) 2007 Elsevier Ltd. All rights reserved.

This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof. (c) 2005 Elsevier Inc. All rights reserved.

Periodic Fourier transform is formally introduced to analyses of the electromagnetic wave propagation in optical waveguides. The transform make the field components periodic and they are then expanded in Fourier series without introducing an approximation of artificial periodic boundary. The proposed formulation is applied to two-dimensional slab waveguide structures, and the numerical results evaluate the validity and show some properties of convergence.

We study the multiple existence of positive solutions for the following strongly coupled elliptic system:
Delta[(1 + alphaupsilon)u] + u(a - u - cupsilon) = 0 in Omega,
Delta[(1 + betau)upsilon] + upsilon(b + du - upsilon) = 0 in Omega,
u = v = 0 on Omega,
where alpha, beta, a, b, c, d are positive constants and Omega is a bounded domain in R-N. This is the steadystate problem associated with a prey-predator model with cross-diffusion effects and u (resp. upsilon) denotes the population density of preys (resp. predators). In particular, the presence of beta represents the tendency of predators to move away from a large group of preys. Assuming that a is small and that beta is large, we show that the system admits a branch of positive solutions, which is S or D shaped with respect to a bifurcation parameter. So that the system has two or three positive solutions for suitable range of parameters. Our method of analysis uses the idea developed by Du-Lou (J. Differential Equations 144 (1998) 390) and is based on the bifurcation theory and the Lyapunov-Schmidt procedure. (C) 2003 Elsevier Inc. All rights reserved.

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions. (C) 2003 Elsevier Inc. All rights reserved.

In this paper we are concerned with the reaction-diffusion equation u(t) = Deltau + f(u) in a ball of RN with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f (u) = \u \ (q-1)u + \u \ (p-1)u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Delta phi + f (phi) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.