DCDS
Limiting profiles of semilinear elliptic equations with large advection in population dynamics
King-Yeung Lam Wei-Ming Ni
Limiting profiles of solutions to a 2$\times$2 Lotka-Volterra competition-diffusion-advection system, when the strength of the advection tends to infinity, are determined. The two species, competing in a heterogeneous environment, are identical except for their dispersal strategies: One is just random diffusion while the other is "smarter" - a combination of random diffusion and a directed movement up the environmental gradient. With important progress made, it has been conjectured in [2] and [3] that for large advection the "smarter" species will concentrate near a selected subset of positive local maximum points of the environment function. In this paper, we establish this conjecture in one space dimension, with the peaks located and the limiting profiles determined, under mild hypotheses on the environment function.
keywords: asymptotic behavior ecology large advection. Semilinear equations
NHM
Traveling fronts of pyramidal shapes in competition-diffusion systems
Wei-Ming Ni Masaharu Taniguchi
It is well known that a competition-diffusion system has a one-dimensional traveling front. This paper studies traveling front solutions of pyramidal shapes in a competition-diffusion system in $\mathbb{R}^N$ with $N\geq 2$. By using a multi-scale method, we construct a suitable pair of a supersolution and a subsolution, and find a pyramidal traveling front solution between them.
keywords: pyramidal shapes Traveling front competition-diffusion system.
DCDS
Periodic solutions for a 3x 3 competitive system with cross-diffusion
Salomé Martínez Wei-Ming Ni
In this paper we study the role of cross-diffusion in the existence of spatially non-constant periodic solutions for the Lotka-Volterra competition system for three species. By properly choosing cross-diffusion coefficients, we show that Hopf bifurcation occurs at a constant steady state. Furthermore, these spatially nonhomogeneous periodic solutions are stable if diffusion rates are in appropriate ranges.
keywords: 3 x 3 competitive system Cross-diffusion periodic solutions.
DCDS
On the first positive Neumann eigenvalue
Wei-Ming Ni Xuefeng Wang
We study the first positive Neumann eigenvalue $\mu_1$ of the Laplace operator on a planar domain $\Omega$. We are particularly interested in how the size of $\mu_1$ depends on the size and geometry of $\Omega$. A notion of the intrinsic diameter of $\Omega$ is proposed and various examples are provided to illustrate the effect of the intrinsic diameter and its interplay with the geometry of the domain.
keywords: first positive eigenvalue Neumann eigenvalue problem domain-monotonicity intrinsic diameter diffusion.
DCDS
On a limiting system in the Lotka--Volterra competition with cross-diffusion
Yuan Lou Wei-Ming Ni Shoji Yotsutani
In this paper we investigate a limiting system that arises from the study of steady-states of the Lotka-Volterra competition model with cross-diffusion. The main purpose here is to understand all possible solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done without the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.
keywords: existence asymptotic behavior cross-diffusion parameter representation.
DCDS
On the global existence of a cross-diffusion system
Yuan Lou Wei-Ming Ni Yaping Wu
We consider a strongly-coupled nonlinear parabolic system which arises from population dynamics. The global existence of classical solutions is established when the space dimension is two and one of the cross-diffusion pressures is zero.
keywords: a priori estimates. Cross-diffusion global existence
DCDS
An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles
Kimie Nakashima Wei-Ming Ni Linlin Su
We study the following Neumann problem

$ d\Delta u+g(x)u^{2}(1-u)=0 \ $ in Ω ,
$ 0\leq u\leq 1 $in Ω and $ \frac{\partial u}{\partial\nu}=0 $ on ∂Ω,

where $\Delta$ is the Laplace operator, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ with $\nu$ as its unit outward normal on the boundary $\partial\Omega$, and $g$ changes sign in $\Omega$. This equation models the "complete dominance" case in population genetics of two alleles. We show that the diffusion rate $d$ and the integral $\int_{\Omega}g\ \d x$ play important roles for the existence of stable nontrivial solutions, and the sign of $g(x)$ determines the limiting profile of solutions as $d$ tends to $0$. In particular, a conjecture of Nagylaki and Lou has been largely resolved.
   Our results and methods cover a much wider class of nonlinearities than $u^{2}(1-u)$, and similar results have been obtained for Dirichlet and Robin boundary value problems as well.

keywords: Diffusion equations limiting behavior. variational method indefinite nonlinearity
DCDS
Pattern formation in a cross-diffusion system
Yuan Lou Wei-Ming Ni Shoji Yotsutani
In this paper we study the Shigesada-Kawasaki-Teramoto model [17] for two competing species with cross-diffusion. We prove the existence of spectrally stable non-constant positive steady states for high-dimensional domains when one of the cross-diffusion coefficients is sufficiently large while the other is equal to zero.
keywords: existence competition stability Density-dependent diffusion steady states.
DCDS
Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations
Yong Jung Kim Wei-Ming Ni Masaharu Taniguchi
Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. If $[\liminf_{|x|\to\infty}u(x),\limsup_{|x|\to\infty}u(x)]$ is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile $u$ has to be identically zero.
keywords: Travelling waves travelling spots reaction-diffusion equations.
DCDS
On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion
Yuan Lou Salomé Martínez Wei-Ming Ni
In this paper we investigate the role of cross-diffusion in the $3\times 3$ Lotka-Volterra competition model. Of particular interest is the existence of non-constant steady states created by cross-diffusion in $3\times 3$ systems. A comparison with $2\times 2$ systems is also included.
keywords: degree theory competition-­diffusion systems a priori estimates. Cross­-diffusion

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